Boundary Value Problems

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 smooth bounded domain and \(f:\overline {U}\to \mathbb {R}\) be a smooth function. Consider the energy

on the admissible class

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

Find and justify the PDE boundary value problem satisfied by \(u\).

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 \(\{(x,y): f(x,y)\ne 0\}\).

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 \(I[u]\).

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 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.

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?

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

Solve Laplace’s equation \(\phi _{xx}+\phi _{yy}=0\) in the domain \[ D=\{(x,y)\in \mathbb {R}^2:x^2+y^2>1,\ (x+2)^2+y^2<9\} \] with boundary conditions: \(\phi (x,y)=a\) when \(x^2+y^2=1\) and \(\phi (x,y)=b\) when \((x+2)^2+y^2=9\).

Consider the following inhomogeneous initial-Neumann problem: \[ \begin {aligned} u_t&=Du_{xx}+\alpha tx, &&0<x<\pi ,\ 0<t,\\ u(x,0)&=1, &&0<x<\pi ,\\ u_x(0,t)&=u_x(\pi ,t)=0, &&0<t; \end {aligned} \] where \(\alpha \) is a constant.

1. Determine the eigenfunctions of the homogeneous equation \[ u_{xx}=\lambda u,\qquad 0\le x\le \pi ,\qquad u_x(0)=u_x(\pi )=0. \]
2. Solve the inhomogeneous initial-Neumann problem by carrying out an eigenfunction expansion of \(u(x,t)\) in terms of the eigenfunctions obtained in part (a). That is, denoting the eigenfunctions by \(v_k\), integer \(k\), set \[ u(x,t)=\sum _{k\ge 0}c_k(t)v_k(x), \] and determine the time-dependent coefficients \(c_k(t)\).
3. Give a physical interpretation of the solution in the limit \(\alpha \to 0\).

Consider the following inhomogeneous initial-Neumann problem: \[ \begin {aligned} u_t&=Du_{xx}+\alpha tx, &&0<x<\pi ,\ 0<t,\\ u(x,0)&=1, &&0<x<\pi ,\\ u_x(0,t)&=u_x(\pi ,t)=0, &&0<t; \end {aligned} \] where \(\alpha \) is a constant.

1. Determine the eigenfunctions of the homogeneous equation \[ u_{xx}=\lambda u,\qquad 0\le x\le \pi ,\qquad u_x(0)=u_x(\pi )=0. \]
2. Solve the inhomogeneous initial-Neumann problem by carrying out an eigenfunction expansion of \(u(x,t)\) in terms of the eigenfunctions obtained in part (a). That is, denoting the eigenfunctions by \(v_k\), integer \(k\), set \[ u(x,t)=\sum _{k\ge 0}c_k(t)v_k(x), \] and determine the time-dependent coefficients \(c_k(t)\).
3. Give a physical interpretation of the solution in the limit \(\alpha \to 0\).

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 \[ u(x)\le \frac {n}{\omega _nr^n}\int _{B_r(x)}u(y)\dd y. \]

Find the real-valued function \(\Phi (z)\) that solves Laplace’s equation in the region between the two noncentric circles \(\lvert z\rvert =1\) and \(\lvert z-1\rvert =\frac {5}{2}\) subject to the boundary conditions \(\Phi =0\) on \(\lvert z\rvert =1\) and \(\Phi =1\) on \(\lvert z-1\rvert =\frac {5}{2}\).

Solve Laplace’s equation \[ u_{xx}+u_{yy}=0 \quad \text {in the domain}\quad D=\{(x,y)\in \mathbb {R}^2:\ (x-1)^2+y^2>1\ \& \ (x-2)^2+y^2<4\} \] subject to the boundary condition \[ u(x,y)=a\text { when }(x-1)^2+y^2=1 \qquad \text {and}\qquad u(x,y)=b\text { when }(x-2)^2+y^2=4 \] [\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer in terms of the original real variables \(x,y\)].

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 \(c\).

Let \(X\) be a Banach space, let \(z\in X\) with \(z\ne 0\), and let \(f\in X'\) with \(f\ne 0\). Consider the linear operator \(T:X\to X\) defined for \(x\in X\) by \[ Tx=f(x)z. \]

1. Briefly explain why \(T\) is compact.
2. Use Neumann series to give a condition on \(f\) and \(z\) that ensures the equation \(x+Tx=y\) admits a unique solution \(x\) for all \(y\in X\).
3. Use the Fredholm alternative to find a condition on \(f\) and \(z\) that ensures the equation \(x+Tx=y\) admits a unique solution \(x\) for all \(y\in X\).

