Fundamental Solutions

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 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 \(I[u]\).

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 \(\partial ^kE=\delta \).

Suppose \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\) and periodic of period \(T>0\). Consider the linear differential equation in \(\mathbb {R}^n\) \[ \begin {cases} \dfrac {dx}{dt}=A(t)x,\\ x(0)=x_0. \end {cases} \tag {2} \] Let \(\Phi (t)\) be the fundamental solution matrix with \(\Phi (0)=I\).

1. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
2. What is the fundamental solution matrix and what are the Floquet exponents for the system \[ A(t)= \begin {pmatrix} a & 1\\ 0 & \dfrac {\dot {p}(t)}{p(t)} \end {pmatrix}, \tag {3} \] where \(p(t)\) is a \(T\)-periodic function?

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 \(t\to \infty \).

Explain the fundamental difference of wave equations in 1D, 2D, and 3D: In 3D — unlike 1D and 2D — any signal (with finite support) propagates leaving no trace behind it.

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”.

Prove the fundamental theorem of algebra:

Every polynomial of degree \(n(\ge 1)\) has exactly \(n\) zeros (each zero being counted according to its multiplicity).

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 \(g:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) 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 \(t\in J\).]

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 \(\lambda _1(D_1)\ge \lambda _1(D_2)\).

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.

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 \(A(t)\) be a continuous, non-constant, real matrix-valued, \(T>0\) periodic function \(A(t+T)=A(t)\). For the system \[ \dot {x}=A(t)x \tag {4} \] define: Monodromy Matrix, Floquet Multipliers and Floquet Exponents.
2. If \(e^{T\gamma }=\rho \) is a Floquet multiplier, then there is a solution of (4) of the form \[ x(t)=e^{\gamma t}p(t) \] where \(p(t)=\pm p(t+T)\).
3. Prove that this equation does not have a fundamental set of bounded solutions. \[ \ddot {x}-(\cos ^2t)\dot {u}+(\sin ^2t)u=0 \]

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.

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

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.

The fundamental solution for the Laplacian in two dimensional space is \(G(\mathbf {x},\mathbf {y})=\frac {1}{2\pi }\ln (\abs {\mathbf {x}-\mathbf {y}})\), where \(\mathbf {x}=(x_1,x_2)\). What is the Green’s function for the Laplacian operator on the first quadrant \(0<x_1,x_2<\infty \), subject to boundary conditions \(G(x_1,0)=0\), \(G_{x_1}(0,x_2)=0\)? Be sure to verify the validity of your solution.

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.

Prove the fundamental theorem of algebra:

Every polynomial of degree \(n(\ge 1)\) has exactly \(n\) zeros (each zero being counted according to its multiplicity).

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\qquad x\in \mathbb {R}^n,\qquad A(t)=A(t+T) \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Consider the equation \(\dot {x}=f(t)A_0x\), \(x\in \mathbb {R}^2\), with \(f(t)\) a scalar \(T\)-periodic function and \(A_0\) a constant matrix with real distinct eigenvalues. Determine the corresponding Floquet multipliers.

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.

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 \(f\in \ell ^{1'}\).

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.

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.

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.

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

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. For a smooth domain \(D\subset \mathbb {R}^n\), solve Poisson’s equation \[ \nabla ^2u=f\text { in }D,\qquad \frac {\partial u}{\partial n}=g\text { on }\partial D \] in terms of an appropriately defined Green’s function.
2. Show that the two-dimensional Green’s function for the Laplacian in an unbounded domain is \[ G(x,x')=\frac {1}{2\pi }\ln \lvert x-x'\rvert . \]
3. Use the method of images to derive the Green’s function for the Laplacian in the half-space \(x\in \mathbb {R}\), \(y>0\) with a Neumann boundary condition on \(y=0\), and evaluate the corresponding solution of part (a) when \(f=0\).

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.

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 \(\mathbb {R}\).

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 \(t>0\).

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.

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 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 \[ f(x)=\frac {1}{2\pi }\int _{-\infty }^{\infty }e^{-i\mu x}F(\mu )\,d\mu , \qquad F(\mu )=\int _{-\infty }^{\infty }e^{i\mu x}f(x)\,dx. \]

Using Green’s functions, solve the problem \[ \frac {d^2u}{dx^2}=\frac {1}{1+x^2},\qquad u(-1)=u(1)=0 \] for \(u(x)\), \(x\in [-1,1]\) (obtain an integral representation for the solution, but do not evaluate the integral).

