Distributions and Weak Solutions

Let \(U\) be a bounded domain in \(\mathbb {R}^d\) and \(\Gamma \) be a closed subset of \(\partial U\). Consider the energy

on the admissible class

Suppose that \(u\) is a minimizer for the calculus of variations problem

find and justify the PDE boundary value problem solved by \(u\). Show that there is at most one solution of this PDE in the admissible class \(\mathcal {A}\).

Let \(U\) be a bounded domain and \(u\) solve the heat equation \[ \begin {cases} \partial _t u=\Delta u+cu & \text {in } U\times (0,\infty ),\\ u(x,0)=g(x),\\ u(x,t)=0 & \text {on } \partial U\times (0,\infty ). \end {cases} \]

Classify, with justification, the optimal range of values for \(c\in \mathbb {R}\) so that for any smooth initial data \(g\) the solution \(u\) above converges to 0 as \(t\to \infty \).

Find the entropy solution of the following scalar conservation law, arising in traffic flow modeling,

with initial data

After the rarefaction hits the shock you should write the explicit ODE IVP for the shock location \(\gamma (t)\) and a piecewise definition of the solution using this shock location \(\gamma (t)\), but you don’t actually need to solve the ODE IVP to get the explicit formula for \(\gamma (t)\).

Let \(f:\mathbb {R}^d\to \mathbb {R}\) smooth. Show that there is at most one smooth solution of \[ \begin {cases} -\Delta u=f & \text {in } \mathbb {R}^d,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0. \end {cases} \]

Give a (non-trivializing) additional condition on \(f\) so that you can show existence of such a solution.

Suppose that \(U\subset \mathbb {R}^n\) is a bounded domain and \(b:U\to \mathbb {R}^n\) is a smooth velocity field. Suppose that \(u(x, t)\) solves the reaction-diffusion- advection equation with a logistic reaction term

Show that if \(u(x, 0) < 1\) in \(U\) then \(u(x, t) < 1\) for all \(t > 0\) and all \(x \in U\).

Let \(U\) a bounded domain in \(\mathbb {R}^n\). Consider a smooth solution \(u\) of the equation \[ \begin {cases} u_t=\Delta u+c(x,t)u & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ),\\ u(x,0)=f(x). \end {cases} \]

Show that, if \(c(x, t) \le -\gamma \) for all \(x\) and \(t\), then

Consider the scalar conservation law,

Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \]

Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.

Consider the wave equation in a two-dimensional quadrant

with boundary conditions \(\partial _x u(0,y,t)=0\) and \(u(x,0,t)=0\) and initial condition \(u_0\), supported in the ball

Describe, precisely and with careful justification, the support of \(u(x, y, t)\) at all positive times \(t > 0\). Note: The support of a function \(f(x,y)\) is the closure of the set \(\{(x,y): f(x,y)\ne 0\}\).

Show that \(u(x)=\frac {1}{4\pi |x|}\) is the fundamental solution of the Laplace equation in \(\mathbb {R}^3\), \(-\Delta u=\delta _{0}\) in the sense of distributions, i.e. \[ \int _{\mathbb {R}^3} u(x)\Delta \varphi (x)\,dx=-\varphi (0) \]

for all \(\varphi \) smooth and compactly supported. You should provide a fully detailed proof, not just a formal computation.

Let \(b(x)\) a smooth vector field on a bounded domain \(U\) in \(\mathbb {R}^n\). Consider a smooth scalar solution \(u\) of the equation \[ \begin {cases} u_t=\Delta u+\nabla \cdot (b(x)u) & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ),\\ u(x,0)=f(x). \end {cases} \]

1. Show that if \(\nabla \cdot b = 0\) then for all \(t > 0\)

   \[ \sup _{x\in U}u(x,t)\le \sup _U |f|. \]
2. Show that if \(\nabla \cdot b \ne 0\) then for all \(t > 0\) \[ \sup _{x\in U}u(x,t)\le e^{At}\sup _U |f| \quad \text {with } A=\sup _U \nabla \cdot b. \]

Let \(U\) be a bounded domain. Consider the energy

on the admissible class

Suppose that there is \(u\in \mathcal {A}\) satisfying

Show that \(u\) must solve the PDE boundary value problem \[ \begin {cases} -\Delta u=\lambda & \text {in } U,\\ u=0 & \text {on } \partial U. \end {cases} \]

Here \(\lambda \in \mathbb {R}\) is a Lagrange multiplier associated with the constraint. Find a formula for \(\lambda \) in terms of \(I[u]\).

Consider the scalar conservation law,

Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1-x, & 0<x<1. \end {cases} \]

Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.

The goal of this problem is to use Picard iterates to find the solution of the initial value problem \[ \frac {dx}{dt}=x^2,\qquad x(0)=1 \tag {1} \]

1. State a theorem, without proof, with necessary conditions on \(f(x,t)\), guaranteeing the existence of a unique solution of the differential equation \(\frac {dx}{dt}=f(x,t)\), \(x(0)=x_0\).
2. Verify that the function \(f=x^2\) satisfies these necessary conditions.
3. Define the Picard iterates for the equation (1) and calculate the first three functions \(x_0(t)\), \(x_1(t)\), and \(x_2(t)\).
4. Prove by induction that \[ x_n(t)=1+t+t^2+\cdots +t^n+O(t^{n+1}) =\sum _{j=0}^{n}t^j+O(t^{n+1}). \tag {2} \] Hint: You will need to show that \[ \sum _{j=0}^{n}t^j\sum _{k=0}^{n}t^k =\sum _{j=0}^{n}jt^j+O(t^{n+1}). \]
5. Find the exact solution of (1). How does it compare to the approximate solution found in the previous part?

1. For a system of equations \(\frac {dx}{dt}=A(t)x\), where \(x\in \mathbb {R}^n\) and \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\), what is Abel’s formula? (State without proof).
2. For the linear differential equation \[ \frac {d^2x}{dt^2}-p(t)\frac {dx}{dt}+q(t)x=0 \tag {4} \] where \(p(t)>\alpha >0\) is a smooth function defined for all \(t\), prove there is at least one solution \(x(t)\) that is unbounded as \(t\to \infty \).

Under what conditions on the parameters \(a\) and \(b\) does the boundary value problem \[ u''+u=\cos (x),\qquad u'(0)=a,\qquad u'(\pi )=b \tag {5} \] have a solution?

1. Show that the wave equation \(u_{tt}=c^2u_{xx}\) \((-\infty <x<\infty ,\ t>0)\) conserves energy \[ E=\int _{-\infty }^{\infty }\left [(u_t)^2+c^2(u_x)^2\right ]d x. \]
2. Show the uniqueness: If two solutions of the wave equation satisfy the same initial conditions, then the solutions are identical.

Obtain the following symmetry properties for the Cauchy problem for the 1D heat equation \[ u_t=ku_{xx}+f(x,t)\quad (-\infty <x<\infty ,\ t>0),\qquad u(x,0)=\phi (x),\qquad u_t(x,0)=\psi (x). \]

1. If the data is even in \(x\) [\(\phi (x)=\phi (-x)\), \(\psi (x)=\psi (-x)\), \(f(x,t)=f(-x,t)\)], then the solution is also even in \(x\) [\(u(x,t)=u(-x,t)\)].
2. If the data is odd in \(x\) [\(\phi (x)=-\phi (-x)\), \(\psi (x)=-\psi (-x)\), \(f(x,t)=-f(-x,t)\)], then the solution is also odd in \(x\) [\(u(x,t)=-u(-x,t)\)].
3. If the data is periodic in \(x\) [\(\phi (x+A)=\phi (x)\), \(\psi (x+A)=\psi (x)\), \(f(x+A,t)=f(x,t)\)], then the solution is also periodic in \(x\) with the same period \(A\) [\(u(x+A,t)=u(x,t)\)].

1. Show the uniqueness of Robin’s problem \[ \begin {aligned} \Delta u&=f(x)\ [x\in D],\\ \nabla u\cdot \mathbf {n}+a(x)u&=h(x)\ [x\in \partial D],\qquad [D\text { is a 3D domain, }a(x)>0]. \end {aligned} \]
2. When \(a\equiv 0\), we have the Neumann problem. Show that it has no solution, unless \[ \int _D fd x=\oint _{\partial D}hd A. \] Describe physical meaning of this fact in the case of steady heat conduction.
3. Prove the uniqueness of the Neumann problem up to an additive constant.

Let \(x_+=xH(x)\), where \(x\in \mathbb {R}\) and \(H(x)\) is the Heaviside function so that \(x_+=x\) for \(x>0\) and \(x_+=0\) for \(x\le 0\).

1. Show that differentiating in the sense of distributions we have for any integer \(k\ge 0\) that \[ \partial (x_+^k)= \begin {cases} kx_+^{k-1}, & \text {if } k\ge 1,\\ \delta , & \text {if } k=0. \end {cases} \]
2. Let \(k\ge 1\) be an integer. Use (a) to find a distribution \(E\) such that \(\partial ^kE=\delta \).

Let \(p(t)>\alpha >0\) be a smooth function defined for all \(t\). Prove that for any solution of \[ x''+p(t)x'+x=0, \tag {4} \] \(x(t)\to 0\) as \(t\to \infty \).

Under what conditions on the parameters \(a\) and \(b\) does the boundary value problem \[ u''+u=\sin (x),\qquad u(0)=a,\qquad u(\pi )=b, \tag {5} \] have a solution? Is the solution, if it exists, unique? If not, why not?

Consider the nonlinear integral equation \[ u(x)-\lambda \int _a^b w(x,y)\sin (u(y))\,dy=\phi (x), \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on the interval \([a,b]\) and \(w(x,y)\) is continuous on \([a,b]\times [a,b]\). Find a constant \(C>0\) such that \(\lvert \lambda \rvert <C\) implies that the nonlinear integral equation has a unique solution \(u\).

Hint: You may assume the inequality \(\lvert \sin (x)-\sin (y)\rvert \le \lvert x-y\rvert \), for any \(x,y\in \mathbb {R}\).

Let \(L\) be a partial differential operator of the form: \[ L=\sum _{j=0}^{m-1} c_j\partial ^j, \] where the \(c_j\) are arbitrary constants. Show that \(x^m(L\delta )=0\) as distributions in \(\mathcal {D}'(\mathbb {R})\). Here \(\delta \in \mathcal {D}'(\mathbb {R})\) is the usual “delta function”.

Consider the distribution \(u\in \mathcal {D}'(\mathbb {R})\) defined by \[ \langle u,\phi \rangle =\int _{-\infty }^{\infty }\lvert x\rvert \phi (x)\,dx, \qquad \phi \in \mathcal {D}(\mathbb {R}). \] Find its derivative, \(\partial u\).

1. Consider a 2-dimensional map \((x,y)\to (u,v)\).

   Explain: If the map is given by an analytic function [\(u+iv=f(x+iy)\)] and at some point \(z_0=x_0+iy_0\) the derivative \(f'(z_0)\ne 0\), then this map preserves small shapes in the vicinity of \(z_0\).

2. Does there exist a conformal transformation \(u+iv=f(x+iy)\) that maps triangle \[ \Delta =\{(x,y):0<x<1,\ 0<y<1,\ x+y<1\} \] to circle \[ D=\{(u,v):u^2+v^2<1\}\,? \]

Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\). Show that there is \(T>0\) such that the initial value problem \[ \begin {aligned} \dot {x}&=f(t,x)\\ x(0)&=x_0 \end {aligned} \] has a solution \(x(t)\in C^1([0,T),\mathbb {R}^n)\). Is \(x(t)\) unique? [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

1. Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuously differentiable function and \(x_0\in \mathbb {R}^d\) be such that \(f(t,x_0)=0\) for all \(t\ge 0\). Define: the constant solution \(x(t)=x_0\) on \(t\ge 0\) is a stable (Liapunov stable) solution for of \[ \dot {x}=f(t,x). \]
2. Let \(A\) be an \(d\times d\) real matrix. State and prove a theorem that gives necessary and sufficient conditions on \(A\) so that the the zero solution \(z(t)=0\) on \(t\ge 0\) is stable for \[ \dot {x}=Ax. \]
3. Let \(A(t)\) be a smooth, \(T\)-periodic for some \(T>0\), \(d\times d\) real matrix function. State but don’t prove a theorem that gives necessary and sufficient conditions on \(A(t)\) so that the the zero solution \(z(t)=0\) is stable for \[ \dot {x}=A(t)x. \]

Let \(g:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuously differentiable function and \(p>1\) a constant such that \[ \abs {g(t,x)}\le \abs {x}^p\quad \text {for all }t\ge 0\text { and }x\in \mathbb {R}^d. \] Let \(A\) be a real matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\).

1. Show that there is \(\delta >0\) such that if \(\abs {x_0}\le \delta \), then the initial value problem \[ \begin {aligned} \dot {x}&=Ax+g(t,x)\\ x(0)&=x_0 \end {aligned} \tag {1} \] has a bounded solution on \([0,\infty )\). [Provide the complete proof. Do not just quote a theorem.]

2. Is the zero solution of (1) asymptotically stable? Explain.

   [Hint: let \(f(t)\), \(\varphi (t)\) be nonnegative continuous functions on the interval \(J=(\alpha ,\beta )\) containing \(t_0\). Let \(c_0\ge 0\). Gronwall’s Lemma says that if \[ f(t)\le c_0+\left |\int _{t_0}^{t}\varphi (s)f(s)\dd s\right | \] for all \(t\in J\) then \[ f(t)\le c_0\exp \left |\int _{t_0}^{t}\varphi (s)\dd s\right | \] for all \(t\in J\).]

Consider Newton’s equation \[ \ddot {x}+\frac {x}{2+x^2}=\epsilon \cos t \] Prove that there is an \(\epsilon _0>0\) such that if \(\abs {\epsilon }<\epsilon _0\) then there is a \(2\pi \)-periodic solution.

Find and sketch the solution of the differential equation \[ u_t+uu_x=0 \] on the domain \(-\infty <x<\infty \), \(t>0\) subject to the initial condition \(u(0,x)=u_0(x)\), where

1. \[ u_0(x)= \begin {cases} 0; & \text {if }x<0,\\ x, & \text {if }0<x<1,\\ 1, & \text {if }x>1; \end {cases} \]
2. \[ u_0(x)= \begin {cases} 1; & \text {if }x<0,\\ 1-x, & \text {if }0<x<1,\\ 0, & \text {if }x>1. \end {cases} \]

For which (if any) of these initial profiles does the solution of the problem \[ u_t+uu_x=\epsilon u_{xx}, \] \(\epsilon >0\), converges to a traveling wave profile? What is the speed of the travelling wave solution?

Let \(D\subset \mathbb {R}^n\) be a bounded, connected domain with smooth boundary. Let \(\phi \) be an eigenfunction for the Laplacian on \(D\) with Dirichlet Boundary conditions (\(\phi =0\) on \(\partial D\)) and with eigenvalue \(\lambda \) \[ \Delta u+\lambda u=0. \tag {2} \]

1. Show that there is a finite constant \(C\) so that every nonzero continuously differentiable function \(u\in C^1(D)\) such that \(u=0\) on \(\partial D\) satisfies \[ \int _D u^2 \le C\int _D \abs {\nabla u}^2. \]
2. By integrating eigenfunctions of (2), conclude that eigenvalues are positive \(\lambda >0\).
3. Denote the least value of the Rayleigh Quotient \[ \lambda _1(D)= \inf _{\substack {u\in C_0^1(D)\\ u\not \equiv 0}} \frac {\int _D \abs {\nabla u}^2}{\int _D u^2}. \] which satisfies \(\lambda \ge 1/C\) by (a). Assuming that \(\lambda _1(D)\) is an eigenvalue of \(D\) with Dirichlet boundary conditions, show that if two such domains satisfy \(D_1\subset D_2\) then \(\lambda _1(D_1)\ge \lambda _1(D_2)\).

Consider the nonlinear integral equation \[ f(x)-\lambda \int _a^b k(x,y;f(y))\,dy=\phi (x) \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on \([a,b]\), \(k\) is continuous on \(C=[a,b]\times [a,b]\times \mathbb {R}\), and \(k\) satisfies a Lipschitz condition of the form \[ \lvert k(x,y;z_1)-k(x,y;z_2)\rvert \le L\lvert z_1-z_2\rvert \] in its ‘functional argument’. Show that the nonlinear integral equation has a unique solution, \(f\), for any \(\lambda \) such that \[ \lvert \lambda \rvert <\frac {1}{L(b-a)}. \]

Let \(\delta \in \mathcal {D}'(\mathbb {R})\) denote the ‘delta function’. Find \(u\in \mathcal {D}'(\mathbb {R})\) that solves the following linear partial differential equation: \[ \partial ^2 u=\delta . \] Is the solution unique?

Hint: You can solve this equation by guessing the distribution and then verifying that is solves the equation.

Let \(p(t)\), \(q(t)\) and \(r(t)\) be continuous functions defined for \(t\in \mathbb {R}\) and \(t_0,x_0,x_1\in \mathbb {R}\). Consider the initial value problem \[ \begin {cases} \ddot {x}+p(t)\dot {x}+q(t)x=r(t)\\ x(t_0)=x_0,\qquad \dot {x}(t_0)=x_1. \end {cases} \tag {1} \]

1. Let \(X\) be a Banach Space with norm \(\norm {\bullet }\). Let \(C\subset X\) be a closed subset. State the Contraction Mapping Theorem (otherwise known as the Banach Fixed Point Theorem) for a map \(K:C\to X\).
2. Carefully reformulate (1) as a fixed point problem. Prove that the fixed point problem is equivalent to the local existence problem for (1).
3. Using the Contraction Mapping Theorem, show that there is a \(\delta >0\) and a unique function \(x(t)\in C^2([t_0,t_0+\delta ])\) that satisfies your fixed point problem, and hence (1), for \(t_0\le t\le t_0+\delta \).

Assuming the local existence result in Problem A1 for equation (1), show that in fact a unique solution exists globally for all \(t\in \mathbb {R}\).