Solve Laplace’s equation \[ \phi _{xx}+\phi _{yy}=0 \quad \text {in the domain}\quad D=\{z:\lvert z-1\rvert >1,\ \lvert z-2\rvert <2\} \] subject to the boundary condition \[ \phi (x,y)=a\text { when }\lvert z-1\rvert =1 \qquad \text {and}\qquad \phi (x,y)=b\text { when }\lvert z-2\rvert =2 \] (\(a\) and \(b\) are real parameters).

Solve Laplace’s equation \[ \phi _{xx}+\phi _{yy}=0 \quad \text {in the domain}\quad D=\{(x,y)\in \mathbb {R}^2:\ x^2+y^2>1,\ x^2+(y+1)^2<4\} \] subject to the boundary condition \[ \phi (x,y)=a\text { when }x^2+y^2=1 \qquad \text {and}\qquad \phi (x,y)=b\text { when }x^2+(y+1)^2=4 \] (\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer in terms of the original real variables \(x\) ans \(y\)).

Let \(T:X\to X\) be a bounded linear operator defined on a Banach space \(X\). Denote by \(R_\lambda \) the resolvent operator associated with \(T\) for some \(\lambda \in \rho (T)\). The resolvent set of \(T\) is denoted by \(\rho (T)\).

1. Use Neumann series to show that for \(\lambda ,\mu \in \rho (T)\), \[ R_\lambda =\sum _{j=0}^{\infty }(\lambda -\mu )^j R_\mu ^{j+1}, \] where the series is absolutely convergent in the operator norm when \[ \lvert \lambda -\mu \rvert <\lVert R_\mu \rVert ^{-1}. \]
2. Deduce from part (a) whether the spectrum \(\sigma (T)\) is open, closed or neither.

Consider the following inhomogeneous initial-Neumann problem: \[ \begin {aligned} u_t &= Du_{xx}+\alpha tx, && 0<x<\pi ,\quad 0<t,\\ u(x,0) &= 1, && 0<x<\pi ,\\ u_x(0,t)&=u_x(\pi ,t)=0, && 0<t; \end {aligned} \] where \(\alpha \) is a constant.

1. Determine the eigenfunctions of the homogeneous equation \[ u_{xx}=\lambda u,\qquad 0\le x\le \pi ,\qquad u_x(0)=u_x(\pi )=0. \]
2. Solve the inhomogeneous initial-Neumann problem by carrying out an eigenfunction expansion of \(u(x,t)\) in terms of the eigenfunctions obtained in part (a). That is, denoting the eigenfunctions by \(v_k\), integer \(k\), set \[ u(x,t)=\sum _{k\ge 0} c_k(t)v_k(x), \] and determine the time-dependent coefficients \(c_k(t)\).
3. Give a physical interpretation of the solution in the limit \(\alpha \to 0\).

Consider the following inhomogeneous initial-Neumann problem: \[ \begin {aligned} u_t &= Du_{xx}+\alpha tx, && 0<x<\pi ,\quad 0<t,\\ u(x,0) &= 1, && 0<x<\pi ,\\ u_x(0,t)&=u_x(\pi ,t)=0, && 0<t; \end {aligned} \] where \(\alpha \) is a constant.

1. Determine the eigenfunctions of the homogeneous equation \[ u_{xx}=\lambda u,\qquad 0\le x\le \pi ,\qquad u_x(0)=u_x(\pi )=0. \]
2. Solve the inhomogeneous initial-Neumann problem by carrying out an eigenfunction expansion of \(u(x,t)\) in terms of the eigenfunctions obtained in part (a). That is, denoting the eigenfunctions by \(v_k\), integer \(k\), set \[ u(x,t)=\sum _{k\ge 0} c_k(t)v_k(x), \] and determine the time-dependent coefficients \(c_k(t)\).
3. Give a physical interpretation of the solution in the limit \(\alpha \to 0\).

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 \(f=0\).

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 \(f\).

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 \(x^0\).