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.

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 \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\qquad x\in \mathbb {R}^n,\qquad A(t)=A(t+T) \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Consider the equation \(\dot {x}=f(t)A_0x\), \(x\in \mathbb {R}^2\), with \(f(t)\) a scalar \(T\)-periodic function and \(A_0\) a constant matrix with real distinct eigenvalues. Determine the corresponding Floquet multipliers.

(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.

1. Solve Poisson’s equation \[ \nabla ^2u=f\text { in }D,\qquad u=g\text { on }\partial D \] in terms of an appropriately defined Green’s function.
2. Show that the two–dimensional Green’s function for the Laplacian in an unbounded domain is \[ G(\mathbf {x},\mathbf {x}')=\frac {1}{2\pi }\ln \lvert \mathbf {x}-\mathbf {x}'\rvert . \]
3. Use the method of images to derive the Green’s function for the Laplacian in the half–space \(x\in \mathbb {R}\), \(y>0\) with a Dirichlet boundary condition on \(y=0\), and evaluate the corresponding solution of part (a) when \(f=0\).

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.

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. By constructing the one-dimensional Green’s function solve the inhomogeneous equation \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=\alpha ,\qquad u(1)=\beta . \]
2. Use Green’s functions to solve \[ \frac {d^2u}{dx^2}-\alpha ^2u=f(x)\quad \text {on }L^2(-\infty ,\infty ) \] with \(\alpha \) real.

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.

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 \(0<t<T\), \[ e(t)\le \left (\int _0^t\left (\int _U f^2(x,s)\,dx\right )^{1/2}\,ds\right )^2. \]

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.

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; \(z=x+iy\).)

Formulate and prove the Fundamental theorem of Algebra.

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.

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 \(\Omega \).

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

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 \[ \begin {aligned} \Delta v &= 0,\quad \text {in }\Omega ,\\ v &= g,\quad \text {on }\partial \Omega . \end {aligned} \]

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?

Liouville’s theorem states that a function analytic and bounded in the complex plane is necessarily a constant.

Prove this theorem. Using it prove the Fundamental theorem of algebra: Every polynomial \(P(z)\) which is not a constant has at least one complex root.

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.

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 \(t_0>0\), \(\lvert (x(t_0),y(t_0),z(t_0))-P\rvert \ge \epsilon \).

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.

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.

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.

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.

Consider the differential operator \(Lu=u''+k^2u\), \(k\ne 0\), subject to homogeneous boundary conditions \(u(0)=u(1)=0\). Show by direct calculation that the associated Green’s function is \[ g(x,y)= \begin {cases} \dfrac {\sin kx\,\sin k(y-1)}{k\sin k}, & 0\le x<y\le 1,\\[1.2ex] \dfrac {\sin ky\,\sin k(x-1)}{k\sin k}, & 0\le y<x\le 1, \end {cases} \] provided \(\sin k\ne 0\). Using the fact that \(L\) is self-adjoint, show that the solution to the equation \[ u''(x)+k^2u(x)=f(x) \] with inhomogeneous boundary data \(u(0)=\alpha \), \(u(1)=\beta \) is \[ u(x)=\int _0^1 g(x,y)f(y)\,dy-\frac {\alpha }{\sin k}\sin k(x-1) +\frac {\beta }{\sin k}\sin kx \] provided \(\sin k\ne 0\). What happens when \(k=m\pi \)?

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 \(\delta \).

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.

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 \(C^2(\Omega )\cap C(\overline {\Omega })\)?

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 the fundamental existence uniqueness theorem for solutions of (2.2) and sketch a proof.

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

Use general existence uniqueness principles to give the form of all fundamental matrix solutions 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.

Derive the variation of constants formula (using the fundamental solution matrix \(\Phi \)) for solutions of \[ x'=A(t)x+h(t). \tag {2.5} \]

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

Construct the Green function for the equation \[ -u''+u=f(x)\qquad (-\infty \le x\le +\infty ),\qquad u(x)\to 0\text { as }x\to \pm \infty . \]

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.

Liouville’s theorem states that a function analytic and bounded in the complex plane is necessarily a constant.

Prove this theorem. Using it prove the Fundamental theorem of algebra: Every polynomial \(P(z)\) which is not a constant has at least one complex root.

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 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 \(F\).

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.