1. Let \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) satisfy \(f(0)=0\). State the definitions: the zero solution of \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0, \end {cases} \] is Stable, Asymptotically Stable and Unstable.
2. Show that the zero solution is asymptotically stable. \[ \begin {cases} \dot {x}=xy-x^3\\ \dot {y}=-y+x^3y \end {cases} \]
3. Suppose that the real \(n\times n\) matrix \(A\) has an eigenvalue with \(\Reop \,\lambda >0\). Prove that the zero solution of \(\dot {x}=Ax\) is unstable.

Consider Rayleigh’s equation \[ \ddot {x}+\dot {x}^3-2\mu \dot {x}+x=0. \] What kind of bifurcations occur to the equilibrium point of system as the parameter \(\mu \) is varied? Determine the bifurcation values and describe how the solutions change near the bifurcation point.

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _nr^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _nr^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. Let \(A\) be an \(n\times n\) real matrix. Find necessary and sufficient conditions on \(A\) so that for all \(x_0\in \mathbb {R}^n\), the solution \(\varphi (t;x_0)\) of (1) remains bounded for \(t\ge 0\). Prove your result. \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \tag {1} \]
2. Suppose that \(f(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) is such that \(f(0)=0\) and the Jacobian matrix \(A=df(0)\) satisfies the condition in (a). Do all solutions \(\psi (t;x_0)\) of the nonlinear equation (2) still remain bounded for \(t\ge 0\), at least if \(x_0\) is close to \(0\)? Explain. \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0. \end {cases} \tag {2} \]

1. Consider the velocity potential \(\phi \) for linear water waves. Let \(h_0\) denote the depth of water in the absence of waves, and \(g\) denote the gravitational constant. The corresponding boundary value problem on \(\mathbb {R}\times [-h_0,0]\) is given by Laplace’s equation \[ \frac {\partial ^2\phi }{\partial x^2} +\frac {\partial ^2\phi }{\partial y^2}=0, \] with \[ \frac {\partial ^2\phi }{\partial t^2} +g\frac {\partial \phi }{\partial y}=0 \qquad \text {on } y=0, \] and \[ \frac {\partial \phi }{\partial y}=0 \qquad \text {on } y=-h_0. \] The linear wave solution takes the form \[ \phi (x,y,t)=Y(y)e^{ikx-i\omega t}. \] Derive the transcendental dispersion relation \[ \omega ^2=gk\tanh (kh_0). \] Determine the group and phase velocities for (i) deep water waves (large \(h_0\)) and (ii) shallow water waves (small \(h_0\)). Which case is non-dispersive?
2. Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \). Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _nr^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _nr^n}\int _{B_r(x)}u(y)\dd y. \]

Consider a string clamped at the end points \(a\) and \(b\), say, with \(u(a,t)=0\) and \(u(b,t)=0\), where \(u(x,t)\) is the string’s deviation from the horizontal rest position. The kinetic energy of the string is \[ E_k(t)=\frac {1}{2}\int _a^b \rho u_t^2d s, \] where \(\rho (x,t)\) is the mass density and \(ds=\sqrt {1+u_x^2}d x\) is an infinitesimal arc length. The potential energy consists of the sum of the energy due to the stretching of the string, and the work done against a load: \[ E_p(t)=\int _a^b \left (d(\sqrt {1+u_x^2}-1)-lu\sqrt {1+u_x^2}\right )d x. \] Here \(d(x,t)\) is the string’s elastic coefficient and \(l(x,t)\) is the load on the string. The classical action for the continuum model is \[ J=\int _{t_1}^{t_2}[E_k(t)+E_p(t)]d t. \]

1. Substitute the integral expressions for \(E_k\) and \(E_p\) into the action \(J\), and consider the variations \(u+\epsilon \psi \) such that \(\psi \) vanishes at the end points \(a,b\) and at the initial and final times \(t_1,t_2\). Calculate the first variation \(\delta J\).
2. After integrating by parts terms of the form \(u_t\psi _t\) and \(u_x\psi _x\), and using the boundary conditions for the variation \(\psi \), show that Hamilton’s principle \(\delta J=0\) yields the following PDE for \(u\): \[ (\rho u_t)_t-(1+u_x^2)^{-1/2} \left [d(1+u_x^2)^{-1/2}u_x\right ]_x-l=0. \]
3. Show that if \(\rho ,d\) are constants, there is no load (\(l=0\)), and the small slope approximation \(\abs {u_x}\ll 1\) holds, then the PDE of part (b) reduces to the classical one-dimensional wave equation. Write down an expression for the wave speed.

Consider the differential operator acting on \(L^2(\mathbb {R})\), \[ L=-\frac {d^2}{dx^2},\qquad 0\le x<\infty \] with self-adjoint boundary conditions \(\psi (0)/\psi '(0)=\tan \theta \) for some fixed angle \(\theta \).

1. Show that when \(\tan \theta <0\) there is a single negative eigenvalue with a normalizable eigenfunction \(\psi _0(x)\) localized near the origin, but none when \(\tan \theta >0\).
2. Show that there is a continuum of eigenvalues \(\lambda =k^2\) with eigenfunctions \(\psi _k(x)=\sin (kx+\eta (k))\), where the phase shift \(\eta \) is found from \[ e^{i\eta (k)}=\frac {1+ik\tan \theta }{\sqrt {1+k^2\tan ^2\theta }}. \]
3. Evaluate the integral \[ I(x,x')=\frac {2}{\pi }\int _0^\infty \sin (kx+\eta (k))\sin (kx'+\eta (k))\dd k, \] and interpret the result with regards the relationship to the Dirac Delta function and completeness, that is, \(\delta (x-x')-I(x,x')=\psi _0(x)\psi _0(x')\). You will need the following standard integral \[ \int _{-\infty }^{\infty } e^{ikx}\frac {1}{1+k^2t^2}\frac {\dd k}{2\pi } =\frac {1}{2\abs {t}}e^{-\abs {x/t}}. \] HINT: you should monitor how the bound state contribution (for \(\tan \theta <0\)) switches on and off as \(\theta \) is varied. Keeping track of the modulus signs \(|\ldots |\) in the standard integral is crucial for this.

Does the partial differential equation \[ \frac {\partial u}{\partial t}-\frac {\partial u}{\partial x}=-u \] subject to the restriction that \(u=\exp (-x^2)\) on the curve \(t=x^2\) have a solution? If it does have a solution, indicate for what ranges of \(x\) and \(t\) this solution is uniquely defined and if it does not have a solution, explain why.

The Ornstein-Uhlenbeck (OU) differential equation is \[ \frac {\partial p}{\partial t} =\frac {\partial }{\partial x}(kxp)+D\frac {\partial ^2p}{\partial x^2}. \]

1. What type of partial differential equation is this? What calculation is needed to justify this statement?
2. If \(t\) has units of time and \(x\) has units of length, what are the units of the parameters \(k\) and \(D\)?
3. Express this equation in nondimensional variables? How many nondimensional parameters remain after nondimensionalization?
4. Find the solution of the OU equation on the domain \(-\infty <x<\infty \), \(t\ge 0\) with initial data \(p(x,0)=\delta (x-x_0)\) for arbitrary \(x_0\). Give an interpretation of this solution in terms of a probability.

Suppose the wave equation \[ \frac {\partial ^2u}{\partial t^2}-c^2\nabla ^2u=0,\qquad x\in \mathbb {R}^n,\qquad t>0, \] subject to initial conditions \(u(x,0)=0\), \(\frac {\partial u}{\partial t}(x,0)=h(x)\), for \(x\in \mathbb {R}^n\) has a solution denoted by \(u_h\). Find (and verify) the solution of the same equation subject to the initial conditions \(u(x,0)=g(x)\), \(\frac {\partial u(x,0)}{\partial t}=h(x)\), for \(x\in \mathbb {R}^n\) in terms of \(u_h\) and \(u_g\).

1. Suppose an initial profile \(u_0(x)\) is monotone with \(\lim _{x\to -\infty }u_0(x, )=u_L\), \(\lim _{x\to \infty }u_0(x)=u_R\). Under what conditions on \(u_L\) and \(u_R\) does the solution of Burger’s equation \[ \frac {\partial u}{\partial t} +u\frac {\partial u}{\partial x}=0, \] evolve into a shock, and what is its asymptotic speed?
2. Consider the viscous Burger’s equation \[ \frac {\partial u}{\partial t} +u\frac {\partial u}{\partial x} =\epsilon \frac {\partial ^2u}{\partial x^2}. \] with \(\epsilon >0\). Under what conditions on \(u_L\) and \(u_R\), where \(\lim _{x\to -\infty }u(x,t)=u_L\), \(\lim _{x\to \infty }u(x,t)=u_R\) does this equation have a travelling wave solution, and what is its speed? Verify your result.

1. Find all values of \(i^i\).
2. Find all solutions of the equation \(\cos z=2\).
3. Find all solutions of the equation \(\tan z=i\).

Consider the van der Pol equation \[ \ddot {x}+x+\epsilon (x^2-1)\dot {x}=\Gamma \cos (\omega t), \qquad 0<\epsilon \ll 1 \] with \(\Gamma =O(1)\) and \(\omega \ne 1/3,1,3\). Using the method of multiple scales show that the solution is attracted to \[ x(t)=\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2\ge 2(1-\omega ^2)^2\) and \[ x(t)=2\left [1-\frac {\Gamma ^2}{2(1-\omega ^2)^2}\right ]^{1/2}\cos t +\frac {\Gamma }{1-\omega ^2}\cos \omega t+O(\epsilon ) \] when \(\Gamma ^2<2(1-\omega ^2)^2\). Explain why this result breaks down when \(\omega =1/3,1,3\).

1. Does a solution of the equation \(u_x+xu_y=u\) with \(u(x,x^2)=x\) exist for \(-\infty <x<\infty \), \(y>0\)? Explain why or why not.
2. Does a solution of the equation \(xu_x-yu_y=u\) with \(u(x,x^2)=x\) exist for \(-\infty <x<\infty \), \(y>0\)? Explain why or why not.

1. Give definitions for the following: invariant set, attracting set, \(\omega \)-limit set.
2. Determine the invariant sets and the attracting set of the dynamical system \[ \begin {aligned} \dot {x} &= -y+x(1-x^2-y^2),\\ \dot {y} &= x+y(1-x^2-y^2),\\ \dot {z} &= \alpha >0. \end {aligned} \] Also sketch the flow.
3. Describe what happens to the flow if we identify the points \((x,y,0)\) and \((x,y,2\pi )\) in the planes \(z=0\) and \(z=2\pi \).
4. By explicitly constructing solutions on the invariant torus \(x^2+y^2=1\), \(0\le z<2\pi \), show that the torus is only an attractor if \(\alpha \) is irrational.

Suppose that the eigenvalues \(\lambda _n\) and normalized eigenfunctions \(\phi _n(x)\) of the Dirichlet problem for the Laplacian in a bounded domain \(D\) are known: that is, \[ \nabla ^2\phi _n=\lambda _n\phi _n\quad \text {in }D,\qquad \phi _n=0\quad \text {on }\partial D \] for all integers \(n\ge 0\)

1. Using Green’s Theorem show that \(\lambda _n<0\).
2. Derive the eigenfunction expansion of the Green’s function \(G(x,y)\) for the Helmholtz equation \[ \nabla ^2G+k^2G=\delta (x-y)\quad \text {in }D,\qquad G=0\quad \text {on }\partial D, \] assuming that \(k^2+\lambda _n\ne 0\) for all \(n\ge 0\).
3. Now suppose that \(k^2+\lambda _n=0\) for some \(n\). Show how to construct a generalized Green’s function by solving \[ (\nabla ^2-\lambda _n)G(x,y)=\delta (x-y)+c\phi _n(x)\phi _n(y) \] via an eigenfunction expansion with a suitable choice of \(c\).

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _n r^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Consider the differential operator acting on \(L^2(\mathbb {R})\), \[ L=-\frac {d^2}{dx^2},\qquad 0\le x<\infty \] with self-adjoint boundary conditions \(\psi (0)/\psi '(0)=\tan \theta \) for some fixed angle \(\theta \).

1. Show that when \(\tan \theta <0\) there is a single negative eigenvalue with a normalizable eigenfunction \(\psi _0(x)\) localized near the origin, but none when \(\tan \theta >0\).
2. Show that there is a continuum of eigenvalues \(\lambda =k^2\) with eigenfunctions \(\psi _k(x)=\sin (kx+\eta (k))\), where the phase shift \(\eta \) is found from \[ e^{i\eta (k)}=\frac {1+ik\tan \theta }{\sqrt {1+k^2\tan ^2\theta }}. \]
3. Evaluate the integral \[ I(x,x')=\frac {2}{\pi }\int _0^\infty \sin (kx+\eta (k))\sin (kx'+\eta (k))\dd k, \] and interpret the result with regards the relationship to the Dirac Delta function and completeness, that is, \(\delta (x-x')-I(x,x')=\psi _0(x)\psi _0(x')\). You will need the following standard integral \[ \int _{-\infty }^{\infty } e^{ikx}\frac {1}{1+k^2t^2}\frac {\dd k}{2\pi } =\frac {1}{2\abs {t}}e^{-\abs {x/t}}. \]

HINT: you should monitor how the bound state contribution (for \(\tan \theta <0\)) switches on and off as \(\theta \) is varied. Keeping track of the modulus signs \(\abs {\cdots }\) in the standard integral is crucial for this.

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _n r^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Consider the distribution \(u\in \mathcal {D}'(\mathbb {R})\) defined for \(\phi \in \mathcal {D}(\mathbb {R})\) by \[ \langle u,\phi \rangle =\int _{-\infty }^{\infty }\lvert x\rvert \phi (x)\,dx. \] Find its derivative \(\partial u\).

Let \(p(t)>0\) be a smooth function that is periodic of least period \(T>0\). Show that every solution approaches zero as \(t\to \infty \). \[ x''+p(t)x'+x=0 \]

Show that for small \(\norm {\epsilon }\), there is a \(2\pi \)-periodic solution near zero to the inhomogeneous equation \[ \ddot {y}+5y=\epsilon \sin ^3 t. \]

A problem in the dynamics of the overhead power wire for an electric locomotive leads to the model \[ \frac {\partial ^2u}{\partial t^2} =\frac {\partial ^2u}{\partial x^2} \quad \text {for }x\ne X(t),\ t>0, \] where \(X\) is a prescribed smooth function with \(0<X'(t)<1\). Here \(X(t)\) is the locomotive position and \(u\) is the displacement of the wire. Across \(x=X\) there are prescribed discontinuities \[ \left [\frac {\partial u}{\partial x}\right ]_{X^-}^{X^+} =-V(X(t),t),\qquad [u]_{X^-}^{X^+}=0. \] Suppose that \(u=u_t=0\) at \(t=0\). By constructing a weak solution that takes into account the discontinuities across \(X\), show that \[ u(x,t)=\int _0^{\tau _0}V(X(\tau ),\tau )\left (1-X'(\tau )^2\right )d\tau , \] where the range of \(\tau \) is taken so that \(y=X(\tau )\) lies within the range \[ \tau -t<y-x<t-\tau ,\qquad \tau >0. \] [Hint: you will need to derive a relationship between discontinuities in \(\partial _tu\) and in \(\partial _xu\) along \(X(t)\). You will also need to consider the domain of dependence in the \((x,t)\)-plane].

Suppose \[ \nabla ^2u-cu=f\quad \text {in }D, \] with \[ u=g\quad \text {on }\partial D, \] where \(D\) is bounded and \(\partial D\) is smooth.

1. Assuming that a solution \(u\) exists, use an energy argument to show that it is unique when \(c>0\).
2. If \(c<0\) and \(f=g=0\), and \(D\) is the region \(r^2=x^2+y^2<1\), show that \[ u=a_0J_0(r\sqrt {-c}),\qquad a_0\text { constant}, \] is a solution as long as \(\sqrt {-c}\) is a zero of the Bessel function \(J_0(x)\), which satisfies \[ \frac {d^2J_0}{dx^2}+\frac {1}{x}\frac {dJ_0}{dx}+J_0=0. \]
3. If \(c<0\) and \(f=g=0\), and \(D\) is the region \(1<x^2+y^2+z^2<4\), show that there are non-trivial solutions if \(c=-n^2\pi ^2\), \(n=1,2,3\ldots \).

Let \(\ell ^{1'}\) be the dual space of \(\ell ^1\). The objective here is to show that \(\ell ^{1'}\) is isomorphic to \(\ell ^\infty \). You may use that a Schauder basis for \(\ell ^1\) (and \(\ell ^\infty \)) is \((e_k)\), where \(e_k=(\delta _{kj})\), i.e. the sequence with zeroes everywhere except for a \(1\) in the \(k\)–th position.

1. Show that any \(f\in \ell ^{1'}\) corresponds to a sequence \(g=(\gamma _k)\in \ell ^\infty \).
2. Show that any sequence \(g=(\gamma _k)\in \ell ^\infty \) corresponds to some \(f\in \ell ^{1'}\).
3. Parts (a) and (b) define a bijective linear mapping \(G:\ell ^{1'}\to \ell ^\infty \). Prove that \(G\) is an isomorphism by showing \(\lVert f\rVert _{\ell ^{1'}}=\lVert G(f)\rVert _{\ell ^\infty }\) for any \(f\in \ell ^{1'}\).

Let \(A\) be an \(n\times n\) real matrix such that \(\Re \lambda <0\) for all eigenvalues \(\lambda \) of \(A\). Show that there are positive constants \(c\) and \(\mu \) such that any solution \(x(t)\) of \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] satisfies \[ \norm {x(t)}\le ce^{-\mu t}\norm {x_0}\qquad \text {for all }t\ge 0. \]

Show that for small \(\norm {\epsilon }\), there is a \(2\pi \)-periodic solution to the forced Duffing’s equation \[ \ddot {y}+3y+\epsilon y^3=\cos t. \]

Show that the equation has a non constant periodic solution. \[ \ddot {x}+(x^2+\dot {x}^2-1)\dot {x}+2x=0 \]

Suppose that \(u(\mathbf {x})\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(\mathbf {x}\in \Omega \) and \(r>0\) is chosen such that \(B_r(\mathbf {x})\subset \Omega \) (ball of radius \(r\) centered at \(\mathbf {x}\)) then \[ u(\mathbf {x})=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(\mathbf {x})}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n r^n}\int _{B_r(\mathbf {x})}u(\mathbf {y})\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Consider the problem \(u''(x)=x\delta '(x)\), where \(\delta \) is the “delta function”. Explain what it means for this problem to have a solution \(u\) in the sense of distributions. Find a distributional solution \(u\) and prove that it does in fact solve the problem in the sense of distributions.