Let \(a(x)\in C^\infty (\overline {U})\), where \(U\) is a smooth bounded connected domain, \(U\subset \mathbb {R}^n\). Assume \(a(x)\) is uniformly bounded away from zero on \(U\). Consider the initial-boundary value problem \[ (*)\quad \begin {cases} u_t=\operatorname {div}(aDu), & \text {in }U\times (0,\infty ),\\ u=0, & \text {on }\partial U\times (0,\infty ),\\ u=g, & \text {on }U\times \{0\}. \end {cases} \]

1. If \(g\) minimizes \(\lambda =\int _U a\lvert Dg\rvert ^2\dd x\) over all \(g\in H_0^1(U)\) satisfying \(\int _U g^2\dd x=1\), characterize \(\int _U u(x,1)^2\dd x\) in terms of \(\lambda \).
2. Let \(J(g)=\int _U u(x,1)^2\dd x\), where \(u\) solves \((*)\) with initial condition \(g\), and denote \[ B=\{g\in L^2(U):\norm {g}_2=1\}. \] Denote by \(g_1\) the minimizer of \(\lambda \) from part (a). Prove \[ \sup _{g\in B} J(g) \] is attained by \(g_1\). Does \(J(g)\) have a minimizer over \(B\)? What is \(\inf _{g\in B}J(g)\)?

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 \(0<E_1\le E_2\le \cdots \).

This problem examines features of the differential operator \[ Lu=u'' \] subject to boundary conditions \(u(-1)+u(1)=0\) and \(u'(-1)=u'(1)\).

1. What is the adjoint of this operator? Is it self-adjoint?
2. What is the nullspace of the operator and what is the nullspace of the adjoint operator?
3. Under what conditions on \(f\), \(a\), and \(b\) does the boundary value problem \[ u''=f,\qquad u(-1)+u(1)=a,\qquad u'(-1)-u'(1)=b \] have a solution?
4. Find the smallest least squares solution for the boundary value problem \[ u''=1,\qquad u(-1)+u(1)=1,\qquad u'(-1)-u'(1)=1. \] What equation does this least squares solution satisfy?

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.

1. Let \[ A= \begin {pmatrix} 3 & 1\\ 1 & 3 \end {pmatrix}. \] Find the maximum value of the quadratic form \(\langle Ax,x\rangle \) for \(\lVert x\rVert \le 1\).
2. Prove that eigenvectors corresponding to distinct eigenvalues of a self-adjoint matrix are orthogonal.
3. Let \(H=-\frac {d^2}{dx^2}\) on \(L^2[0,1]\) with boundary conditions \(\psi (0)=\psi (1)=0\). Find the eigenvalues and eigenvectors of \(H\). Show that the eigenvectors of \(H\) corresponding to distinct eigenvalues are orthogonal.

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.

For the differential operator \(Lu=u''+(x^2-1)u'\) on the interval \([0,1]\) with boundary conditions \(u(0)=u'(1)\) and \(u(1)=u'(0)\) find the adjoint operator and its domain.

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 \(u(1)=1\).

1. Find the spherically symmetric eigenfunctions and corresponding eigenvalues for the Laplacian on a spherical domain of radius \(R\) with Dirichlet boundary conditions \(u(R)=0\).
2. Use these eigenfunctions to solve the heat equation on a spherical domain of radius \(R\), with \(u(R,t)=U_0\) and \(u(r,0)=0\).
3. Estimate the time \(t\) at which \(u(0,t)\) is \(\frac {U_0}{2}\). How does this time depend on the radius \(R\)?

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. Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.
2. Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_x&=0,\quad t>0,\quad x\in \mathbb {R}\\ u(0,x)&=g(x),\quad x\in \mathbb {R}. \end {aligned} \] Also solve this problem explicitly.

Let \(\mathbb {R}_+=(0,\infty )\). Consider the problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0, && \text {in } \mathbb {R}_+\times \{t>0\},\\ u(x,0) &= g(x),\quad u_t(x,0)=h(x), && x\in \mathbb {R}_+ && \text {(initial conditions)},\\ u_x(0,t) &= 0, && t>0 && \text {(boundary condition)}. \end {aligned} \] Assume that \(g,h\in C^1(\overline {\mathbb {R}_+})\), and that \(g_x(0)=h_x(0)=0\).

1. Find an explicit formula for the solution \(u(x,t)\) (d’Alembert’s formula is a good place to start).
2. Suppose that for \(0\le t<\frac {1}{2}\) and \(x>0\), we know \(u(x,t)=g(x+t)\), and assume \(\operatorname {supp}g\subset (1,2)\). Thus, initially a wave is travelling toward the origin \(x=0\). What happens to the wave after it hits the origin, say for \(t>2\)? Is it absorbed, reflected, damped, inverted? Calculate \(u(x,3)\).