Construct the Green function for the equation \[ u''(x)=f(x)\quad \text {on the interval }0\le x\le 1 \] if \(u\) satisfies the auxiliary conditions \[ \int _0^1 xu(x)\,dx=0\qquad \text {and}\qquad u(0)+u'(1)=0. \]

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

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.

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 \[ -\Delta u=f. \]

Let \(S\) be the selfadjoint realization in the Hilbert space \(\mathcal {H}=L_2(0,\infty )\) for the singular Sturm-Liouville operator \(\mathcal {L}=-u''\) on \(0<x<\infty \) with boundary condition \(u'(0)=0\). Assume that \(S\) is defined by the domain \[ D(S)=\{u\in \mathcal {H}:u''\in \mathcal {H}\text { and }u'(0)=0\} \] and the relation \(Su=-u''\) for all \(u\in D(S)\).

1. Show that the Green’s function \(G\) for the operator \(S\) is given by the equation below, where \(\zeta =re^{i\theta }\) and \(\sqrt {\zeta }=\sqrt {r}e^{i\theta /2}\) on \(0<\theta <2\pi \): \[ G(x,y,\zeta )= \begin {cases} i e^{i\sqrt {\zeta }y}\cos (\sqrt {\zeta }x)/\sqrt {\zeta }, & 0\le x\le y<\infty ,\\ i e^{i\sqrt {\zeta }x}\cos (\sqrt {\zeta }y)/\sqrt {\zeta }, & 0\le y\le x<\infty . \end {cases} \]
2. Prove that the operator \(S\) has a pure continuous spectrum consisting of all non-negative real numbers.

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 \[ u''-\alpha ^2u=f(x),\qquad (\alpha >0). \]

Liouville’s theorem states that a bounded entire function \(f(z)\) must be a constant.

1. Prove Liouville’s theorem.
2. Use Liouville’s theorem to prove the Fundamental Theorem of Algebra: every polynomial \(P_n(z)\) of degree \(n\ge 1\) has a root.

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.

Let \(S\) be the selfadjoint realization in the Hilbert space \(\mathcal {H}=L_2(0,\infty )\) for the singular Sturm-Liouville operator \(\mathcal {L}=-u''\) on \(0<x<\infty \) with boundary condition \(u(0)=0\). Assume that \(S\) is defined by the domain \[ D(S)=\{u\in \mathcal {H}:u''\in \mathcal {H}\text { and }u(0)=0\} \] and the relation \(Su=-u''\) for all \(u\in D(S)\). Complete the following:

1. Show that the Green’s function \(G\) for the operator \(S\) is given by the equation below, where \(\xi =re^{i\theta }\) and \(\sqrt {\xi }=\sqrt {r}e^{i\theta /2}\) on \(0<\theta <2\pi \). \[ G(x,y,\xi )= \begin {cases} e^{i\sqrt {\xi }y}\sin (\sqrt {\xi }x)/\sqrt {\xi } & \text {for }0\le x\le y<\infty ,\\ e^{i\sqrt {\xi }x}\sin (\sqrt {\xi }y)/\sqrt {\xi } & \text {for }0\le y\le x<\infty . \end {cases} \]
2. Prove that the operator \(S\) has a pure continuous spectrum consisting of all non-negative real numbers.

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

Construct the Green’s function for the equation \(u''(x)=f(x)\) on the interval \(0\le x\le 1\) if \(u\) satisfies the auxiliary conditions \[ \int _0^1 xu(x)\,dx=0\qquad \text {and}\qquad u(0)+u'(1)=0. \]

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.

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 \(t\longrightarrow \infty \).

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.

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: \[ R_z=(A-z)^{-1}. \] \[ \bigl ((E_{b+}+E_{b-}-E_{a+}-E_{a-})u,u\bigr ) =\lim _{\epsilon \to 0+}\frac {1}{\pi i} \int _a^b\bigl ((R_{\lambda +i\epsilon }-R_{\lambda -i\epsilon })u,u\bigr )\,d\lambda . \]

The singular Sturm-Liouville operator \(Lu=-u''\) on \(-\infty <x<\infty \) is known to have a unique selfadjoint realization \(T\) in the Hilbert space \(H=L_2(-\infty ,\infty )\). It is defined by \(D(T)=H\cap \{u:u''\in H\}\) and \(Lu=-u''\) for all \(u\in D(T)\). (Do not prove this). Calculate the Green’s function (resolvent kernel) of \(T\).