Let \(A\) be a real \(m\times n\) matrix, and let \(b\in \mathbb {R}^m\) be a column vector. Consider the problem \[ A^*Ax=A^*b. \tag {*} \]

1. Explain the relationship between \((*)\) and the problem \[ \min _{x\in \mathbb {R}^n}\lVert Ax-b\rVert ^2. \]
2. Under what conditions on \(A\), \(A^*\), and \(b\) is it guaranteed that \((*)\) has a solution \(x\)? Under what conditions is it guaranteed that \((*)\) has a unique solution?
3. Given \(\varepsilon >0\), consider the problem \[ (A^*A+\varepsilon I)x_\varepsilon =A^*b. \tag {**} \] Formulate an associated minimization problem, describe existence and uniqueness, and explain how \(\lim _{\varepsilon \to 0+}x_\varepsilon \) is related to solutions of \((*)\).

Consider the problem \(u''(x)=f(x)\), on \((0,1)\), with boundary conditions \(u'(0)=u'(1)=0\).

1. Write down the weak formulation of the problem.
2. Using a piecewise linear finite element basis with uniform grid size \(h\), use the Galerkin method to find a finite-dimensional (matrix) approximate representation for the problem. Do not solve the equations.
3. Neither of the problems (a) and (b) have unique solutions in general. Explain why not, and describe what can be done to correct the deficiency.

Define \[ Ku(x)=\sum _{n=0}^{10}\left (\int _{-1}^{1}e^{-ny}u(y)\,dy\right )x^n, \qquad x\in [-1,1]. \]

1. Find the adjoint \(K^*\) of the operator \(K\).
2. Assuming \(\lambda \ne 0\) is fixed and real, use the Fredholm Alternative to characterize the set of functions \(g\) such that the problem \((K-\lambda I)u=g\) on \([-1,1]\) has a solution.
3. From the structure of the operator \(K\), what can be said about the set \[ \sigma _p=\{\lambda \in \mathbb {C}:(K-\lambda I)u=0\text { has a solution }u\ne 0\}? \] There is no need to actually calculate \(\sigma _p\).

Construct the Green’s function for the problem \[ u''(x)+\alpha ^2u(x)=f(x),\quad \text {on }(0,1),\qquad u(0)=u(1)=0. \] Write the solution \(u\) in terms of the Green’s function, and determine values of \(\alpha \) for which the Green’s function does not exist.

Consider a modified predator-prey model in the first quadrant. Show that it has at least one nonconstant periodic solution. \[ \begin {aligned} \dot {x} &= x(8-x)-\frac {9xy}{x+1},\\ \dot {y} &= y\left (1-\frac {y}{x}\right ) \end {aligned} \]

Suppose that \(u(\mathbf {x})\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(\mathbf {x}\in \Omega \) and \(r>0\) is chosen such that \(B_r(\mathbf {x})\subset \Omega \) (ball of radius \(r\) centered at \(\mathbf {x}\)) then \[ u(\mathbf {x})=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(\mathbf {x})}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n r^n}\int _{B_r(\mathbf {x})}u(\mathbf {y})\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Consider the problem \(Lu=1\) on \((0,\pi )\), where \(Lu=u''+u\) with boundary conditions \(u'(0)=u'(\pi )=0\).

1. Write down the weak formulation of the problem.
2. Find the \(L^2[0,\pi ]\) adjoint of \(L\), and determine whether or not \(L\) is self-adjoint.
3. Find the nullspace of \(L\). Use Fredholm theory to characterize the set of functions \(f\) such that \(Lu=f\) has a solution. Verify that \(f=1\) satisfies your condition, and find an infinite set of solutions to the problem \(Lu=1\).

Define the operator \(K\) on \(L^2[0,1]\) by \[ Ku(x)=\int _0^1 k(x,y)u(y)\,dy, \] for \(x\in [0,1]\), where \(k(x,y)\) is a real-valued, bounded, continuous function, with \(k(x,y)>0\).

1. Find the \(L^2\) adjoint \(K^*\) for \(K\), and find a condition on \(k\) which guarantees that \(K\) is self-adjoint.
2. Describe the properties of the set \[ \{\lambda \in \mathbb {C}:(K-\lambda I)u=0\text { for some }u\ne 0\} \] (cardinality, bounds, etc..).
3. Consider the problem \(Ku=f\). Let \(X\) be the set of all \(f\in L^2[0,1]\) such that a unique solution \(u=K^{-1}f\) exists. Can we be assured that there is a constant \(C\) such that \(\lVert K^{-1}f\rVert \le C\lVert f\rVert \), for all \(f\in X\)? Why or why not?

Let \(f(x)=|x|-1\) on the interval \([-1,1]\).

1. Treating \(f\) as a distribution, calculate \[ \int _{-1}^{1} f'(x)f'(x)\,dx \quad \text {and}\quad \int _{-1}^{1} f(x)f''(x)\,dx. \]
2. Explain why \[ \int _{-1}^{1} f'(x)f''(x)\,dx \quad \text {and}\quad \int _{-1}^{1} f(x)f'''(x)\,dx \] do not make sense.
3. In general, what property is required of \(g(x)\) so that \[ \int _{-1}^{1} g(x)f^{(n)}(x)\,dx \] is well-defined for a given \(n\)th derivative?

Show that there is a solution that goes to \(\infty \) as \(t\to \infty \). \[ y''-\cos ^2(t)y'+\sin ^2(t)y=0. \]

Show that the planar system \[ \begin {aligned} \dot {x} &= x(x^2+y^2-3x-1)-y,\\ \dot {y} &= y(x^2+y^2-3x-1)+x \end {aligned} \] has at least one nonconstant periodic solution.

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\subset \Omega \) (ball of radius \(r\) centered at \(y\)) then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi }u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. Specify the weak formulation for the differential equation \[ u''+\lambda \delta (x-1/2)u=f(x) \] subject to boundary conditions \(u(0)=u(1)=0\).
2. Find all values of \(\lambda \) for which this differential equation with \(f(x)=0\) has a nontrivial solution and verify that the corresponding solution is a weak solution.
3. Suppose \(\lambda \) is such that this differential equation with \(f(x)=0\) has only the trivial solution. Find the solution with \(f(x)=1\).

1. Find all conditions on \(\alpha ,\beta ,\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \tag {1} \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Consider the integral equation \[ \phi (x)-\lambda \int _0^\pi \sin (x+t)\phi (t)\,dt=\cos (x)+\sin (x), \qquad 0\le x\le \pi . \]

1. Find the unique solution when \(\lambda \ne \pm 2/\pi \). Why is a unique solution guaranteed to exist?
2. Show that there is no solution when \(\lambda =2/\pi \).
3. Show that when \(\lambda =-2/\pi \) there is a one-parameter family of solutions of the form \[ \phi (x)=f_p(x)+\alpha f_h(x). \] Determine \(f_h(x)\).

According to Maxwell’s Equations, the static magnetic field \(H=(H_1,H_2,H_3)^T\) in an inhomogeneous medium satisfies \[ \nabla \cdot (\mu H)=0,\qquad \nabla \times H=0, \] where \(\mu \) is a space-dependent permeability. Suppose the interface \(z=0\) separates a medium in which \(\mu =\mu _+\) from one in which \(\mu =\mu _-\) with \(\mu _\pm \) constants.

1. Define the weak solution: for every given simply connected, smooth domain \(\Omega _3\subset \mathbb {R}^3\) and arbitrary embedded smooth two dimensional surface with boundary \(\Omega _2\subset \mathbb {R}^3\), assuming enough regularity, integrate by parts the equations \[ \int _{\Omega _3}\psi \nabla \cdot (\mu H)\dd V=0,\qquad \int _{\Omega _2}\psi \nabla \times H\cdot \dd S=0, \] where \(\psi \) is an arbitrary, suitable test function.
2. Taking \(\Omega _3\) to be a cylindrical domain that cuts the interface vertically, use Green’s Theorem to show that \[ [\mu H_3]_-^+=0. \]
3. Taking \(\Omega _2\) to be a rectangular surface that cuts the interface vertically, use Stokes’ Theorem to show that \[ [H_1]_-^+=0=[H_2]_-^+. \]

Let \(u\) solve the initial value problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0, && \text {in }\mathbb {R}\times (0,\infty ),\\ u=g,\quad u_t=h, &&& \text {on }\mathbb {R}\times \{0\}. \end {aligned} \] Suppose \(g,h\) have compact support. Define the kinetic energy \(k(t)=\frac 12\int _{-\infty }^{\infty }u_t^2(x,t)\dd x\) and the potential energy \(p(t)=\frac 12\int _{-\infty }^{\infty }u_x^2(x,t)\dd x\). Prove

1. The total energy \(E=k(t)+p(t)\) is constant in \(t\).
2. Assume that there are two distinct solutions \(u,v\) to the initial value problem. By considering the energy of the difference solution \(U=u-v\), prove that \(u=v\), that is, the solution is unique.

Consider the first-order equation \[ u=x_1u_{x_1}+x_2u_{x_2}+\frac 12\left (u_{x_1}^2+u_{x_2}^2\right ), \] for the function \(u(x_1,x_2)\).

1. Write down the characteristic equations \[ \begin {aligned} \dot {p} &= -D_xF-D_zF\,p,\\ \dot {z} &= D_pF\cdot p,\\ \dot {x} &= D_pF, \end {aligned} \] associated with this problem.
2. Verify that the Cauchy data \(u(x_1,0)=\frac 12(1-x_1^2)\), \(x_1\in \mathbb {R}\), is noncharacteristic, and find an admissible set of initial conditions.
3. Find a solution \(u(x_1,x_2)\) to the original equation which satisfies the given Cauchy data.

In Kirchhoff-Love Plate Theory, the vertical displacement \(w(x)\) of an isotropic static plate under pure bending satisfies \[ Lw:=D\left (\frac {\partial ^4w}{\partial x_1^4} +2\frac {\partial ^4w}{\partial x_1^2\partial x_2^2} +\frac {\partial ^4w}{\partial x_2^4}\right )=-f, \] where \(D>0\) is a known constant bending stiffness, and \(-f(x)\) is a given applied transverse load. Assume that the plate lies within a simply connected bounded region \(U\subset \mathbb {R}^2\) with smooth boundary \(\partial U\). Define the bilinear form \[ B(w,v)=\int _U (Lw)v\dd x. \]

1. Suppose that the boundary of the plate is clamped, that is, the displacement \(w\) satisfies \[ \begin {aligned} w &= 0, && \text {on }\partial U,\\ \frac {\partial w}{\partial \eta } &= 0, && \text {on }\partial U, \end {aligned} \] where \(\eta \) is the unit outward normal. Let \(X\) be the space of smooth functions satisfying the above boundary conditions. Show that \(B\) is symmetric and positive semidefinite over \(X\times X\).
2. Using an elementary estimate for solutions of Poisson’s equation, show that \(B\) is bounded and coercive over \(X\times X\) in a well-chosen norm.
3. By extending \(X\) appropriately, use the Lax-Milgram Theorem to prove that the boundary value problem above admits a unique weak solution for all sufficiently smooth \(f\).

Let \(U\subset \mathbb {R}^n\) be a bounded open set with smooth boundary \(\partial U\), and consider the problem \[ \begin {cases} -\Delta u+c(x)u=\lambda u+f, & \text {in }U,\\ u=0, & \text {on }\partial U, \end {cases} \] where \(c(x)\in L^\infty (U)\), and \(f\in C_0^\infty (\partial U)\).

1. In the case \(f=0\), by examining the bilinear form over \(H_0^1(U)\) associated with the operator \(-\Delta +c-\lambda \), determine a general condition on \(\lambda \in \mathbb {R}\) which guarantees that \(u=0\) is the only weak solution of the problem.
2. Still assuming \(f=0\), suppose that \(\lambda =\lambda _1,\lambda _2\) have been chosen such that there exist nonzero weak solutions \(u_1,u_2\in H_0^1(U)\) respectively (that is, \((\lambda _1,u_1)\) and \((\lambda _2,u_2)\) are eigenpairs). Prove that \(\lambda _j\) must be real, and if \(\lambda _1\ne \lambda _2\), then \(u_1\) and \(u_2\) are orthogonal.
3. Suppose \(\lambda _1\) is an eigenvalue as in part (b), and let \(E_1\subset H_0^1(U)\) be the set of all associated eigenfunctions. Setting \(\lambda =\lambda _1\), for general \(f\in C_0^\infty (U)\), find the least restrictive constraint on \(f\) which guarantees a weak solution to the problem exists. Is the solution unique?

Let \(U\subset \mathbb {R}^n\) be an open bounded connected set with smooth boundary \(\partial U\). Consider the initial boundary-value problem for the heat equation: \[ \begin {cases} u_t-\Delta u=0, & \text {in }U\times (0,T],\\ u=0, & \text {on }\partial U\times [0,T],\\ u=g, & \text {on }U\times \{t=0\}. \end {cases} \]

1. Use the mean value property for the heat equation \[ u(x,t)=\frac {1}{4r^n}\iint _{E(x,t;r)} u(y,s)\frac {\lvert x-y\rvert ^2}{(t-s)^2}\dd y\dd s, \] where \[ E(x,t;r)=\{(y,s)\in \mathbb {R}^{n+1}:s\le t,\ \phi (x-y,t-s)\ge 1/r^n\} \] and \(\phi (x,t)\) is the fundamental solution of the heat equation, to prove the strong maximum principle for the heat equation.
2. Show that the strong maximum principle implies the principle of infinite propagation speed for the problem above.

Assume that \(u\in C^2(\mathbb {R}^n\times [0,\infty ])\) satisfies the initial value problem for the wave equation \[ \begin {cases} u_{tt}-\Delta u=0, & \text {in }\mathbb {R}^n\times (0,\infty ),\\ u=g,\ u_t=h, & \text {on }\mathbb {R}^n\times \{t=0\}. \end {cases} \] Recall that when \(n=3\), using spherical means and the Euler-Poisson-Darboux equation, one can derive Kirchhoff’s formula \[ \begin {aligned} u(x,t) &= \frac {\partial }{\partial t}\left (t\fint _{\partial B(x,t)}g\dd S\right ) +t\fint _{\partial B(x,t)}h\dd S\\ &= \fint _{\partial B(x,t)} \bigl (th(y)+g(y)+Dg(y)\cdot (y-x)\bigr )\dd S(y),\qquad t>0. \end {aligned} \]

1. Outline the procedure (the “method of descent”) for deriving a similar formula (Poisson’s formula) for the initial value problem in the case \(n=2\).
2. Explain how a disturbance originating at a point \(x\in \mathbb {R}^2\) at time \(t=0\) propagates in relation to the cone \(C=\{(y,t):t>0,\ \lvert x-y\rvert <t\}\). In particular, using the Kirchhoff and Poisson formulas, show that there is a qualitative difference in propagation behavior between the cases \(n=2\) and \(n=3\).

Assume that \(u\in C^2(\mathbb {R})\), and \(f\in C_0^2(\mathbb {R})\).

1. Assume \(u\) satisfies \(u''=0\). Without using an explicit representation for the solution, show that the mean value property \[ u(x)=\frac 12\bigl (u(x+r)+u(x-r)\bigr )=\frac {1}{2r}\int _{x-r}^{x+r}u(t)\dd t, \] for each interval \((x-r,x+r)\), is satisfied, and conversely show that the mean value property implies \(u''=0\).
2. Assuming \(u''=0\) in the bounded interval \((a,b)\), use part (a) to prove the strong maximum principle for \(u\).
3. Find a radially symmetric fundamental solution \(\Phi (x)\) for the problem \(u''=0\), and use it to prove that \[ u(x)=\int _{-\infty }^{\infty }\Phi (x-y)f(y)\dd y \] satisfies \(-u''=f\) in \(\mathbb {R}\).

Given \(f\in C^\infty (\mathbb {R}^n)\), and \(g\in C^\infty (\mathbb {R}^{n-1})\), consider the eikonal equation \[ \lvert Du\rvert =f(x),\qquad \text {in }\mathbb {R}^n, \] with the boundary condition \(u=g\) on the plane \(\Gamma =\{x_n=0\}\).

1. Write down the characteristic equations \[ \begin {aligned} \dot {p} &= -D_xF-D_zF\,p,\\ \dot {z} &= D_pF\cdot p,\\ \dot {x} &= D_pF, \end {aligned} \] associated with this problem.
2. Given an initial point \(x^0\in \Gamma \), if possible, determine a set of initial conditions for the characteristic equations which satisfy the compatibility conditions (in other words, the initial conditions are admissible).
3. Determine conditions on \(g\) and \(f\) which guarantee a unique local solution to the boundary value problem near \(x^0\).

Compute the unique entropy solution of \[ \begin {aligned} u_t+\left (\frac {u^2}{2}\right )_x &= 0 && \text {in }\mathbb {R}\times (0,\infty ),\\ u &= g && \text {on }\mathbb {R}\times \{0\}, \end {aligned} \] where \[ g(x)= \begin {cases} 0, & \text {if }x<0,\\ x, & \text {if }0\le x\le 1,\\ 0, & \text {if }x>1. \end {cases} \] Draw a sketch documenting your answer, illustrating what happens at all times \(t>0\).

Determine whether the planar system \[ \begin {aligned} \dot {x} &= x-x^3-xy^2,\\ \dot {y} &= 2y-y^5-yx^4 \end {aligned} \] has any nonconstant periodic solutions.

Suppose that \(u\in C^2\) solves \[ u_{tt}-\operatorname {div}(aDu)=0,\qquad \text {in }\mathbb {R}^n\times (0,\infty ), \] where \(a\in C^1(\mathbb {R}^n)\) satisfies \(\frac 12\le a(x)\le 1\) for all \(x\).