1. What is the weak formulation for the boundary value problem \(u''=f(x)\) subject to boundary conditions \(u(0)=u(1)=0\)?
2. Let \(\phi (x)=N_2(x+1)\) where \(N_2(x)\) is the piecewise linear function \[ N_2(x)= \begin {cases} x, & 0<x<1,\\ 2-x, & 1\le x<2,\\ 0, & \text {elsewhere}. \end {cases} \] Divide the unit interval \([0,1]\) into \(N\) uniform subintervals with endpoints at \(x_k=\frac {k}{N}\), \(k=0,\ldots ,N\). Use \(\phi (x)\) to define a finite element basis for continuous piecewise linear functions on the unit interval which interpolates function values at \(x_k\). Suppose \(u_k=u(x_k)\) is known. Express \(u(x)\) in terms of \(\phi (x)\).
3. Use the Galerkin method to find equations for \(u_k=u(x_k)\) that approximately solve the boundary value problem in part (a), using the piecewise linear finite element basis.

Discuss the flow pattern associated with the complex potential \[ \Omega (z)=Q_0z+\frac {\overline {Q_0}a^2}{z}+\frac {i\gamma }{2\pi }\log z \] with \(a>0\), \(\gamma \) real and \(Q_0=U_0+iV_0\). Sketch the flow when \(\gamma =0\). Given that the complex force on a cylindrical obstacle with boundary \(C\) is \[ \overline {F}=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz, \] where \(\rho \) is the density of fluid, determine the lift when \(\gamma \ne 0\).

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 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.

Consider a linear chain of \(2N\) atoms with the same spacing \(a\) and force constant \(\beta \) but of two different masses \(m,M\), \(M>m\), placed alternately. Label the light atoms by even integers \(2n\), \(n=0,\ldots ,N-1\) and the heavy atoms by odd integers \(2n-1\), \(n=1,\ldots ,N\). Denoting their displacements from equilibrium by the variables \(U_{2n}\) and \(V_{2n-1}\), Newton’s law of motion gives \[ \begin {aligned} m\ddot {U}_{2n} &= \beta [V_{2n-1}+V_{2n+1}-2U_{2n}],\\ M\ddot {V}_{2n-1} &= \beta [U_{2n}+U_{2n-2}-2V_{2n-1}]. \end {aligned} \] Determine the modes of vibration of the system and sketch the resulting pair of dispersion curves. [Assume periodic boundary conditions \(U_0=U_{2N}\) and \(V_1=V_{2N+1}\).]

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 \(u(R,\theta )=f(\theta )\).

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.]

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.

The “best” least squares solution of the boundary value problem \(u''+u=0\), \(u(0)=\alpha \), \(u(\pi )=0\) is the unique exact solution of what problem?

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 \[ \int _{\mathbb {R}^2}\frac {x_3\,dy_1\,dy_2} {\bigl [(y_1-x_1)^2+(y_2-x_2)^2+x_3^2\bigr ]^{3/2}}=2\pi . \] ]

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 Sturm-Liouville problem \[ x^2(x^2u')'+\lambda u=0 \] on \(1/2\le x\le 1\) with boundary conditions \(u(1/2)=u(1)=0\).

1. State without proof the main theorem on eigenfunction expansions which applies to this example.
2. Use the change of variables \(w(t)=u(1/t)\) to transform the differential equation into \(d^2w/dt^2+\lambda w=0\). Then calculate the eigenvalues \(\lambda _n\) and eigenfunctions \(u_n\), by citing without proof a result for the Sturm-Liouville problem \(y''+\lambda y=0\), \(y(a)=0=y(b)\).
3. Sturm oscillation theory and the Prüfer transformation are used in the general theory to produce the candidate eigenvalues and eigenfunctions. Sketch briefly how this is accomplished, without proofs.

Consider a linear chain of \(2N\) atoms with the same spacing \(a\) and force constant \(\beta \) but of two different masses \(m,M\), \(M>m\), placed alternately. Label the light atoms by even integers \(2n\), \(n=0,\ldots ,N-1\) and the heavy atoms by odd integers \(2n-1\), \(n=1,\ldots ,N\). Denoting their displacements from equilibrium by the variables \(U_{2n}\) and \(V_{2n-1}\), Newton’s law of motion gives \[ \begin {aligned} m\ddot {U}_{2n} &= \beta [V_{2n-1}+V_{2n+1}-2U_{2n}],\\ M\ddot {V}_{2n-1} &= \beta [U_{2n}+U_{2n-2}-2V_{2n-1}]. \end {aligned} \] Determine the modes of vibration of the system and sketch the resulting pair of dispersion curves. [Assume periodic boundary conditions \(U_0=U_{2N}\) and \(V_1=V_{2N+1}\).]

