Elliptic Equations

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

Suppose that \(u(x)\in C^2(D)\) is a harmonic function on the smooth bounded domain \(D\subset \mathbb {R}^n\) so that \(\nabla ^2u=0\) in \(D\).

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

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

Green’s integral identity is \[ \int _\Omega (v\nabla ^2u-u\nabla ^2v)d x =\int _{\partial \Omega }(vu_\nu -uv_\nu )d\sigma \] where \(u_\nu \) refers to the outward normal derivative.

The simple pendulum consists of a point particle of mass \(m\) suspended from a fixed point by a massless rod of length \(L\), which is allowed to swing in a vertical plane. If friction is ignored then the equation of motion is \[ \ddot {x}+\omega ^2\sin x=0,\qquad \omega ^2=\frac {g}{L}, \] where \(x\) is the angle of inclination of the rod with respect to the downward vertical and \(g\) is the gravitational constant.

1. Using conservation of energy, show that the angular velocity of the pendulum satisfies \[ \dot {x}=\pm \sqrt {2}(C+\omega ^2\cos x)^{1/2}, \] where \(C\) is an arbitrary constant. Express \(C\) in terms of the total energy of the system.
2. Sketch the phase diagram of the pendulum equation in the \((x,\dot {x})\)-plane. Illustrate the one-parameter family of curves given by part (a) for different values of \(C\). Take \(-3\pi \le x\le 3\pi \). Indicate the fixed points of the system and the separatrices - curves linking the fixed points. Give a physical interpretation of the underlying trajectories in the two distinct dynamical regimes \(\abs {C}<\omega ^2\) and \(\abs {C}>\omega ^2\).
3. Show that in the regime \(\abs {C}<\omega ^2\), the period of oscillations is \[ T=4\sqrt {\frac {L}{g}}K(\sin x_0/2), \] where \(\dot {x}=0\) when \(x=x_0\) and \(K\) is the complete elliptic integral of the first kind, which is defined by \[ K(\alpha )=\int _0^{\pi /2}\frac {1}{\sqrt {1-\alpha ^2\sin ^2u}}d u. \]
4. For small amplitude oscillations, the pendulum equation can be approximated by the linear equation \[ \ddot {x}+\omega ^2x=0. \] Solve this equation for the initial conditions \(x(0)=A\), \(\dot {x}(0)=0\) and sketch the phase-plane for different values of \(A\). Compare with the phase-plane for the full nonlinear equation in part (b).

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 \(c\).

A function \(u\in C^2(\Omega )\), \(\Omega \subset \mathbb {R}^n\), is subharmonic (superharmonic) in \(\Omega \) if \(\nabla ^2u\ge 0\) (\(\nabla ^2u\le 0\)) in \(\Omega \).

1. Show that if \(u\) is subharmonic then for every ball \(B_R(\mathbf {x})\Subset \mathbf {R}\) \[ u(\mathbf {x})\le \frac {1}{\omega _n R^{n-1}}\int _{\partial B_R(\mathbf {x})}u(\sigma )\dd \sigma . \] Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n R^n}\int _{B_R(\mathbf {x})}u(\mathbf {y})\dd y. \] How are the inequalities changed if \(u\) is superharmonic?
2. Show that if \(u\in C(\overline {\Omega })\) is subharmonic (superharmonic), the maximum (minimum) of \(u\) is attained only on \(\partial \Omega \), unless \(u\) is constant.
3. Show that if \(u\) is harmonic in \(\Omega \) then \(u^2\) is subharmonic.
4. Let \(u\) be subharmonic in \(\Omega \) and \(F:\mathbb {R}\to \mathbb {R}\) smooth. Under what conditions on \(F\) is \(F(u)\) subharmonic?

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

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

Consider the second-order equation \[ \nabla ^2u-cu=f\qquad \text {in }D\subset \mathbb {R}^n \] with \(c\) constant, \(u,f\in C^2(D)\) and \[ u=g\qquad \text {on }\partial D, \] where \(D\) is bounded and \(\partial D\) is smooth.

1. Assuming a solution \(u\) exists, show that it is unique when \(c>0\). Establish this using two different methods: (i) an integral method and (ii) a Maximum Principle.
2. Suppose that \(n=2\), \(c<0\), \(f=g=0\), and \(D\) is the region \(r^2=x^2+y^2<1\). Show that for \(a_0\) constant, \[ u=a_0J_0(r\sqrt {-c}), \] is a solution as long as \(\sqrt {-c}\) is a zero of the Bessel function \(J_0(x)\), which satisfies \[ \frac {d^2J_0}{dx^2}+\frac 1x\frac {dJ_0}{dx}+J_0=0. \]
3. Suppose that \(n=3\), \(c<0\), \(f=g=0\), and \(D\) is the region \(1<x^2+y^2+z^2<4\). Show that there are non-trivial solutions if \(c=-n^2\pi ^2\), \(n=1,2,3\ldots \).

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 \(f=0\).

(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. Show that if the real symmetric matrix \(A\) has real eigenvalues \(\lambda _i\) and orthogonal eigenvectors \(\mathbf {x}_i\), then for any vector \(\mathbf {y}=\sum _i c_i\mathbf {x}_i\), the smallest eigenvalue satisfies \[ \lambda _0\le \frac {\mathbf {y}^TA\mathbf {y}}{\mathbf {y}^T\mathbf {y}}. \]
2. Show that the eigenfunctions \(\phi \) and eigenvalues \(-\lambda \) of the problem \[ \nabla ^2\phi +\lambda \phi =0\quad \text {in a region }D, \] with \[ \frac {\partial \phi }{\partial n}+\alpha \phi =0\quad \text {on }\partial D, \] where \(\partial /\partial n\) is the outward normal derivative, satisfy \[ \lambda \int _D \phi ^2d x =\int _D \lvert \nabla \phi \rvert ^2d x+\alpha \int _{\partial D}\phi ^2d s. \] Assuming that the eigenfunctions \(\phi \) form a complete orthonormal set, derive an upper bound for the smallest eigenvalue in terms of an appropriate energy integral (Rayleigh quotient).

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 \(f=0\).

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 \(\epsilon \to 0^+\).

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

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

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

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

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 \(u\in H^{k+2}(\Omega ')\).]

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.

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 \[ \Delta u=f\qquad \text {on }\T ^n. \]

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?

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

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

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

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

The (generalized) Hilbert transform \(H\) of a periodic function \(f\) can be defined as follows: For any simply connected two-dimensional domain \(\Omega \), let \(\phi \) satisfy \(\nabla ^2\phi =0\) with \(\nabla \phi \cdot \mathbf {n}=f\) on \(\partial \Omega \). Then \(Hf=\nabla \phi \cdot \mathbf {t}\) on \(\partial \Omega \). (\(\mathbf {n}\) is the exterior unit normal, \(\mathbf {t}\) is the unit tangent vector.)

1. Prove that \(H(Hf)=-f\).
2. Find a relationship between the Fourier transform and the Hilbert transform, using a unit circle as the domain \(\Omega \).
3. What is the Hilbert transform of \(f(x)=a\sin (x)+b\cos (2x)\)?

Let \(G=(0,1)\), \(V=H_0^1(G)\), and \(H=L^2(G)\). Define \[ a(u,v)=\int _0^1 \partial u\,\partial \overline {v}\,dx. \] Use what you know about coercive and elliptic sesquilinear forms and the spectral theorem for operators defined by them to find a basis for \(L^2(G)\). Be sure to present a fairly complete discussion.

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