1. Assunimg \(\operatorname {supp}_x u(x,t)\subset B(0,R)\subset \mathbb {R}^n\) for all \(0\le t\le T\), show that the energy \[ E(t)=\frac 12\int _{\mathbb {R}^n}u_t^2+a\lvert Du\rvert ^2\dd x,\qquad 0\le t\le T, \] is conserved.
2. Using an appropriately defined local energy, establish that solutions exhibit finite propagation speed: if for some \(r>0\), \(u(x,0)\) and \(u_t(x,0)\) are zero for \(x\in B(x_0,r)\subset \mathbb {R}^n\), then \(u\) is zero in the cone \(K=\{(x,t):0\le t\le r,\lvert x-x_0\rvert <r-t\}\).

The time-dependent Schrödinger equation can be written \[ i u_t=-\Delta u+Vu, \] where \(u(x,t)\), \(x\in \mathbb {R}^n\), \(t\in \mathbb {R}\), is the complex-valued “wave function,” and \(V(x)\ge 0\) is a given continuous real-valued potential function.

1. Using complex exponential plane waves, describe the dispersion relation for this equation in case \(V\equiv 1\). Is this problem dispersive?
2. Assuming \(V\) is bounded, and the problem is formulated over the smooth bounded connected spatial domain \(U\subset \mathbb {R}^n\), with homogeneous Dirichlet boundary conditions, prove that the solutions to the time dependent problem can be expressed as a discrete sum involving solutions to the time-independent problem \[ E_k\phi _k=\Delta \phi _k+V\phi _k,\qquad \text {in }U \] where \(0<E_1\le E_2\le \cdots \).

Consider Burger’s equation \[ \begin {aligned} u_t+\left (\frac {u^2}{2}\right )_x &= 0, && \text {in }\mathbb {R}\times (0,\infty ),\\ u &= g, && \text {on }\mathbb {R}\times \{0\}. \end {aligned} \]

1. Construct an integral solution satisfying the Rankine-Hugoniot condition which exhibits “nonphysical shocks.”
2. Write down the explicit entropy condition for this problem and explain why it rules out the solution in part (a).
3. Construct another integral solution which satisfies the entropy condition, and explain why this condition does not allow shocks.

Show that the planar system \[ \begin {aligned} \dot {x} &= y,\\ \dot {y} &= (1-3x^2-y^2)y-x \end {aligned} \] has at least one nonzero periodic solution.

Traffic flow may be modeled by the equation \[ \frac {\partial }{\partial x}\bigl (G(\rho )\bigr )+\frac {\partial }{\partial t}\rho =0 \] where \(\rho (x,t)\) is the density of traffic flow along a highway. If \(\rho _{\max }\) is the maximum density of cars in a bumper to bumper situation, then \[ G(\rho )=c\rho \left (1-\frac {\rho }{\rho _{\max }}\right ) \] where \(c>0\) is the free speed constant. Suppose that the initial concentration is given by \[ \rho (x,o)= \begin {cases} \frac 12\rho _{\max }, & \text {if }x<0,\\ 0, & \text {if }x\ge 0. \end {cases} \] Describe the weak solution. Does a shock form? Interpret your result in terms of traffic flow.

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\subset \Omega \) (ball of radius \(r\) centered at \(y\)) then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi }u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. Find the weak formulation for the boundary value problem \[ u''-\delta (x)u=1 \] subject to boundary conditions \(u(-1)=u(1)=0\).
2. Find the solution to this boundary value problem.

The fourth quadrant \(x\ge 0\), \(y\le 0\) is occupied by an impermeable solid, around which air flows. Treating the flow as incompressible and irrotational use conformal mapping to map to a simpler problem and give one non-trivial solution for the complex potential \(f(z)\) in the original coordinates. Find the equation in polar coordinates for the streamline passing through \(z=-1\). What happens to the fluid velocity as one approaches the corner?

The operator \(Lu=-u''\) on \(L^2(-\infty ,\infty )\) has a Green’s function defined by \(-G''-\lambda G=\delta (x-\xi )\) given by \[ G(x,\xi ;\lambda )=-\frac {e^{i\sqrt {\lambda }|x-\xi |}}{2i\sqrt {\lambda }}. \] Assume the representation \[ -\frac {1}{2\pi i}\int _{C_\infty }G(x,\xi ;\lambda )\,d\lambda =\delta (x-\xi ) \] and show how it leads to the Fourier Transform Theorem that \[ f(x)=\frac {1}{2\pi }\int _{-\infty }^{\infty }e^{-i\mu x}F(\mu )\,d\mu , \qquad F(\mu )=\int _{-\infty }^{\infty }e^{i\mu x}f(x)\,dx. \]

Let \[ A= \begin {pmatrix} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \end {pmatrix}. \]

1. Find the eigenvalues and eigenvectors of \(A\), and the range of the function \[ \phi (x)=\frac {\langle Ax,x\rangle }{\langle x,x\rangle } \] where \(x=(x_1,x_2,x_3)\) is a real-valued vector.
2. Compute \(\exp (A)\).
3. Consider the equation \[ Ax=\mu x \] Find all solutions for \(\mu =1\) and for \(\mu =3\).

Let \[ f(x)=\operatorname {signum}(x)= \begin {cases} 1, & x>0,\\ 0, & x=0,\\ -1, & x<0. \end {cases} \]

1. Find its first and second derivatives using the theory of distributions.
2. Compute \[ \begin {aligned} I_0&=\int _{-1}^{\infty }\log (x+10)f(x)\,dx,\\ I_1&=\int _{-1}^{\infty }\log (x+10)f'(x)\,dx,\\ I_2&=\int _{-1}^{\infty }\log (x+10)f''(x)\,dx. \end {aligned} \]

Find a solution \(u(x,y)\) of Laplace’s equation on the domain \(-\infty <x<\infty \), \(0<y<\infty \) for which \(u(x,0)=x^{1/2}\) for \(0<x<\infty \). What is \(u(x,0)\) for \(-\infty <x<0\)?

Use Fourier transforms to solve the integral equation \[ \int _{-\infty }^{\infty } k(x-y)u(y)\,dy-u(x)=f(x) \] with \(k(x)=H(x)\), the Heaviside function.

1. Let \(f(x)=|x|\). Find its first and second derivatives using operational calculus, that is, the theory of distributions.
2. Using Green’s functions, find the solution to \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=u(1)=0. \] Use the reproducing property of the delta distribution to formally verify that your solution satisfies the equation.

Show that the system \[ \begin {aligned} \dot {x}&=x+y-x^3\\ \dot {y}&=-x+y-y^3 \end {aligned} \] has at least one nonzero periodic solution.

Traffic flow may be modeled by the equation \[ \frac {\partial }{\partial x}\bigl (G(\rho )\bigr )+\frac {\partial }{\partial t}\rho =0 \] where \(\rho (x,t)\) is the density of traffic flow along a highway. If \(\rho _{\max }\) is the maximum density of cars in a bumper to bumper situation, then \[ G(\rho )=c\rho \left (1-\frac {\rho }{\rho _{\max }}\right ) \] where \(c>0\) is the free speed constant. Suppose that the initial concentration is given by \[ \rho (x,o)= \begin {cases} \frac 12\rho _{\max }, & \text {if }x<0,\\ 0, & \text {if }x\ge 0. \end {cases} \] Find the shock curve and describe the weak solution. Interpret your result in terms of traffic flow.

Let \(D\subset \mathbb {R}^d\) be a bounded domain with smooth boundary \(\partial D\). Assume that the complete, orthonormal set of eigenfunctions \(\{\phi _n\}_{n=1}^{\infty }\) for the negative Dirichlet Laplacian on \(D\) is known, in other words, \[ -\Delta \phi _n=\lambda _n\phi _n,\quad \text {in }D,\ n=1,2,3,\ldots , \] with \(\phi _n(x)=0\) for \(x\in \partial D\), and \(\int _D\phi _n^2\dd x=1\).

1. Given \(f\in L^2(D)\), find a closed-form series solution \(u(x,t)\) for the initial boundary value problem for the heat equation \[ \begin {aligned} u_t-\Delta u&=0,\quad t>0,\ x\in D\\ u(x,t)&=0,\quad x\in \partial D,\\ u(x,0)&=f,\quad x\in D. \end {aligned} \]
2. Show that there exists a constant \(C\) such that \(\left \lvert \int _D u(x,t)\phi _n(x)\dd x\right \rvert \le Ce^{-\lambda _n t}\), for all \(n\). Find \(\lim _{t\to 0}u(x,t)\).
3. Give an example (the case \(d=1\) will suffice) which shows that the heat equation does not exhibit finite propagation speed.

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\) (the ball of radius \(r\), centered at \(y\)) satisfies \(B_r(y)\subset \Omega \), then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi } u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain a maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. Write down the set of linear equations which if solved give the Green’s function for the operator \(Lu=d^3u/dx^3\) on the interval \([0,1]\) with boundary conditions \(u(0)=u'(0)=u(1)=0\). (There is no need to explicitly solve these equations).
2. Letting \(G(x,y)\) denote the Green’s function which solves (a), and using a particular solution of \(d^3u/dx^3=0\), find the general solution to \(d^3u/dx^3=f(x)\) on the interval \([0,1]\) with boundary conditions \(u(0)=u'(0)=0\) and \(u(1)=1\).

Consider the scalar equation \[ \ddot {x}+\dot {x}=-\epsilon (x^2-x),\qquad 0<\epsilon \ll 1 \] Using the method of multiple scales show that the \(O(1)\) solution is \[ x_0(t,\tau )=A(\tau )+B(\tau )e^{-t}, \] where \(\tau =\epsilon t\), and identify any resonant terms at \(O(\epsilon )\). Show that the non–resonance condition for the amplitude \(A\) is \[ A_\tau =A-A^2 \] and hence determine the asymptotic behavior of \(x_0\). Comment on the domain of validity of the asymptotic expansion.

Consider the scalar differential equation \[ \ddot {x}+x=-\epsilon f(x,\dot {x}) \] with \(\lvert \epsilon \rvert \ll 1\). Let \(y=\dot {x}\).

1. Show that if \(E(x,y)=(x^2+y^2)/2\) then \[ \dot {E}=-\epsilon f(x,y)y. \] Hence show that an approximate periodic solution of the form \(x=A\cos t+O(\epsilon )\) exists if \[ \int _0^{2\pi } f(A\cos t,-A\sin t)\sin td t=0. \]
2. Let \(E_n=E(x(2\pi n),y(2\pi n))\) with \(x(t)=A_n\cos t+O(\epsilon )\) for \(2\pi n\le t<2\pi (n+1)\). Show that to lowest order \(E_n\) satisfies a difference equation of the form \[ E_{n+1}=E_n+\epsilon F(E_n) \] and write down \(F(E_n)\) explicitly as an integral. Hence deduce that a periodic orbit with approximate amplitude \(A^*=\sqrt {2E^*}\) exists if \(F(E^*)=0\) and this orbit is stable if \[ \epsilon \frac {dF}{dE}(E^*)<0. \]
3. Using the above result, find the approximate amplitude of the periodic orbit of the van der Pol equation \[ \ddot {x}+x+\epsilon (x^2-1)\dot {x}=0 \] and verify that it is stable.

Consider the scalar linear equation \[ a\frac {\partial u}{\partial x}+b\frac {\partial u}{\partial y}=\alpha u \] with \(a,b,\alpha \) differentiable functions of \(x,y\). Suppose that \(u\) is prescribed on some arc \(\Gamma \) (Cauchy data).

1. By introducing a test function \(\psi \) that vanishes on an arbitrary curve \(\gamma \) and applying Green’s theorem to the domain \(D\) bounded by \(\partial D=\Gamma \cup \gamma \), derive the integral equation for a weak solution to the Cauchy problem.
2. Extend the analysis to the case where \(u\) is discontinuous across an open curve \(C_0\) within the domain \(D\). Hence show that \(C_0\) is a characteristic projection.
3. Write down the generalization of part (b) to the case of the quaslinear equation \[ \frac {\partial P}{\partial x}+\frac {\partial Q}{\partial y}=c \] where \(P,Q,c\) are differentiable functions of \(x,y,u\), and hence derive the Rankine–Hugonoit condition.

Paint flowing down a wall has thickness \(u(x,t)\) satisfying (for \(t>0\)) \[ \frac {\partial u}{\partial t}+u^2\frac {\partial u}{\partial x}=0. \]

1. Show that the characteristics are straight lines and that the Rankine–Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {\left [u^3/3\right ]_-^+}{[u]_-^+}. \]
2. A stripe of paint is applied at \(t=0\) so that \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1\\ 1, & 0<x<1 \end {cases} \] For sufficiently small \(t\), determine \(u\) in the domains \(x<0\), \(0<x<t\), \(t<x<S(t)\) and \(S(t)<x\), where the shock is \(x=S(t)=1+t/3\)
3. Explain why the solution changes at \(t=3/2\) and show that thereafter \(\dot {S}=S/3t\).

(a) Consider the three–dimensional wave equation \[ \frac {\partial ^2u}{\partial t^2}=c^2\nabla ^2u \] with Cauchy data \[ u(\mathbf {x},0)=0,\qquad \frac {\partial u}{\partial t}(\mathbf {x},0)=g(\mathbf {x}) \]

1. Show that radially symmetric solutions are in the form of outgoing and incoming waves \[ u(r,t)=\frac {1}{r}F(r\pm ct). \] [Hint: Perform the substitution \(v=ur\).]
2. Writing the general solution to the Cauchy problem as a superposition of outgoing waves \[ u(\mathbf {x},t)=\int _{\mathbb {R}^3}\frac {\delta (r-ct)}{r}f(x',y',z')\dd x'\dd y'\dd z', \] where \(r^2=(x-x')^2+(y-y')^2+(z-z')^2\), derive the retarded potential solution \[ \begin {aligned} u(\mathbf {x},t)=ct\int _0^{2\pi }\int _0^\pi &f(x+ct\sin \theta \cos \phi ,\ y+ct\sin \theta \sin \phi ,\\ &z+ct\cos \theta )\sin \theta \dd \theta \dd \phi \end {aligned} \] and show that \(f=g/(4\pi c)\).
3. Assuming that the initial data \(g\) is only nonzero in a bounded domain \(D\), use a graphical construction to derive Huygen’s principle.

On the interval \([0,1]\) use eigenfunctions to find the resolvent of the integral equation \[ u(x)-\lambda \int _0^1 x^2y^2u(y)\,dy=f(x). \] When does a solution not exist? Write down the pseudoresolvent in this case.

1. Using Green’s functions, find the solution to the equation \[ u''=f(x),\qquad u(0)=0,\qquad u(1)=1. \]
2. Find the necessary and sufficient condition on \(f(x)\) for a solution to \[ u''+m^2\pi ^2u=f(x),\qquad u(0)=u(1)=0 \] to exist.

Find a solution \(u(x,y)\) of Laplace’s equation in 2 dimensions for which \(u(x,0)=|x|^{1/2}\).

Solve the integral equation \[ \int _{-\infty }^{\infty } k(x-y)u(y)\,dy-u(x)=f(x) \] with \(k(x)=H(x)\), the Heaviside function.

1. Let \(G\subset \mathbb {R}^N\) be an open set. Define what is meant by a distribution on \(G\). Let \(f\in L^2(G)\) be a real or complex valued function. Show that \(f\) may be identified with a distribution \(T_f\).
2. Let \(T\) be a distribution on \(G\) and let \(\alpha =(\alpha _1,\cdots ,\alpha _N)\) be a multiindex. Define the concept of the \(\alpha \)-th distributional derivative of \(T\).
3. Let \[ r:\mathbb {R}^N\to \mathbb {R} \] be defined by \[ \begin {aligned} r(x)&=1,\quad \lvert x\rvert <1\\ r(x)&=0,\quad \lvert x\rvert \ge 1, \end {aligned} \] where \(\lvert \cdot \rvert \) is the Euclidean norm in \(\mathbb {R}^N\). Show \(r\) may be thought of as a distribution on \(\mathbb {R}^N\) and compute its distributional gradient \(\nabla r\).

1. Let \(G\) be an open subset of \(\mathbb {R}^N\) and let \(f\in L^2(G)\). Show that \(f\) defines a continuous linear functional on \(H^1(G)\).
2. State the Lax-Milgram theorem for subspaces \(V\subset H^1(G)\).

3. Show that the Lax-Milgram theorem may be applied to the form \[ a(u,v)=\sum _{i=1}^{N}\int _G \partial _i u\,\partial _i\overline {v} +\int _G u\overline {v} \] and \(f\) as above.

   Assume that \(G\) is a bounded open set with “very smooth” boundary and let \(u\) be the Lax-Milgram solution just determined. What partial differential equation (in the sense of distributions) will be solved by \(u\) and what boundary conditions does \(u\) satisfy (please give an argument to prove the latter)?

The dilation equation is \[ \phi (x)=\sum _k c_k\phi (2x-k) \] and the associated wavelet is \[ W(x)=\sum _k(-1)^k c_{1-k}\phi (2x-k). \]

1. Show that the function \[ N_2(x)= \begin {cases} x, & 0<x<1,\\ 2-x, & 1\le x<2,\\ 0, & \text {elsewhere} \end {cases} \] satisfies the dilation equation and determine the values of \(c_k\). Sketch the associated wavelet. Briefly explain whether or not \(N_2(x)\) can act as a scaling function for a multiresolution analysis?
2. Prove that if the functions \(\{\phi (x-k)\}_k\) form an orthonormal set then \[ \int _{-\infty }^{\infty }W(x)W(x-m)\,dx = \delta _{0,m}\int _{-\infty }^{\infty }\phi ^2(x)\,dx. \]
3. Prove that \[ \int _{-\infty }^{\infty } W(x)\phi (x-m)\,dx=0 \] for all \(m\).