Use the implicit function theorem for mappings between Banach spaces to discuss the unique solvability of the scalar boundary value problem \[ u''+2u+u^2=h(t),\qquad u(0)=u(\pi ), \tag {6} \] for continuous functions \(h\) of small maximum norm. Results needed from Sturm-Liouville theory may be assumed but should be quoted.

Consider the space of real-valued functions \(H^2(0,1)\) and define the quadratic form \[ a(u,v)=\int _0^1 u''v''\,dx+b\int _0^1 u'v'\,dx+c\int _0^1 uv\,dx, \] where \(b\) and \(c\) are positive constants. Show that \(a\) is a form to which the Lax-Milgram theorem may be applied and state the result thus obtained. Let \(f\in L^2(0,1)\), and restrict the form \(a\) to the space \(H^2_0(0,1)\). Show that \(f\) may be thought of as a continuous linear functional on \(H^2_0(0,1)\), and apply the Lax-Milgram theorem to the pair \(a,f\). Determine a boundary value problem which is solved in a distributional sense by the Lax-Milgram solution and show that this solution is at least three times continuously differentiable.

Let \(G\) be a bounded open set in \(\mathbb {R}^N\) with smooth boundary. State and indicate a proof of Poincaré’s inequality for functions \(u\in H^1_0(G)\). Using the inequality show that \[ \norm {u}_0^2=\int _G \lvert \nabla u\rvert ^2\,dx \] defines a norm equivalent to the \(H^1\) norm. Consider the form \[ a(u,v)=\int _G \nabla u\cdot \overline {\nabla v} +\int _G c(x)\cdot \nabla u\,\overline {v}\,dx, \tag {7} \] where the components of the vector function \(c\) belong to \(L^\infty (G)\).

Provide a condition on the function \(c\) in order that \(a\) is a sesquilinear form satisfying the conditions of the Lax-Milgram theorem. Determine a boundary value problem solved by the Lax-Milgram solution.

1. Let \(H\) be a Hilbert space and let \[ A:D(A)\to H \] be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving necessary and sufficient conditions for the unique solvability of the problem \[ \begin {aligned} u'+Au&=0,\\ u(0)&=u_0\in D(A). \end {aligned} \] Be sure to define pertinent terms.

2. Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_x&=0,\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \] Solve this problem explicitly.

   Also state a result about the initial value problem \[ \begin {aligned} u_t-u_x&=f(t,x),\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \]

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. What is the weak formulation for the boundary value problem \(u''=f(x)\) subject to boundary conditions \(u(0)=u(1)=0\)?
2. Let \(\phi (x)=N_2(x+1)\) where \(N_2(x)\) is the piecewise linear function given in Problem A.2. Divide the unit interval \([0,1]\) interval \(N\) uniform subintervals with endpoints at \(x_k=\frac {k}{N}\), \(k=0,\ldots ,N\). Use \(\phi (x)\) to define a finite element basis for continuous piecewise linear functions on the unit interval which interpolates function values at \(x_k\). Suppose \(u_k=u(x_k)\) is known. Express \(u(x)\) in terms of \(\phi (x)\).
3. Use the Galerkin method to find equations for \(u_k=u(x_k)\) that approximately solve the boundary value problem in part (a), using the piecewise linear finite element basis.

Give the heat equation on a semi-infinite rod with initial temperature \(g(x)\) and Dirichlet boundary conditions. Apply the method of odd extension of the heat kernel formula for \((-\infty ,\infty )\) to derive a formula for the solution.

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 \(u'(0)=0\), \(u(\pi )=0\).

Suppose \(\Omega \) is some closed, bounded domain in two spatial dimensions.

1. Under what conditions does the equation \(\nabla ^2\phi =0\) in \(\Omega \), \(\mathbf {n}\cdot \nabla \phi =f\) on \(\partial \Omega \), have a solution? Is it unique?
2. Find an integral equation for \(\phi \) on the boundary \(\partial \Omega \). Be as explicit as possible.
3. Under what conditions does this integral equation have a solution?

Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_x&=0,\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \] Solve this problem explicitly.

Also state a result about the initial value problem \[ \begin {aligned} u_t-u_x&=f(t,x),\qquad t>0,\ x\in \mathbb {R},\\ u(0,x)&=g(x),\qquad x\in \mathbb {R}. \end {aligned} \]

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 \[ u=0\text { when }\theta =0,\qquad u=5\text { when }\theta =\frac {3\pi }{4}. \]

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.