1. Show that a test function \(\psi (x)\) is of the form \(\psi =(x\phi )'\) where \(\phi \) is a test function if and only if \[ \int _{-\infty }^{\infty }\psi (x)\,dx=0 \quad \text {and}\quad \int _0^{\infty }\psi (x)\,dx=0. \]
2. Solve the following equation in the sense of distribution: \[ x^2\frac {d\phi }{dx}=0. \]

Consider the integral equation \[ \phi (x)-\lambda \int _0^\pi \sin (x+t)\phi (t)\,dt = \cos (x)+\sin (x), \qquad 0\le x\le \pi . \]

1. Find the unique solution when \(\lambda \ne \pm 2/\pi \).
2. Use the Fredholm alternative theorem to show that there is no solution when \(\lambda =2/\pi \).
3. Use the Fredholm alternative theorem to show that when \(\lambda =-2/\pi \) there is a one-parameter family of solutions of the form \[ \phi (x)=\frac {\cos (x)+\sin (x)}{2}+\alpha (\cos (x)-\sin (x)). \]

Solve Laplace’s equation \(\Delta \phi =0\) in the domain between the two nonconcentric circles \(|z|=1\) and \(|z-1|=5/2\), subject to the boundary conditions \(\phi =a\) on \(|z|=1\) and \(\phi =b\) on \(|z-1|=5/2\).

For each \(n\in \N \), consider the Cauchy problem \[ \begin {aligned} -\Delta u_n &= 0, && \text {in } U,\\ u_n &= \frac {1}{n^2}\sin nx, && \text {on } \{(x,y):y=0\},\\ \frac {\partial u_n}{\partial y} &= \frac {1}{n}\sin nx, && \text {on } \{(x,y):y=0\}, \end {aligned} \] where \(U=\{(x,y):0<y<1\}\). Find a sequence \(\{u_n\}\) of solutions to these problems, prove that \(\{u_n\}\) does not converge to zero, and explain why this implies that the Cauchy problem above is not “well posed”.

Let \(U\subset \mathbb {R}^n\) be open and bounded, with smooth boundary. For \(T>0\), denote \(U_T=U\times (0,T]\). Let \(\Gamma _T=\overline {U_T}-U_T\) (closure taken in \(\mathbb {R}^n\times \mathbb {R}\)). Prove that functions \(u(x,t)\) satisfying \(u\in C^{2,1}(U_T)\cap C(\overline {U_T})\) and \[ \begin {aligned} \Delta u-u &\ge u_t, && \text {in } U_T,\\ u &\ge 0, && \text {in } \overline {U_T}, \end {aligned} \] also satisfy the maximum principle \[ \max _{\overline {U_T}} u=\max _{\Gamma _T} u. \]

Consider the wave equation in three spatial dimensions \[ u_{tt}-\Delta u=0 \quad \text {in } \mathbb {R}^3\times (0,\infty ). \]

1. Explain the concept of domain of dependence for solutions of the wave equation, and describe what this has to do with finite propagation speed.
2. Sketch a proof, either using a representation formula or energy methods, that solutions \(u\in C^2\) of the wave equation have an explicitly defined domain of dependence.

Denote \(U=\{x\in \mathbb {R}^2:x_1>0\}\), and \(\Gamma =\{x\in \mathbb {R}^2:x_1=1\}\). Use the method of characteristics to solve the first-order problem \[ \begin {aligned} x_1u_{x_1}+uu_{x_2}-x_2 &= 0, && \text {in } U,\\ u(1,x_2) &= 2x_2, && \text {on } \Gamma . \end {aligned} \] (Hint: to solve the characteristic equations, note that the linear combinations \(x_2+z\) and \(x_2-z\) decouple.) Determine the region around \(\Gamma \) upon which the solution is well-defined.

Let \(U=\{x\in \mathbb {R}^2:0<\lvert x\rvert <1\}\), and denote \(\Gamma =\{\lvert x\rvert =1\}\). Consider \[ \begin {aligned} -\Delta u &= f, && \text {in } U,\\ u &= 0, && \text {on } \{x=0\},\\ \frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma , \end {aligned} \] where \(\eta \) is the unit outward normal, and \(f\in L^2(U)\).

Prove a Poincaré-type inequality for this problem, and use it to show that there exists a unique weak solution \(u\) in an appropriate Hilbert space.

Let \(U\) be an open subset of \(\mathbb {R}^2\). Prove that if \(u\in C^2(U)\) satisfies \[ \Delta u-u=0 \quad \text {in } U, \] then \[ u(x)=\frac {1}{I(r)}\int _{\partial B(x,r)} u\,dS_y \] for every ball \(B(x,r)\subset U\), where \[ I(r)=\frac {1}{\pi }\int _0^\pi e^{r\cos t}\,dt. \] Assume without proof that \(I(r)\) is the unique solution to the modified Bessel equation \[ r\frac {d^2\varphi }{dr^2}+\frac {d\varphi }{dr}-r\varphi =0,\quad r>0, \] which satisfies \(\varphi (0)=1\).

Let \(U\subset \mathbb {R}^n\) be open and bounded, with smooth boundary. For \(T>0\), denote \(U_T=U\times (0,T]\). Let \(\Gamma _T=\overline {U_T}-U_T\) (closure taken in \(\mathbb {R}^n\times \mathbb {R}\)). Recall that for solutions \(u(x,t)\) of \[ \begin {aligned} u_t-\Delta u &= f, && \text {in } U_T,\\ u &= g, && \text {on } \Gamma _T, \end {aligned} \] we define the energy \[ e(t)=\int _U u^2(x,t)\,dx. \]

1. Use energy methods to show that the problem above has at most one solution \(u\in C^{2,1}(U_T)\) (\(C^2\) in the \(x\) variables, and \(C^1\) in \(t\)).
2. Assuming \(g=0\), and \(u\in C^{2,1}(U_T)\), prove that for \(0<t<T\), \[ e(t)\le \left (\int _0^t\left (\int _U f^2(x,s)\,dx\right )^{1/2}\,ds\right )^2. \]

Consider the scalar problem \[ \begin {aligned} u_t+H(Du) &= 0, && \text {in } \mathbb {R}^n\times (0,\infty ),\\ u &= x_1, && \text {on } \mathbb {R}^n\times \{t=0\}. \end {aligned} \] Assume the Hamiltonian \[ H(p)=\frac {1}{4}\sum _{j=1}^n\frac {1}{a_j}(p_j-b_j)^2, \] where the coefficients \(a_j,b_j\) are fixed real numbers, with \(a_j>0\).

1. Use the Legendre transform \(H^*(q)=\sup _p\{p\cdot q-H(p)\}\) to find the Lagrangian.
2. Setting the coefficients \(a_j=1\), \(b_j=0\) for all \(j\), use the Hopf-Lax formula \[ u(x,t)=\min _y\left \{tL\left (\frac {x-y}{t}\right )+g(y)\right \} \] to find an explicit weak solution to the problem above.

Let \(U=\{x\in \mathbb {R}^2:\frac {1}{2}<|x|<1\}\), and denote \(\partial U=\Gamma _1\cup \Gamma _2\) where \[ \Gamma _1=\{|x|=\tfrac {1}{2}\},\quad \Gamma _2=\{|x|=1\}. \] Consider \[ \begin {aligned} -\Delta u &= f, && \text {in } U,\\ \alpha u+\frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma _1,\\ \frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma _2, \end {aligned} \] where \(\alpha >0\), \(\eta \) is the unit outward normal on \(\partial U\), and \(f\in L^2(U)\).

Prove that there exists a constant \(C>0\) such that \[ C\int _U u^2\,dx \le \int _U |Du|^2\,dx+\int _{\Gamma _1} u^2\,dS \] for all \(u\in C^1(\overline {U})\). Polar coordinates may be helpful. Explain how this estimate could be used in a proof of existence and uniqueness for solutions of the problem above.

1. Find all conditions on \(\alpha \), \(\beta \), \(\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Solve the Laplace equation \(\Delta \phi =0\) subject to the boundary conditions: \[ \begin {aligned} \phi& =a &&\text {on } \{z: |z|=2,\ z\ne 2\},\\ \phi& =b &&\text {on } \{z: |z-1|=1,\ z\ne 2\}. \end {aligned} \] (\(a\) and \(b\) are real constants; \(z=x+iy\).)

Let \(A\in \mathbb {R}^{n\times m}\), \(B\in \mathbb {R}^{r\times m}\), and \(b\in \mathbb {R}^n\) be given. Assume \(R(B^*)=N(A)\). For fixed \(\mu \ge 0\), consider the problem \[ \min _{x\in \mathbb {R}^m} \|Ax-b\|_2^2+\mu \|Bx\|_2^2. \]

1. Find the necessary conditions (analogous to the normal equations for the standard least-squares problem) and prove that the necessary conditions have a solution when \(\mu =0\).
2. Prove that the necessary conditions have a unique solution if \(\mu >0\).

Consider the distribution \(g\) defined by \(\langle g,\phi \rangle =\int _{-\infty }^{\infty } e^{-|x|}\phi (x)\,dx\), for all test functions \(\phi \).

1. Calculate the second derivative \(g''\) explicitly. Prove \(g''\) is a distribution of order zero.
2. Find a differential equation for which \(g\) is a Green’s function.

Consider the fourth-order differential equation \[ \begin {aligned} u^{(4)} &= f, && \text {on } (0,1),\\ u(0) &= u(1), && u'(0)=u'(1),\\ u''(1) &= -2, && u^{(3)}(1)=1. \end {aligned} \]

1. Find the least restrictive constraint(s) on \(f\in L^2(0,1)\) under which a solution to the problem exists.
2. With fixed \(f\) chosen such that a solution exists, is the solution unique? If the solution is unique, prove it. Otherwise provide a counterexample.

Prove that the nonlinear integral equation \[ u(x)-\int _0^1 \frac {u^2(y)}{4+x^2-y}\,dy=\frac {x^3}{3},\qquad 0\le x\le 1, \] has exactly one continuous solution \(u\) satisfying \(\max _{0\le x\le 1}|u(x)|\le 1\). Describe, but do not carry out, an iterative procedure for obtaining an approximation to this solution.

1. Introducing the discrete Fourier transform \(U(k,t)=\sum _{n=-\infty }^{\infty }e^{ikn}u_n(t)\), \(0\le k<2\pi \), show that the general solution to the discretized heat equation \[ \frac {du_n}{dt}=\frac {1}{h^2}\{u_{n+1}-2u_n+u_{n-1}\},\qquad -\infty <n<\infty , \] can be written in the form \[ u_n(t)=\int _0^{2\pi } U(k,0)e^{-\epsilon (k)t}e^{-ikn}\,\frac {dk}{2\pi } \] and calculate the function \(\epsilon (k)\).
2. Using the fact that \(U(k,t)\) and \(u_n(t)\) form a transform pair for fixed \(t\), prove the identity \[ \sum _{n=-\infty }^{\infty }e^{i(k-k')n}=2\pi \delta (k-k'). \] Hence, find \(u_n(t)\) given the initial data \(u_n(0)=\sin (2\pi n/p)\) for some fixed integer \(p\).

Variational Principles. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary \(\partial \Omega \) and unit outward normal \(\eta \). Consider the problem \[ \begin {aligned} \Delta u-\epsilon u &= f,\quad \text {in }\Omega ,\\ \frac {\partial u}{\partial \eta } &= 0,\quad \text {on }\partial \Omega , \end {aligned} \] where \(f\in L^2(\Omega )\) satisfies \(\int _\Omega f=0\).

1. Prove that for each real constant \(\epsilon >0\), there exists a unique weak solution \(u_\epsilon \in H^1(\Omega )\).
2. In the case \(\epsilon =0\), find an additional condition on \(u\) which guarantees a unique weak solution \(u_0\), such that the solutions \(u_\epsilon \) from part (a) converge to \(u_0\) as \(\epsilon \to 0^+\). Prove existence and uniqueness of \(u_0\) (with the extra condition), and prove \(u_\epsilon \to u_0\) in \(H^1(\Omega )\) as \(\epsilon \to 0^+\).

Heat Equation. Given the constant \(\beta \in \mathbb {R}\), consider the initial value problem \[ \begin {aligned} u_t &= \beta \Delta u,\quad x\in \mathbb {R}^n,\quad t>0,\\ u(x,0) &= g(x),\quad x\in \mathbb {R}^n. \end {aligned} \]

1. Assume \(\beta >0\) and let \(g\in C(\mathbb {R}^n)\) be bounded. Prove that there exists a bounded solution \(u(x,t)\) which is \(C^\infty \) for all \(t>0\). Under what condition are solutions unique?
2. Assume \(\beta <0\) and let \(g\in C^\infty (\mathbb {R}^n)\) be bounded. Are solutions \(u(x,t)\) necessarily bounded and continuous for \(t>0\)? Prove, or provide a counterexample.

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be open. Suppose \(u\in C(\Omega )\) satisfies the mean value property: for every ball \(B_r(\xi )=\{x\in \mathbb {R}^n:\lvert x-\xi \rvert <r\}\) with \(B_r(\xi )\subset \Omega \), \[ u(\xi )=\frac {1}{\omega _n}\int _{\lvert x\rvert =1}u(\xi +rx)\,dS_x, \] where \(\omega _n\) is the measure of the \((n-1)\)-dimensional unit sphere in \(\mathbb {R}^n\). Prove that \(\Delta u=0\) in \(\Omega \).

Conservation Laws. Consider the first-order equation \[ (G(u))_x+u_y=0, \] where \(G\in C^\infty \), with initial condition \(u(x,0)=h(x)\).

1. Using the method of characteristics, derive the general solution of the initial value problem. Assuming \(h\in C^\infty \), find a condition on \(G'(h(x))\) which implies the solution \(u(x,y)\) will develop a singularity in finite time \(y>0\).
2. Set \(G(u)=\frac {1}{2}u^2\). Given \(u_0\in \mathbb {R}\), find explicit weak solution(s) for \[ h(x)= \begin {cases} u_0, & x<0,\\ 0, & x>0. \end {cases} \] Show that your solutions are weak solutions. Which properties do your solutions exhibit: shocks, rarefaction, nonuniqueness?

Distributions. Let \(\Omega \subset \mathbb {R}^n\) be an open set.

1. Let \(\{f_n\}\subset C(\Omega )\), with \(f_n\) converging to \(f\in C(\Omega )\) pointwise, i.e., \(f_n(x)\to f(x)\) for all \(x\in \Omega \). Does it follow that \(f_n\) converges to \(f\) in the sense of distributions? Prove or give a counterexample.
2. Let \(\{\mu _n\}\subset \mathcal {D}'(\Omega )\) with \(\mu _n\) converging to \(\mu \in \mathcal {D}'(\Omega )\) in the sense of distributions. Let \(\alpha \) be a fixed multi-index. Does it follow that the derivatives \(D^\alpha \mu _n\) converge to \(D^\alpha \mu \) in the sense of distributions? Prove or give a counterexample.
3. Find a sequence of \(L^1(\mathbb {R})\) functions which do not converge in \(L^1(\mathbb {R})\) but which do converge in the sense of distributions.

Wave Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary. Consider the initial boundary value problem \[ \begin {aligned} u_{tt} &= \Delta u,\quad x\in \Omega ,\quad t>0,\\ u(x,0) &= g(x),\quad x\in \Omega ,\\ u_t(x,0) &= 0,\quad x\in \Omega ,\\ u(x,t) &= 0,\quad x\in \partial \Omega ,\quad t>0. \end {aligned} \] Show that there exists a sequence of initial conditions \(\{g_j\}\subset C^2_0(\overline {\Omega })\) and a sequence of frequencies \(\{\omega _j\}\subset \mathbb {R}\) with \(\omega _j^2\to \infty \), such that each solution \(u^{(j)}(x,t)\) corresponding to initial condition \(g_j\) satisfies \[ u^{(j)}_{tt}(x,t)=-\omega _j^2u^{(j)}(x,t),\quad x\in \Omega ,\quad t>0. \] Moreover, prove that there is at most a countable set of \(\omega \) for which \(u_{tt}=-\omega ^2u\), no matter what initial conditions \(g\) are chosen.

Problem.

1. Let \(D\) be an open set in \(\mathbb {R}\times \mathbb {R}^N\) and let \[ f:D\to \mathbb {R}^N \] be a continuous function such that \(f(t,x)\) satisfies a local Lipschitz condition with respect to the \(x\) variable. Consider the initial value problem \[ x'=f(t,x),\quad x(t_0)=x_0,\quad (t_0,x_0)\in D. \] State what is known about the existence and continuability of solutions of this problem. Be particularly detailed about the behavior of solutions as the time variable approaches the endpoints of existence intervals.
2. Apply the above results by sketching the solution picture for the scalar differential equation \[ x'=\frac {1}{tx}. \] (I.e. find \(D\) and then consider typical cases.)

Problem. Let \(\Omega \) be an open subset of \(\mathbb {R}^N\).

1. Define what is meant by a distribution on \(\Omega \).
2. Show that all functions in \(L^1_{\mathrm {loc}}(\Omega )\), are, by an appropriate identification, distributions.
3. Give an example of a distribution which is not of this type.
4. Give the derivative formula for differentiating distributions.
5. Give a definition of the \(L^2\) Sobolev spaces \(H^m(\Omega )\) and \(H^m_0(\Omega )\). (Other notations used are the \(L^2\) Sobolev spaces \(W^{m,2}(\Omega )\) and \(W^{m,2}_0(\Omega )\).)

Problem. State the Lax-Milgram theorem and give a brief sketch of a proof. Use it to establish the existence and uniqueness of solutions (in the sense of distributions) of the boundary value problem \[ \begin {aligned} -\Delta u &= h,\quad \text {in }\Omega ,\\ u &= 0,\quad \text {on }\partial \Omega , \end {aligned} \] where \(h\in L^2(\Omega )\). Also, in your discussion, tell, in what sense the boundary data are assumed.

Problem. State the Phillips version of the Hille-Yosida theorem and illustrate it by a discussion of the heat equation \[ u_t-\Delta u=h(t,x). \]

Global Existence. Let \(\varphi _t(x_0)\) be the flow of the differential equation \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\in C^1(E)\), \(E\subset \mathbb {R}^n\) is open, and \(x_0\in E\). Prove that if the system has a Liapunov function \(V\) with \(\dot {V}(x)=\left .\frac {d}{dt}V(\varphi _t(x))\right |_{t=0}\le 0\) for all \(x\in E\), and the set \(S=\{x\in E:V(x)\le 1\}\) is compact, then solutions \(\varphi _t(x_0)\) with \(x_0\in S\) exist for all positive times \(t>0\).

Variational Principles. Suppose \(\Omega \in \mathbb {R}^n\) is a smooth, bounded domain. Show that there is an \(\epsilon >0\) depending on \(n\) and \(\Omega \) such that if \(f,g\in C(\overline {\Omega })\) are continuous functions such that \(f\le \epsilon \) then the equation \[ \begin {aligned} \Delta u+f(x)u &= g(x),\quad \text {for }x\in \Omega ,\\ u(x) &= 0,\quad \text {for }x\in \partial \Omega , \end {aligned} \] has a weak solution \(u\in H^1_0(\Omega )\).

Heat Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with smooth boundary \(\partial \Omega \). Let \(f\in C^2(\overline {\Omega })\), \(g\in C^\infty (\partial \Omega )\), and assume \(f=g\) on \(\partial \Omega \). Let \(u(x,t)\) solve \[ \begin {aligned} u_t &= \Delta u,\quad x\in \Omega ,\quad t>0,\\ u(x,0) &= f(x),\quad x\in \overline {\Omega },\\ u(x,t) &= g(x),\quad x\in \partial \Omega ,\quad t>0. \end {aligned} \] Prove that \[ \lim _{t\to \infty }\int _\Omega \lvert u(x,t)-v(x)\rvert ^2\,dx=0 \] where \(v\) solves \[ \begin {aligned} \Delta v &= 0,\quad \text {in }\Omega ,\\ v &= g,\quad \text {on }\partial \Omega . \end {aligned} \]

Conservation Laws. A model for flow of fluid in an infinite pipe says that the density \(\rho (x,t)\) satisfies \[ \frac {\partial \rho }{\partial t}+\rho \frac {\partial \rho }{\partial x}=-\rho ^2,\qquad \rho (x,0)= \begin {cases} 1, & \text {if }x\le 0,\\ \frac {1}{2}, & \text {if }x>0. \end {cases} \]

1. Find a solution \(\rho \).
2. Define what it means for your solution \(\rho \) from part (a) to be a weak solution of the system. (Do not carry out the calculations to prove that \(\rho \) is a weak solution.)

Distributions. Suppose \(f(\xi ),g(\xi )\in C^2_0(\mathbb {R})\) are compactly supported twice continuously differentiable functions. Then \[ u(x,t)=f(x+t)+g(x-t) \tag {2} \] is a classical solution of the wave equation \[ u_{tt}-u_{xx}=0\qquad \text {for }-\infty <x,t<\infty . \tag {3} \] Show that if \(f(\xi ),g(\xi )\in C_0(\mathbb {R})\) are merely compactly supported continuous functions, then (2) still defines a solution to (3) in the sense of distributions.

Wave Propagation. Consider the equation \[ u_{tt}-u_{xx}+\lambda u=0,\quad x\in \mathbb {R},\quad t>0, \] where \(\lambda >0\) is fixed.

1. Explain what is meant by a dispersive solution, and find a dispersion relation for the equation.
2. Define an “energy” functional \(E(t)\) for this equation and show that each solution \(u(x,t)\) with finite initial energy conserves energy. State conditions under which uniqueness of solutions holds.

Construct the Green’s function for the equation \[ u''(x)+\alpha ^2u(x)=f(x),\qquad u(0)=u(1),\qquad u'(0)=u'(1) \] and express the solution in terms of the Green’s function. For what values of \(\alpha \) does the Green’s function fail to exist?

Find the solution of Laplace’s equation on the circular domain \(0\le r\le R\), \(0\le \theta <2\pi \), with Dirichlet boundary data \(u(R,\theta )=f(\theta )\).

Picard’s Existence Theorem.

1. Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Show that there is an \(\epsilon >0\) so that the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0 \end {cases} \] has a unique solution \(y\in C^1([t_0-\epsilon ,t_0+\epsilon ],\mathbb {R}^n)\).
2. Suppose that the hypothesis on \(f\) was replaced by \(f(x,t)\in C^0(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\). Would all or part of the conclusions in (a.) continue to hold?

Constant Coefficients. Suppose that \(A\) is a real \(n\times n\) matrix all of whose eigenvalues \(\lambda \) have strictly negative real part \(\operatorname {Re}\lambda <0\). Suppose \[ \frac {dx}{dt}=Ax,\qquad x(0)=x_0. \tag {1.} \]

1. Show that there is a decreasing function \(\delta (t)>0\) such that \(\delta (t)\to 0\) as \(t\to \infty \) and such that every solution of (1.) satisfies \[ \lvert x(t)\rvert \le \delta (t)\lvert x_0\rvert \qquad \text {as }t\to \infty . \]
2. Define: \(z\) is Liapunov Stable. Define: \(z\) is Asymptotically Stable. Show that the zero solution \(z(t)=0\) of the autonomous system is asymptotically stable.

Nonhomogeneous Equations. Let \(v_0,x_0\in \mathbb {R}^n\) and \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i\le -\epsilon _0\) for all \(i=1,\ldots ,n\) and for some \(\epsilon _0>0\). Show that solution of the initial value problem (2.) is bounded for \(t\ge 0\) as long as it exists. Argue that therefore it exists for all time. \[ \begin {cases} \dfrac {dx}{dt}=Ax+e^{-t}(x+v_0),\\ x(0)=x_0. \end {cases} \tag {2.} \]

Periodic Coefficients. Prove that if \(\lvert \epsilon \rvert \ne 0\) is small enough, then all solutions are bounded \[ \ddot {u}+[1+\epsilon \sin (3t)]u=0. \] [Convert to a system of equations. Give a condition for the boundedness of all solutions. Verify that the condition is satisfied in this case.]

Three Species Predator-Prey Model. Consider the following Lotka-Volterra model for populations of three species \(x,y,z\ge 0\). \[ \begin {cases} \dot {x}=x(4-y-z),\\ \dot {y}=y(1-z),\\ \dot {z}=z(x+y-5);\\ x(0)=x_0,\quad y(0)=y_0,\quad z(0)=z_0. \end {cases} \] The only stationary point for this system with all positive coordinates is at \(P=(2,3,1)\). Show that there is an open \(U\subset \mathbb {R}^2\) such that \((2,3)\in U\), a function \(f(x,y)\in C^1(U)\) and \(\epsilon >0\) so that for initial points which are \(\epsilon \)-close to \(P\), that is for \(\lvert (x_0,y_0,z_0)-P\rvert <\epsilon \), if \(z_0=f(x_0,y_0)\) then the solution converges to \(P\) as \(t\to \infty \), but if \(z_0\ne f(x_0,y_0)\) then the solution eventually goes away from \(P\), that is for some \(t_0>0\), \(\lvert (x(t_0),y(t_0),z(t_0))-P\rvert \ge \epsilon \).

Elliptic Equations. Suppose \(\Omega \subset \mathbb {R}^n\) is a smooth bounded domain. Let \(a^i(x)\in C^\infty (\overline {\Omega })\) be smooth functions. Give conditions on \(a^i(x)\) so that for all \(f\in L^2(\Omega )\) there exists a weak solution to the boundary value problem \[ \begin {cases} \Delta u+\displaystyle \sum _{i=1}^n a^i(x)D_i u=f(x), & \text {for }x\in \Omega ,\\ u(x)=0, & \text {for }x\in \partial \Omega . \end {cases} \] [Here \(\Delta =\sum _{i=1}^n\frac {\partial ^2}{\partial x_i^2}\) is the Laplacian and \(D_i=\frac {\partial }{\partial x_i}\) is the partial derivative.]

Heat Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded smooth domain, and \(0<T<\infty \), \(c\in \mathbb {R}\) be constants. Let \(\psi (x)\in C^\infty _0(\overline {\Omega })\). Let \(u\in C^{2,1}(\overline {\Omega }\times [0,T])\) be a solution to \[ \begin {cases} \dfrac {\partial u}{\partial t}=\Delta u+cu, & \text {if }(x,t)\in \Omega \times [0,T],\\ u(x,0)=\psi (x), & \text {if }x\in \Omega ,\\ u(x,t)=0, & \text {if }(x,t)\in \partial \Omega \times [0,T]. \end {cases} \tag {3.} \]

1. Show that the solution \(u(x,t)\) is bounded on \(\overline {\Omega }\times [0,T]\) in terms of \(c\), \(\sup _{x\in \Omega }\lvert \psi (x)\rvert \) and \(t\). [You may assume \(c\le 0\) but it is true for any \(c\).]
2. Show that such solutions of (3.) depends continuously on \(\psi \).

Vibrating Plate. Let \(\Omega \subset \mathbb {R}^n\) be a smooth bounded domain. Let \(\varphi ,\psi \in C^\infty _0(\Omega )\). Suppose that there exists a solution \(u(x,t)\in C^4(\overline {\Omega }\times [0,T])\) to the initial-boundary value problem for the vibrating clamped plate equation \[ \begin {cases} \dfrac {\partial ^2u}{\partial t^2}+\Delta \Delta u=0, & \text {if }x\in \Omega \text { and }t\ge 0,\\ u(x,0)=\varphi (x),\\ \dfrac {\partial u}{\partial t}(x,0)=\psi (x), & \text {for }x\in \Omega ,\\ u(x,t)=\dfrac {\partial u}{\partial n}(x,t)=0, & \text {if }x\in \partial \Omega \text { and }0\le t\le T. \end {cases} \tag {4.} \]

1. Show that the total energy \[ \mathcal {E}(t)=\int _\Omega \left \{\left (\frac {\partial u}{\partial t}(x,t)\right )^2+ \left (\Delta u(x,t)\right )^2\right \}\,dx \] is bounded in terms of \(\varphi \), \(\psi \) and \(t\).
2. Show that such a solution of (4.) is unique.

Conservation Laws. Find a solution to (5.) and prove that it is a weak solution. \[ \begin {cases} \dfrac {\partial u}{\partial t}+(1+u^2)\dfrac {\partial u}{\partial x}=0, & \text {for }x\in \mathbb {R}\text { and }t\ge 0,\\ u(x,0)= \begin {cases} 1, & \text {if }x<0,\\ 0, & \text {if }x\ge 0, \end {cases} & \text {for }x\in \mathbb {R}. \end {cases} \tag {5.} \]

Distributions. Let \(k(x,y)=\frac {1}{2\pi }\log \lvert x-y\rvert \), where \(x,y\in \mathbb {R}^2\).

1. Show that \(k(\cdot ,y)\) is a distribution on \(\mathbb {R}^2\) for each \(y\in \mathbb {R}^2\).
2. Show that as distributions, \(\Delta _x k(x,y)=\delta _y(x)\) for all \(y\in \mathbb {R}^2\). [\(\delta _y(x)=\delta (x-y)\) and \(\delta \) is the Dirac delta function.]

Fundamental Elliptic Estimate. Let \(\Omega \subset \mathbb {R}^n\) be a bounded smooth domain and let \(H^k(\Omega )\) denote the Sobolev space of functions whose distributional derivatives up to the \(k\)-th order are in \(L^2(\Omega )\). Show that if for some constant \(\lambda \), \(u\in H^1(\Omega )\) is a weak solution to \[ \Delta u+\lambda u=0 \] then \(u\in C^\infty (\Omega )\). [You may assume the fundamental elliptic interior estimate: If \(\Omega '\Subset \Omega \) is a compactly contained subdomain, then there is a finite constant \(C=C(\Omega ,\Omega ',k,n)\) so that \[ \lVert u\rVert _{k+2,\Omega '}\le C\bigl (\lVert \Delta u\rVert _{k;\Omega }+\lVert u\rVert _{0;\Omega }\bigr ) \] for all \(u\in H^{k+2}(\Omega )\). Moreover, if \(f\in H^k(\Omega )\) and \(u\in H^1(\Omega )\) is a weak solution to \(\Delta u=f\), then \(u\in H^{k+2}(\Omega ')\).]

1. Prove that \(x\delta (x)=0\) in the sense of distribution.
2. Formulate the differential equation \(xu'-u=0\) as a problem in the sense of distribution and find its general solution, also in the sense of distribution.

Peano Existence Theorem. Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Consider the system of ODE’s \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \qquad (\spadesuit ) \]

1. Find an example that shows that the solution of \((\spadesuit )\) may not be unique. Give another example that shows that the solution may not exist for all time.
2. Prove the Peano Existence theorem: Suppose \(f(x,t)\in C(\mathbb {R}^n\times \mathbb {R})\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Then there is an \(\epsilon >0\) and a function \(y\in C^1([t_0,t_0+\epsilon ],\mathbb {R}^n)\) which solves \((\spadesuit )\).

Picard’s Existence Theorem. Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Suppose that there is a continuous function \(m(t)\in C(\mathbb {R})\) such that \(\lvert f(x,t)\rvert \le m(t)(1+\lvert x\rvert )\) for all \((x,t)\in \mathbb {R}^n\times \mathbb {R}\). Show that the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \] has a unique solution \(y\in C^1(\mathbb {R},\mathbb {R}^n)\) which exists for all time.

Periodic Coefficients. Prove that the equation \[ \ddot {u}+(\sin ^2 t)\dot {u}+(1+\sin 2t)u=0 \] does not have a fundamental set of periodic solutions.

Conservation Laws. Consider the first order equation \[ \begin {cases} \dfrac {\partial u}{\partial t}+(\tan u)\dfrac {\partial u}{\partial x}=0, & \text {for }x\in \mathbb {R}\text { and }t\ge 0,\\ u(x,0)=\varphi (x), & \text {for }x\in \mathbb {R}. \end {cases} \qquad (\diamond ) \]

1. Find a solution \(u(x,t)\) to \((\diamond )\) where \(\varphi (x)=\operatorname {Arctan}(x)\).
2. For \[ \varphi (x)= \begin {cases} 0, & \text {if }x<0,\\ \frac {\pi }{4}, & \text {if }x>0, \end {cases} \] find two weak solutions to \((\diamond )\). Prove that one of them is a weak solution.
3. State the entropy (Rankine-Hugoniot) condition. Which weak solution of part (b.) satisfies this condition?

Distributions. Let \(\Omega \subset \mathbb {R}^n\) be an open set and \(\mathcal {D}'(\Omega )\) the space of distributions on \(\Omega \).

1. Let \(\delta \in \mathcal {D}'(\mathbb {R}^n)\) be the Dirac delta function. Fix a multiindex \(\beta \). Show that the derivative \(D^\beta \delta \) is a distribution.
2. Let \(C^\infty _0(\Omega )\) denote the space of compactly supported smooth functions on \(\Omega \). Briefly explain why there exist nonzero \(\varphi \in C^\infty _0(\Omega )\).
3. Suppose \(\rho \in C^\infty _0(\mathbb {R}^n)\) is such that \(\rho \ge 0\), the support is contained in the unit ball around zero, \(\operatorname {spt}\rho \subset B_1(0)\), and \(\int _{\mathbb {R}^n}\rho \,dx=1\). Let \(\eta _h=h^{-n}\rho (x/h)\). Show that as distributions, \(\eta _h\to \delta \) as \(h\to 0\).

Heat Equation. Let \(\Omega \subset \mathbb {R}^n\) be a bounded smooth domain, \(0<T<\infty \) be a constant, \(c(x)\in C^\infty (\overline {\Omega })\) such that \(c(x)\le 0\) for all \(x\) and \(\psi (x,t)\in C^\infty (\overline {\Omega }\times [0,T])\). Let \(u\in C^{2,1}(\overline {\Omega }\times [0,T])\) be a solution to \[ \begin {cases} \dfrac {\partial u}{\partial t}=\Delta u+c(x)u, & \text {if }(x,t)\in \Omega \times [0,T],\\ u(x,t)=\psi (x,t), & \text {if }(x,t)\in \Omega \times \{0\}\text { or if }(x,t)\in \partial \Omega \times [0,T]. \end {cases} \qquad (\heartsuit ) \]

1. Prove the Maximum Principle: If \(\psi (x,t)\ge 0\) then the solution satisfies \(u(x,t)\ge 0\).
2. Show that such solutions of \((\heartsuit )\) are unique.

Wave Equation. Let \(f(x,t)\in C^\infty (\mathbb {R}^3\times \mathbb {R})\) and \(R>0\) be fixed. Suppose that the support of \(f\) is contained in the cylinder \[ \operatorname {spt} f\subset \{(x,t)\in \mathbb {R}^3\times \mathbb {R}:\lvert x\rvert \le R\text { and }t\ge 0\}. \] Show that there exists a solution \(u(x,t)\) to the inhomogeneous wave equation \((\spadesuit )\). Show that the solution satisfies \(u(x,t)=0\) whenever \(\lvert x\rvert \ge ct+R\). \[ \begin {cases} \dfrac {\partial ^2u}{\partial t^2}=c^2\Delta u+f(x,t), & \text {if }x\in \mathbb {R}^3\text { and }t\ge 0,\\ u(x,0)=0,\\ \dfrac {\partial u}{\partial t}(x,0)=0, & \text {for }x\in \mathbb {R}^3. \end {cases} \qquad (\spadesuit ) \]

Laplace’s Equation. Let \(\varphi (y_1,y_2)\in C_0(\mathbb {R}^2)\) be continuous with compact support. Let \[ u(x)=\frac {x_3}{2\pi }\int _{\mathbb {R}^2} \frac {\varphi (y_1,y_2)\,dy_1\,dy_2} {\bigl [(y_1-x_1)^2+(y_2-x_2)^2+x_3^2\bigr ]^{3/2}}, \] for \(x\in \mathbb {R}^3_+=\{(x_1,x_2,x_3)\in \mathbb {R}^3:x_3>0\}\), the open upper three space. Show that \(u\in C^2(\mathbb {R}^3_+)\cap C(\overline {\mathbb {R}^3_+})\) and solves the boundary value problem for Laplace’s equation \[ \begin {cases} \Delta u=0, & \text {if }x\in \mathbb {R}^3_+,\\ u(x_1,x_2,0)=\varphi (x_1,x_2), & \text {if }(x_1,x_2)\in \mathbb {R}^2. \end {cases} \] [You may assume \[ \int _{\mathbb {R}^2}\frac {x_3\,dy_1\,dy_2} {\bigl [(y_1-x_1)^2+(y_2-x_2)^2+x_3^2\bigr ]^{3/2}}=2\pi . \] ]

Fundamental Elliptic Estimate. Let \(\T ^n=S^1\times \cdots \times S^1\) be the \(n\)-dimensional torus. Let \(u\in C^\infty (\T ^n)\), which is the same as saying \(u\) is a complex valued smooth function on \(\mathbb {R}^n\) which is \(2\pi \)-periodic in each coordinate. Then for \(k\in \mathbb {R}\), the Sobolev norm is given by \[ \lVert u\rVert _k^2=\sum _\alpha (1+\lvert \alpha \rvert ^2)^k\lvert u_\alpha \rvert ^2 \] where \(u\sim \sum _\alpha u_\alpha e^{i\alpha \cdot x}\) is the Fourier series expansion of \(u\) such that the sum runs through all multiindices \(\alpha \in \mathbb {Z}^n\). Let \(H^k(\T ^n)\) be the \(\lVert \cdot \rVert _k\)-completion of the trigonometric polynomials on \(\T ^n\) (finite sums of the form \(p(x)=\sum _{\lvert \alpha \rvert \le N}c_\alpha e^{i\alpha \cdot x}\).)