Using conformal mapping find the electrostatic potential \(\phi (x,y)\) in the region between the circumferences \[ \gamma =\{z:\ |z|=1\}\qquad \text {and}\qquad \Gamma =\{z:\ |z-1|=2.5\} \] satisfying the boundary conditions \[ \phi \equiv 0\text { on }\gamma ,\qquad \phi \equiv 1\text { on }\Gamma . \]

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.

Derive the boundary value problem for a minimizer \(w_0\) of the functional \[ \min _{\substack {w\in C^2[0,1]\\ w(0)=0}} \frac {\int _0^1\left (\phi (x)(w'')^2+w^2\right )\,dx} {\int _0^1\phi (x)(w')^2\,dx+\beta w(1)^2} \] where \(\phi (x)>0\), \(\forall x\in [0,1]\), \(\beta \) is a real parameter.

Apply the above result to the initial boundary value problem \[ \begin {aligned} u_t-u_{xx}&=0,\quad t>0,\quad x\in (0,\pi )\\ u(0,x)&=g(x),\quad x\in (0,\pi )\\ u(t,0)&=0=u(t,\pi ),\quad t\ge 0. \end {aligned} \]

Suppose \(f(\theta )\) is a \(2\pi \) periodic function. Define its Hilbert Transform as follows: Let \(u(r,\theta )\) be the solution of Laplace’s equation on the unit circle satisfying Neumann data \(u_r(1,\theta )=f(\theta )\). The Hilbert transform of \(f\) is \(H(f)=u_\theta (1,\theta )\).

Use separation of variables to find the Hilbert Transform of \(f\) in terms of its Fourier coefficients.

Apply the method of odd extensions on the heat kernel formula for \((-\infty ,\infty )\) to derive a formula for the solution of the diffusion equation on \((0,\infty )\) with Dirichlet boundary conditions.

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 \[ k(x,y)= \begin {cases} x-1, & \text {for }0\le y\le x\le 1,\\ y-1, & \text {for }0\le x\le y\le 1. \end {cases} \]

Consider the Dirichlet problem in a bounded plane domain \(\Omega \): \[ u_{xx}+u_{yy}=0\quad \text {in }\Omega , \] \[ u=f\quad \text {on }\partial \Omega , \] where \(f\) is continuous on \(\partial \Omega \).

1. Prove that this Dirichlet problem has at most one solution. Define clearly what you mean by a solution.

2. Prove that if \(\Omega \) is a bounded domain for which the Dirichlet problem is solvable for all \(f\in C(\partial \Omega )\) then the mapping \(f\in C(\partial \Omega )\longrightarrow u\in C(\overline {\Omega })\) is continuous in the uniform norms. (\(\overline {\Omega }\) denotes the closure of \(\Omega \).)

DE Qualifying Examination 1992

Consider the integral \[ u(x,t)=\frac {1}{2\sqrt {\pi t}}\int _{-\infty }^{\infty } f(\xi )e^{-(x-\xi )^2/4t}\,d\xi . \]

1. Prove that if \(f(x)\) is continuous and bounded for all \(x\) then \(u\) is a solution of the initial value problem \[ u_t=u_{xx}\quad \text {for all }-\infty <x<\infty ,\ t>0. \] \[ u(x,0)=f(x)\quad \text {for all }-\infty <x<\infty . \] Formulate clearly what you mean by a solution.
2. Show how the results of part a) can be used to solve the initial-boundary value problem \[ u_t=u_{xx}\quad \text {for all }0<x<1,\ t>0, \] \[ u(x,0)=g(x)\quad \text {for }0\le x\le 1, \] \[ u(0,t)=u(1,t)=0\quad \text {for }t\ge 0. \] Define what you mean by a solution and state what conditions on \(g\) guarantee that such a solution exists.

DE Qualifying Examination 1992

Consider the wave equation \(u_{tt}=u_{xx}\).

1. State and derive the D’Alembert formula for the solutions \(u\) of this equation.
2. Use the method of separation of variables to solve the initial-boundary value problem \[ u_{tt}=u_{xx}\quad \text {for }0<x<1,\ t>0, \] \[ u(x,0)=f(x),\quad u_t(x,0)=0\quad \text {for }0\le x\le 1, \] \[ u(0,t)=u(1,t)=0\quad \text {for }t\ge 0. \] Define what you mean by a solution.
3. Show how the solution of b) can be written in the D’Alembert form.