1. Show that for \(k=2\), there is a constant \(c(k,n)<\infty \) so that whenever \(u\in H^{k+2}(\T ^n)\) then \[ \lVert u\rVert _{k+2}\le c\bigl (\lVert \Delta u\rVert _k+\lVert u\rVert _0\bigr ). \tag {\dagger } \]
2. Supposing that \((\dagger )\) holds for all \(k\), prove that if \(f\in C^\infty (\T ^n)\) such that \(\int _{\T ^n} f\,dx=0\) then there is \(u\in C^\infty (\T ^n)\) such that \[ \Delta u=f\qquad \text {on }\T ^n. \]

1. Determine the conditions on \(f(x)\), \(\alpha \) and \(\beta \) for which there solutions of \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=\alpha ,\qquad u'(1)-u(1)=\beta . \]
2. Find the “best” solution of \[ \frac {d^2u}{dx^2}=1,\qquad u(0)=1,\qquad u'(1)-u(1)=1. \]

Find the weak formulation for the boundary value problem \[ u''-\delta (x)u=1,\qquad u(-1)=u(1)=0, \tag {0.1} \] and then solve the problem. Verify that your solution is a weak solution. Why is it not a strong solution?

Consider the error function \[ E(\mathbf {w})=\sum _{p=1}^{P}(y_p-\mathbf {w}\cdot \mathbf {x}_p)^2 \] where \(\mathbf {w}\) and \(\mathbf {x}_p\) are \(N\)-component vectors, \(N\ge P\), and \(y_p,\mathbf {x}_p\) are prescribed.

1. Show that if \(\mathbf {w}_0\) is a minimizer of \(E\) then \[ [X^TX]\mathbf {w}_0=X^T\mathbf {y} \] where \(X_{pj}=[\mathbf {x}_p]_j\), \(j=1,\ldots ,N\), \(X^T\) is the transpose of the matrix \(X\), and \(\mathbf {y}=(y_1,\ldots ,y_P)^T\).
2. Suppose that the weight vector \(\mathbf {w}\) is updated at each time step \(m\) using the gradient decent algorithm \[ w_j(m+1)=w_j(m)-\eta \frac {\partial E(\mathbf {w}(m))}{\partial w_j(m)}. \] Show that \(\mathbf {w}(m)\to \mathbf {w}_0\) as \(m\to \infty \) provided that the rate parameter \(\eta \) satisfies \[ |1-\eta \lambda _j|<1,\qquad j=1,\ldots ,N \] where \(\lambda _j\) are the eigenvalues of the symmetric \(N\times N\) matrix \(X^TX\). [Hint: Explicitly calculate \(\partial E/\partial w_j\), use part (a) to rewrite the difference equation in terms of \(\Delta \mathbf {w}(m)=\mathbf {w}(m)-\mathbf {w}_0\), and then diagonalize. Assume that \(X^TX\) is invertible so that \(\mathbf {w}_0\) is unique].

Suppose that the real scaling function \(\phi \) satisfies the dilation equation \[ \phi (x)=\sum _k c_k\phi (2x-k). \] Also assume that \(\{\phi (x-k),\, k\in \mathbb {Z}\}\) forms an orthonormal set with respect to the inner product \(\langle u,v\rangle =\int _{-\infty }^{\infty }u(x)v(x)\,dx\). Define the corresponding wavelet function according to \[ \psi (x)=\sum _k(-1)^k c_{1-k}\phi (2x-k). \]

1. Prove that \[ \sum _{k\in \mathbb {Z}} c_kc_{k-2p}=2\delta _{0,p}. \]
2. Show that \(\{2^{j/2}\psi (2^jx-k),\, k\in \mathbb {Z}\}\) forms an orthonormal set.
3. Briefly explain signal decomposition and reconstruction within the framework of multiresolution analysis.

Show that the conformal transformation \[ w=\frac {z-a}{z+a},\qquad a=\sqrt {c^2-\rho ^2} \] where \(0<\rho <c\), maps the domain bounded by the circle \(|z-c|=\rho \) and the imaginary axis onto an annular region centered about the origin with outer radius \(|w|=1\) and inner radius \(|w|=\delta \). Find the radius \(\delta \).

Let \(G\subset \mathbb {R}^N\) be an open set. Define what is meant by a distribution on \(G\). Also define the concept of distributional derivatives of any order \(\alpha \), where \(\alpha \) is a multiindex.

Give a definition of the Sobolev spaces \(H^m(G)\), \(H^m_0(G)\), where \(m\) is a nonnegative integer. Use properties of distributions to prove the generalized integration by parts formula \[ \inner {\partial ^\alpha f}{g} = (-1)^{\lvert \alpha \rvert }\inner {f}{\partial ^\alpha g}, \qquad f\in H^m(G),\ g\in H^m_0(G),\ \lvert \alpha \rvert \le m, \] and \(\inner {\cdot }{\cdot }\) is the \(L^2(G)\) inner product.

1. State the theorem of Rellich-Kondrachov for bounded open sets \(G\subset \mathbb {R}^N\).
2. Let \[ f:\overline {G}\times \mathbb {R}\to \mathbb {R} \] be a continuous bounded function such that \[ \lvert f(x,u)\rvert \le M,\qquad (x,u)\in \overline {G}\times \mathbb {R} \] where \(M\) is a constant. Show that the boundary value problem: \[ -\Delta u=f(x,u),\qquad u\in H^1_0(G), \tag {8} \] has a solution by converting (8) into an equivalent fixed point problem in \(L^2(G)\) and by applying Schauder’s fixed point theorem (give reasons why its hypotheses hold!).

1. Find all conditions on \(\alpha \), \(\beta \), \(\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Remark: For this problem you may use without proof that \[ \frac {1}{2\pi }\int _{-\infty }^{\infty }e^{i\mu x}\,d\mu =\delta (x). \]

1. Suppose \(f(x)\) has the Fourier transform \(F(\mu )\). Evaluate \[ \int _{-\infty }^{\infty } f(x)\overline {f(x-k)}\,dx \] in terms of \(F(\mu )\).
2. Use this identity to prove that the set \(\{\operatorname {sinc}(t-k)\}_{k=-\infty }^{\infty }\) forms an orthonormal set.

Find an integral representation for the solution of the telegrapher’s equation \[ u_{tt}+au_t+bu=u_{xx} \] with \(u(0,t)=f(t)\) and \(u_x(0,t)=g(t)\) specified for all time.

Let \(E\) be an open subset of \(\mathbb {R}^n\) containing \(x_0\) and assume that \(\mathbf {f}\) is a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\).

1. Use the method of successive approximations to prove that the initial value problem \(x'=\mathbf {f}(x)\), \(x(0)=x_0\), has a unique solution on some interval \([-a,a]\), and give an estimate for \(a\).
2. Indicate how the result in (a) can be used to prove a similar result for the nonautonomous case.

Consider the equation \[ a(x,y)u_x+b(x,y)u_y=c(x,y,u) \] where \(a\), \(b\) and \(c\) are suitably smooth.

1. Define what is meant by an integral surface for the above equation.
2. Write down the characteristic equations and show how they can be used to construct an integral surface that contains the curve \(x=\phi (s)\), \(y=\psi (s)\), \(u=\eta (s)\), where \(\phi \), \(\psi \) and \(\eta \) are \(C^1\).
3. What additional conditions, if any, must be placed on the data to guarantee a locally unique solution? Explain carefully.
4. Demonstrate the above method by solving \[ \begin {aligned} u_x+2u_y&=-u,\\ u(0,y)&=f(y) \end {aligned} \] where \(f\) is of class \(C^1\).

Let \(\Omega \) be a bounded domain in \(\mathbb {R}^n\). Consider \[ \begin {aligned} \Delta u&=f &&\text {in }\Omega ,\\ u&=0 &&\text {on }\partial \Omega \end {aligned} \] where \(f\in L^2(\Omega )\).

1. Define the concept of a weak \(H^{1,2}_0(\Omega )\) solution to the above problem.
2. Use Poincaré’s inequality to prove existence of a weak \(H^{1,2}_0(\Omega )\) solution.
3. Assume that \(\Omega \) is smooth. Prove that any two solutions in \(C^2(\Omega )\cap C^1(\overline {\Omega })\) must agree. How would you prove this result for any two solutions in \(C^2(\Omega )\cap C(\overline {\Omega })\)?

1. State d’Alembert’s formula for the solutions \(u\) of the wave equation \(u_{tt}=u_{xx}\).
2. Consider the initial boundary value problem \[ \begin {aligned} u_{tt}&=u_{xx} &&\text {for }0<x<\pi ,\ t>0,\\ u(x,0)&=f(x),\qquad u_t(x,0)=0 &&\text {for }0<x<\pi ,\\ u(0,t)&=0,\qquad u(\pi ,t)=0 &&\text {for }t\ge 0. \end {aligned} \] Use d’Alembert’s formula and the parallelogram rule to define a weak solution solution to this problem. HINT: divide the region \(R=\{(x,t):0<x<\pi ,\ t>0\}\) into subregions. What compatibility conditions are needed upon \(f\) to ensure that the above solution is \(C\), is \(C^1\) and is \(C^2\)?
3. Show that the energy \[ E(t)=\frac {1}{2}\int _0^\pi (u_t^2+u_x^2)\,dx \] is conserved, and use this to prove that a \(C^2\) solution to the above initial boundary value problem is unique.

Under what conditions does a solution of the boundary value problem \(u''+u=f(x)\) with \(u(0)=\alpha \), \(u'(\pi /2)=\beta \) exist?

Find the representation of the delta function and the appropriate transform pair for the differential operator \(Lu=-u''\) on the interval \(0<x<\pi \) subject to boundary conditions \(u'(0)=0\), \(u(\pi )=0\).

State a general principle about the existence intervals for solutions of (2.2) and the behavior of solutions at the endpoints of these intervals.

Use the above and a Gronwall inequality argument (note: \(\lvert f(t,x)\rvert \le \lvert f(t,x)-f(t,0)\rvert +\lvert f(t,0)\rvert \)) to show that solutions of (2.2) are defined on all of \(\mathbb {R}\).

Define what is meant by a fundamental matrix solution \(\Phi (t)\) of (2.3).

Let \(\Psi (t)\) be a fundamental matrix solution of (2.4). Prove that \[ \Psi ^T(t)\Phi (t) \] is a nonsingular constant matrix.

Let \(G\subset \mathbb {R}^N\) be an open set. Define what is meant by a distribution on \(G\). Let \(f\in L^2(G)\) be a real function. Show that \(f\) may be identified with a distribution \(T_f\).

Let \(T\) be a distribution on \(G\) and let \(\alpha =(\alpha _1,\ldots ,\alpha _N)\) be a multiindex. Define the concept of the \(\alpha \)-th distributional derivative of \(T\).

Let \[ r(t)= \begin {cases} 0, & t\le -1,\\ 1, & -1<t<1,\\ 0, & t\ge 1. \end {cases} \] Think of \(r\) as a distribution on \(\mathbb {R}\) and compute all distributional derivatives of \(r\).

Provide the higher dimensional analogue of the previous exercise, i.e., let \[ r:\mathbb {R}^N\to \mathbb {R} \] be defined by \[ r(x)=1,\quad \lvert x\rvert <1, \qquad r(x)=0,\quad \lvert x\rvert \ge 1. \] Compute the distributional gradient \(\nabla r\) of \(r\).

Show that in the sense of distributions there exists a unique solution \(u\in H^1_0(G)\) of the equation \[ -\Delta u+u=f. \]

Solve the differential equation in the sense of distributions \[ xu'=\delta (x^4-1),\qquad x\in [-2,2],\qquad u(-2)=u(2)=0. \] Discuss why two boundary conditions are provided for the first order differential equation.

Find the bounded solution \(u(x,y)\) of the Laplace equation in the sector \[ D=\{(r,\theta ): 0<\theta <3\pi /4\}\qquad (x+iy=z=re^{i\theta }) \] satisfying the boundary conditions \[ u=0\text { when }\theta =0,\qquad u=5\text { when }\theta =\frac {3\pi }{4}. \]

Consider the problem \[ \begin {aligned} \frac {dx}{dt} &= f(x,t),\\ x(t_0) &= x_0 \end {aligned} \tag {1.1} \] where \(f\in C(D,\mathbb {R}^n)\) and \(D\subset \mathbb {R}^n\times \mathbb {R}\) is open. Suppose that \(f\) is Lipschitz continuous in \(x\) for \((x,t)\in D\).

1. Prove the existence of a local (in \(t\)) solution to (1.1) using successive approximations.
2. Prove the uniqueness of this solution and show that it can be extended until it meets \(\partial D\).
3. Prove that the solution is continuous in \((x_0,t_0)\).

Consider \[ x'=f(x) \tag {3.1} \] where \(x\in \mathbb {R}^n\) and \(f\in C^1(\mathbb {R}^n,\mathbb {R}^n)\). Suppose that the solution through any point \(x_0\in \mathbb {R}^n\) exists for all \(t\in \mathbb {R}\).

1. Define the positive and negative semiorbits of (3.1) through \(x_0\in \mathbb {R}^n\).
2. Define the \(\alpha \)- and \(\omega \)-limit sets of a point \(x_0\in \mathbb {R}^n\).
3. Consider the nonlinear spring equation \[ x''+g(x)=0 \tag {3.2} \] where \(g(x)\) is an odd function of \(x\) and \(g'(x)>0\). Describe the \(\alpha \)- and \(\omega \)-limit set for any point on the positive \(x\)-axis.
4. Consider \[ x''+x'+g(x)=0 \tag {3.3} \] where \(g\) is as in (c). Define what is meant by a Liapunov function for this equation and derive one (verify that it is a Liapunov function). Describe the \(\omega \)-limit set for any point in the plane.

Consider the problem \[ \begin {aligned} Lu &\equiv -\frac {d}{dx}\left (p\frac {du}{dx}\right )+qu = \lambda u,\\ \alpha u(0)+\beta \frac {du}{dx}(0) &= 0,\\ \gamma u(1)+\delta \frac {du}{dx}(1) &= 0 \end {aligned} \tag {5.1} \] where \(x\in [0,1]\), \(p\in C^1[0,1]\), \(q\in C[0,1]\), and \(\alpha ,\beta ,\gamma ,\delta \) are constants.

1. Show that (5.1) is self-adjoint if and only if \(p\) and \(q\) are real, \(\gamma \bar {\delta }=\bar {\gamma }\delta \), and \(\alpha \bar {\beta }=\bar {\alpha }\beta \).
2. Suppose hereafter that \(p=q=1\), \(\alpha =1\), \(\beta =0\), \(\gamma =0\), \(\delta =1\). Compute the eigenvalues and eigenfunctions for \(L\) with the given boundary conditions.
3. Compute the Green’s function for \(L\) (with coefficients and boundary conditions as in (b)).
4. Show how to solve the nonhomogeneous problem \[ Lu=f(x) \] with \(L\) and the boundary conditions as in (c). What conditions on \(f\) suffice to make \(u\) a classical solution?

Consider the linear partial differential equation \[ Lu\equiv \sum _{\lvert \alpha \rvert \le k} a_\alpha (x)\partial ^\alpha u=f \tag {6.1} \] where \(\alpha =(\alpha _1,\ldots ,\alpha _n)\) is a multi-index. Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with piecewise smooth boundary and suppose throughout this problem that the coefficients in the equations are smooth.

1. Define what is meant by a weak solution of (6.1).
2. Define the principal part of \(L\), the characteristic variety of \(L\), and a characteristic surface for \(L\).

3. Obtain conditions on \(a\), \(b\) and \(c\) under which \[ Lu=a(x,y)u_{xx}+2b(x,y)u_{xy}+c(x,y)u_{yy}=0 \] is

   1. hyperbolic
   2. elliptic
   3. parabolic.
4. Define what is meant by a well-posed problem for a partial differential equation with boundary and/or initial data specified. Give an example of a well-posed problem for one of the three types of equation in (c) and briefly justify your assertions.

Consider the problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0,\\ u(x,0) &= F(x),\\ u_t(x,0) &= 0,\\ u(0,t)=u(\pi ,t) &= 0 \end {aligned} \tag {7.1} \]

1. Use separation of variables to formally solve (7.1) when \(F\in C[0,\pi ]\).
2. Suppose that \(F\in C^4[0,\pi ]\) and that \(F''(0)=F''(\pi )=0\). Prove that the series you obtain in (a) defines a classical solution of (7.1) under the foregoing conditions on \(F\).

Consider the equation \[ a(x,y)u_x+b(x,y)u_y=c(x,y,u) \tag {8.1} \] where \(a\), \(b\) and \(c\) are suitably smooth.

1. Define what is meant by an integral surface for (8.1).
2. Write down the characteristic equations for (8.1) and show how they can be used to construct an integral surface of (8.1) that contains the curve \[ x=\varphi (s)\qquad y=\psi (s)\qquad u=\eta (s), \] where \(\varphi ,\psi \) and \(\eta \) are \(C^1\).
3. What additional conditions, if any, must be placed on the data to guarantee a locally unique solution? Explain carefully.
4. Consider the equation \[ u_t+uu_x=0 \tag {8.2} \] with initial data given by \[ u(x,0)=u_0(x)= \begin {cases} 1, & x\le 0,\\ 1-x, & x\in (0,1),\\ 0, & x\ge 1. \end {cases} \tag {8.3} \] Does (8.2)–(8.3) have a unique classical solution for \(t\ge 0\)? A unique weak solution for \(t\ge 0\)? Explain.

1. Let \(\Omega \subset \mathbb {R}^n\) be open and let \(u\in C(\Omega )\). Define what it means to say that \(u\) is subharmonic in \(\Omega \).
2. Let \(\Omega \) be open, bounded and connected and let \(u\in C^2(\Omega )\cap C^0(\bar {\Omega })\) be subharmonic. State the weak form of the maximum principle for \(u\).
3. Consider the Dirichlet problem \[ \begin {aligned} \Delta u &= f && \text {in }\Omega ,\\ u &= 0 && \text {on }\partial \Omega \end {aligned} \tag {9.1} \] where \(f\in C^0(\Omega )\) and \(\Omega \) is as in (b). Describe in as much detail as you can the major steps in proving the existence of a classical solution of (9.1) via Perron’s Method.

Let \(\Omega \subset \mathbb {R}\) be open, and let \(C_0^\infty (\Omega )\) be the space of \(C^\infty \) functions with support in \(\Omega \). Define convergence of a sequence \(\varphi _n\) as follows: \(\varphi _n\to 0\) iff \(\exists \) a compact set \(K\subset \Omega \) such that

1. Support \(\varphi _n\subset K\) for all \(n\)
2. \(\displaystyle \lim _{n\to \infty }\sup _{x\in K}\lvert \varphi _n^{(p)}(x)\rvert =0\) for \(p\in Z^+\).

1. Define what is meant by a distribution on \(C_0^\infty (\Omega )\).
2. Define \[ \mu _f(\varphi )=\int _{-\infty }^{\infty } f(x)\bar {\varphi }(x)\,dx \] where \(f\in L^1_{\mathrm {loc}}(\Omega )\) and \(\varphi \in C_0^\infty (\Omega )\). Does this define a distribution on \(C_0^\infty (\Omega )\)? Give reasons.
3. Give a concrete example of a regular distribution and verify that is is a distribution.
4. Show how to define the derivatives of a regular distribution.

Consider the integral equation \[ u(x)=f(x)+\frac {1}{2}\int _0^1 xtu(t)\,dt,\qquad \text {for }0\le x\le 1 \] where \(f(x)\) is a prescribed function.

1. Show that the equation has at most one solution.
2. Derive a formula for the solution.
3. Calculate the solution for the special case \(f(x)=5x/6\).

The so called “Brachistochrone problem” admits such formulation. “A particle is sliding along a trough (in a vertical plane from the point \((0,0)\) to some given point \((x_1,y_1)\) \((x_1>0,\ y_1<0)\) under the effect of gravity without friction (having zero initial velocity). Find the form \(y=Y(x)\) of the trough which minimizes the time of descent of this particle.”

1. Find the functional that ought to be minimized.
2. Using this functional, write the boundary value problem that ought to be solved for finding the solution of the “Brachistochrone problem”. You do not need to solve this boundary value problem.

Define: \(z\) is a uniformly Liapunov stable solution of (2).

Define what is meant by a weak solution of (3).

Assume \(f(x,t)=0\). Show by example that if \[ \varphi (x)= \begin {cases} \alpha , & \text {if }x<0,\\ \beta , & \text {if }x\ge 0. \end {cases} \] where \(\alpha <\beta \) are constants, then the IVP (3) may have more that one weak solution. (Check that your examples are weak solutions.)

Under what additional condition are the weak solutions unique?

Show that \(F\) is a distribution on \(\mathbb {R}\).

Let \(\zeta \in C_0(\mathbb {R})\) be continuous with compact support such that the integral \[ \int _{-\infty }^{\infty }\zeta (x)\,dx=1. \] Show that as distributions, the convolutions \(\zeta _\epsilon *F\to F\) as \(\epsilon \to 0\), where \[ \zeta _\epsilon (x):=\frac {1}{\epsilon }\zeta \left (\frac {x}{\epsilon }\right ). \]

Find \(\dfrac {d^k}{dx^k}F\) for all integers \(k>0\).

Find a condition relating \(\Omega \) and \(V\) which implies there exists a unique weak solution \(u\in H^1_0(\Omega )\) for problem (6). Prove the existence and uniqueness under your condition.

Assume that \(u\in H^1(\mathbb {T}^n)\) is a weak solution to \(Lu=f(x)\) on \(\mathbb {T}^n\), where \(f\in C^\infty (\mathbb {T}^n)\). Show that \(u\in C^\infty (\mathbb {T}^n)\).

Use a Green’s function to find the representation of a possible solution to \[ y''+\omega ^2y=f(x),\qquad y(0)=y'(1)=0 \] for arbitrary values of the parameter \(\omega \). For what values of \(\omega \) does the solution exist? If the solution does not exist find the least square solution. Explain the meaning of the least square solution.

Define and find the Green’s function for the discrete operator \(Lu=-\Delta u-\lambda u\) defined on the space of doubly infinite sequences \(u=\{u_n\}\), \(-\infty <n<\infty \), with \(\sum _{n=-\infty }^{\infty }u_n^2<\infty \), and with \((\Delta u)_n=u_{n+1}-2u_n+u_{n-1}\).

Let \[ E_0=E\cap \{u:u(0)=u(\pi )=0\}, \] and restrict \(f\) to \(E_0\). Show that, except for a countable set of values of \(\lambda \) (describe this set !), the following holds: There exists \(\delta =\delta (\lambda )\) such that for all \(h\in X\), \(\norm {h}_X<\delta \), there exists a unique \(u\in E_0\) such that \[ f(u)=h, \tag {2.2} \] i.e. the problem \[ \begin {cases} u''+\lambda u+u^3=h,\\ u(0)=0=u(\pi ) \end {cases} \tag {2.3} \] has a unique solution.

Use properties of the Leray-Schauder degree to deduce that (2.13) has a \(T\)-periodic solution for all \(\epsilon \).

Let \(G\subset \mathbb {R}^N\) be an open set and let \(m\ge 0\) be an integer. Let \(f\in C^m(G)\) be real or complex valued function. Show that \(f\) may be identified with a distribution \(T_f\) and compute the distributional derivatives of \(T_f\) up to order \(m\).

Let \[ r(x)= \begin {cases} x^2, & x\ge 0,\\ 0, & x\le 0. \end {cases} \] Compute all distributional derivatives of \(r\).

Show that in the sense of distributions there exists a unique solution \(u\in W_0^{1,2}(G)\) of the equation \[ -\Delta u+u=f. \]

Let \(G\) be bounded. Show that in the sense of distributions there exists a unique solution \(u\in W_0^{1,2}(G)\) of the equation \[ -\Delta u=f. \]

Suppose \(D\) is the rectangle defined by \(\lvert x-x_0\rvert \le a\), \(\lvert y-y_0\rvert \le b\) in the plane \(\mathbb {R}^2\). Suppose that \(f\) is a real valued function defined on \(D\) which is continuous, hence bounded by some nonnegative number \(M\) on \(D\).

1. Show that if \(\alpha =\min \{a,b/M\}\), then there exists a function \(y(x)\), defined on \(\lvert x-x_0\rvert \le \alpha \), such that \[ y'(x)=f(x,y(x))\quad \text {and}\quad y(x_0)=y_0. \]
2. Show that the solutions \(y\) guaranteed in part (i) may not be unique; that is, give an example in which (i) holds and more than one solution \(y(x)\) exists through the point \((x_0,y_0)\) in your example.

Suppose \(u(x,t): x\in \mathbb {R}\), \(t\in [0,\infty )\) is of class \(C^\infty \) and that \(u\) is a solution of \[ \begin {cases} u_{tt}=u_{xx}\\ u(x,0)=g(x)\\ u_t(x,0)=h(x). \end {cases} \] Suppose \(g\) and \(h\) are in \(C_0^\infty \). (\(C^\infty \) functions with compact support.) Define \[ k(t)=\frac 12\int _{-\infty }^{\infty }u_t^2(x,t)\,dx\quad \text {(kinetic energy)} \] and \[ p(t)=\frac 12\int _{-\infty }^{\infty }u_x^2(x,t)\,dx\quad \text {(potential energy)} \] Prove that \(k(t)+p(t)\) is constant and that for sufficiently large times \(t\), \(k(t)=p(t)\). Hint: d’Alembert’s solution.

Consider the Sturm-Liouville operator \[ \mathcal {L} u=\frac {1}{w(x)}\{-(p(x)u')'+q(x)u\} \] where \(p(x)>0\), \(w(x)>0\) and \(1/p(x)\), \(q(x)\) and \(w(x)\) are locally Lebesgue integrable on an arbitrary interval \(I\) (open or closed, bounded or unbounded).

1. Define what is meant by a solution of \(\mathcal {L} u=\lambda u\) on \(I\).
2. Show that the initial value problem for the differential equation \(\mathcal {L} u=\lambda u\) on \(I\) with \(u(x_0)=c_1\), \((pu')(x_0)=c_2\) at \(x_0\in I\) is equivalent to an integral equation for the vector \((u_1,u_2)=(u,pu')\). Prove the equivalence.

Let \(f(x)\) be in the Hilbert space \(L^2(-\infty ,\infty )\). Construct and use a Green’s function to find the solution in \(L^2(-\infty ,\infty )\) of the equation \[ u''-\alpha ^2u=f(x),\qquad (\alpha >0). \]

The \(z\)-transform is defined on sequences in \(\ell ^2\) by \[ U(z)=\sum _{-\infty }^{\infty }u_nz^n, \] \[ u_n=\frac {1}{2\pi i}\int _C U(z)z^{-n-1}\,dz \] where the contour \(C\) is a closed contour enclosing the origin, and integration is counterclockwise. If \(z\) is restricted to lie on the unit circle then \(U(z)=U(e^{i\theta })=f(\theta )\).

1. Find the relationship between the \(z\)-transform and the Fourier series representation of \(f(\theta )\).
2. Use the \(z\)-transform to solve the system of equations \[ \frac {du_n}{dt}=\frac {1}{2h}(u_{n+1}-u_{n-1}) \] with \(u_n(0)=\delta _{n0}\).

Suppose \(u\) is of class \(C^2\) and \(\Delta u\ge 0\) in a bounded domain \(\Omega \) with \(C^2\) boundary. State and prove a maximum principle.

Consider the boundary value problem \(u''=f(x)\), \(u'(0)=0\), \(u(1)=0\) where \(f(x)\) is a prescribed function.

1. Show that the problem has at most one solution.
2. Show that the solution can be written as \[ u(x)=\int _0^1 k(x,y)f(y)\,dy \] for all continuous functions \(f(x)\) where \[ k(x,y)= \begin {cases} x-1, & \text {for }0\le y\le x\le 1,\\ y-1, & \text {for }0\le x\le y\le 1. \end {cases} \]

Find the general solution, in the sense of distribution theory, of the differential equation \[ x^2u'(x)=\delta (x^2-1). \]

Find the adjoint operator, its domain and the least squares solution of the problem \(Lu\equiv 4u''+u=e^x\) for \(0\le x\le \pi /2\) if \(u(0)=\alpha \) and \(u(\pi /2)=\beta \).

Find the spectral representation of the delta function for the operator \(L\) defined by \(Lu=-u''\) on \(x\in (-\infty ,0]\) with boundary condition \(u(0)=0\). Then solve the problem \(Lu=f\) if \(f(x)=1\) for \(-2\le x\le -1\) and \(f(x)=0\) elsewhere.

Find the Green’s function for the Poisson equation \(\Delta u=f\) in the infinite strip \(0\le y\le 1\), \(-\infty <x<\infty \) with boundary data \(u(x,0)=0\) and \(u(x,1)=1\). Then solve the problem (in quadrature) if \(f(x,y)=y(1-y)e^{-x^2}\).

Show that solutions of initial value problems for (2) are defined on \((-\infty ,\infty )\).

Let \(\operatorname {trace} A(t)\equiv 0\), \(t\in (-\infty ,\infty )\) and let \(\overline {X}(t)\) be a matrix solution of (2) show that \(\det \overline {X}(t)=\text {constant}\), \(t\in (-\infty ,\infty )\).

Let \(\overline {X}(t)\) be a matrix solution of (2) with \(\det \overline {X}(t_0)\ne 0\), some \(t_0\). Characterize all other matrix solutions of (2) and provide a characterization of all solutions of (2).

Assume that \(\overline {X}(t)\) is a matrix solution of (2) with \(\overline {X}(0)\) nonsingular and \(\overline {X}(t)\) uniformly bounded on \([0,\infty )\). Also assume \[ \liminf _{t\to \infty }\operatorname {Re}\int _{0}^{t}\operatorname {trace} A(s)\,ds >-\infty . \] Prove that \(\overline {X}^{-1}(t)\) is uniformly bounded on \([0,\infty )\) and no solution \(\varphi (t)\) of (2), not identically zero, can satisfy \(\varphi (t)\longrightarrow 0\) as \(t\longrightarrow \infty \).

Define what is meant by a distribution on \(G\).

Let \(f:G\longrightarrow \mathbb {R}\) be a measurable function such that \(\int _{G_0}\lvert f\rvert \,dx<\infty \) where \(G_0\) is an open subset of \(G\) with \(\overline {G_0}\) compact, i.e. \(f\in L^1_{\operatorname {loc}}(G)\), show that each such \(f\) defines a unique distribution on \(G\). Give an example of a distribution which is not defined by such an \(f\).

If \(T\) is a distribution on \(G\), define \(\partial ^\alpha T\), where \(\alpha \) is a multi-index. Show that if \(f\in C^1(G)\), then \(f\) and its partial derivatives \(\partial f/\partial x_i\), \(1\le i\le n\), define unique distributions \(T\), \(T_i\), \(1\le i\le n\), on \(G\) and \(\partial ^\alpha T=T_i\); \(\alpha =(0,\ldots ,1,0,\ldots ,0)\), where the \(1\) appears in the \(i\)th place.

Let \[ f=x_+^\lambda = \begin {cases} 0, & x\le 0,\\ x^\lambda , & x>0, \end {cases} \] where \(-1<\lambda <0\). Then \(f\in L^1_{\operatorname {loc}}(\mathbb {R})\), hence \(f\) may be thought of as a distribution.

Compute \(\partial f\). Note that \(\partial f/\partial x\notin L^1_{\operatorname {loc}}(\mathbb {R})\).

Let \(U:\mathbb {R}^2\longrightarrow \mathbb {R}\) be given by \[ U(x,t)=f(x+t)+g(x-t), \] where \(f,g\in L^1_{\operatorname {loc}}(\mathbb {R})\). Show that \[ \partial _x^2 U=\partial _t^2 U \] in the sense of distributions.

Evaluate \(\Delta _3(1/r^2)\), \(r^2=x^2+y^2+z^2\) in \(\mathbb {R}^3\), where \(\Delta _3\) is the distributional Laplacian.

Give a characterization of these spaces in terms of \(L^2(G)\) functions and their distributional derivatives.

Let \(E\) be an open subset of \(\mathbb {R}^n\). Let \(f\) be a continuously differentiable function mapping \(E\) into \(\mathbb {R}^n\), and let \(x_0\) belong to \(E\).

1. Use the method of successive approximations to prove that the initial value problem \(x'=f(x)\), \(x(0)=x_0\) has a unique solution on some interval \([-a,a]\), and give an estimate for \(a\).
2. Suggest how the result in a) can be used to prove a similar result for the nonautonomous case.

The quasilinear Cauchy problem \[ xu_x+yuu_y+xy=0,\qquad u=5\quad \text {when }xy=1, \] has the solution \[ u(x,y)=-1+\sqrt {38-2xy}. \]

1. Derive this solution by the method of characteristics. (Hint: Along a characteristic \(x(s)\), \(y(s)\), \(u(s)\) find \(d(xy)/ds\) and then \(du/dz\) where \(z=xy\).)
2. Verify that \(u(x,y)\) is a solution in the plane domain defined by \(xy<19\).

Consider the Dirichlet problem \[ \Delta u=f\quad \text {in }\Omega , \] \[ u=0\quad \text {on }\partial \Omega , \] where \(\Omega \subset \mathbb {R}^n\) is a bounded open set and \(f\in L_2(\Omega )\).

1. Define the concept of a weak \(H_0^1(\Omega )\) solution of the above problem.
2. Use the Poincaré inequality, \[ \int _\Omega u^2\,dx\le c\int _\Omega \lvert \nabla u\rvert ^2\,dx \quad \text {for all }u\in H_0^1(\Omega ), \] to prove the existence of a weak \(H_0^1(\Omega )\) solution.

Let \(A\) be a selfadjoint operator in a Hilbert space \(\mathcal {H}\) and let \(\{E_\lambda \}\) be the spectral family of \(A\). Prove that if a real number \(\mu \) is in the resolvent set of \(A\) then there exists an interval \(I=\{\lambda :\lvert \lambda -\mu \rvert <\delta \}\) such that \(E_\lambda =\text {const.}\) for all \(\lambda \in I\). (You may use without proof the Stieltjes-Hellinger-Stone theorem: \[ R_z=(A-z)^{-1}. \] \[ \bigl ((E_{b+}+E_{b-}-E_{a+}-E_{a-})u,u\bigr ) =\lim _{\epsilon \to 0+}\frac {1}{\pi i} \int _a^b\bigl ((R_{\lambda +i\epsilon }-R_{\lambda -i\epsilon })u,u\bigr )\,d\lambda . \]