[ 偏微分方程 2020 Spring ] 習題

Assignment 1 ( Done March 12, 2020 )

Problem 1
Derive the minimal surface equation
$$\left(1+u_y^2\right)u_{xx}-2u_xu_yu_{xy}+\left(1+u_x^2\right)u_{yy}=0$$
by the variational method.

Problem 2
Show that the problem
$$\left\{\begin{array}{ll} u_t=-u_{xx},& 0<x<\pi\\ u(0,t)=u(\pi,t)=0,& t\geq 0\\ u(x,0)=f(x), &0\leq x\leq \pi, \end{array} \right.$$
is not well-posed.


Assignment 2 ( Done March 19, 2020 )

Problem 1
Solve the quasilinear equation
$$\left \{ \begin{array}{l} uu_{x}+u_{y}=1, \\ u(2s^{2},2s)=0,\quad s>0. \end{array} \right.$$

Problem 2
Solve the quasilinear equation
$$\left \{ \begin{array}{l} (y^{2}-u^{2})u_{x}-xyu_{y}=xu,\quad x>0,\\ u(x\text{$,$}y=x)=x. \end{array} \right.$$


Problem 3

Let $u$ be a solution of
$$a(x,y)u_{x}+b(x,y)u_{y}=-u$$
of class $C^{1}$ in the closed unit disk $\Omega$ in the $xy$-plane. Let $a(x,y)x+b(x,y)y\geq 0$ on the boundary of $\Omega$. Prove that $u$ vanishes identically.


Problem 4

 (a) Solve Euler's equation
$$xu_{x}+yu_{y}+zu_{z}=\alpha u\quad(\alpha =\text{const.}\neq 0),$$
$$u(x,y,z=1)=h(x,y).$$

Show that $u$ is a homogeneous function of degree $\alpha$; that is, $u$ satisfies
$$u(\lambda x,\lambda y,\lambda z)=\lambda ^{\alpha }u(x,y,z)$$
for any $\lambda >0$.
(b) Show that, for $\alpha <0$, the only solution of (a) of class $C^{1}$ in a neighborhood of the origin is the trivial solution $u\equiv 0$.


Assignment 3 ( Done March 26, 2020 )

Problem 1
Check $$w_{\nu \nu }+w_{\eta \eta }=-\frac{1}{3\eta }w_{\eta }.$$

Problem 2

Determine the general solution of
$$4u_{xx}+5u_{xy}+u_{yy}+u_{x}+u_{y}=2.$$

Assignment 4 ( Done April 9, 2020 )

Problem 1

Show that there is at most one solution of the initial value problem
$$\left\{\begin{array}{ll} u_{tt}-c^2u_{xx}=0,& 0<x<L, t>0\\ u(x,0)=f(x),& 0\leq x\leq L\\ u_t(x,0)=g(x), & 0\leq x\leq L \end{array} \right.$$
and the boundary value problem
$$\left\{\begin{array}{ll} u_{tt}-c^2u_{xx}=0,& 0<x<L,\, t>0\\ u(0,t)=\alpha(t),& t\geq 0\\ u(L,t)=\beta(t), & t\geq 0, \end{array} \right.$$

where $c$ is a positive constant and we assume the compatibility conditions $\alpha(0)=f(0)$ and $\beta(0)=f(L)$.

Problem 2

Let $u$ be the solution of the wave equation
$$u_{tt}-c^2u_{xx}=0.$$
Suppose $\widetilde{u}$ is the solution of
$$\left\{\begin{array}{ll} u_{tt}-c^2u_{xx}=\widetilde{\varphi}(x,t),& x\in\mathbb{R},\, t\in\mathbb{R}^+\\ u(x,0)=\widetilde{f}(x),& x\in\mathbb{R}\\ u_t(x,0)=\widetilde{g}(x), & x\in\mathbb{R} \end{array} \right.$$
where the following are satisfied

• $\Vert f(\cdot)-\widetilde{f}(\cdot)\Vert_\infty <\epsilon$
• $\Vert g(\cdot)-\widetilde{g}(\cdot)\Vert_\infty <\epsilon$
• $\Vert \varphi(\cdot,t)-\widetilde{\varphi}(\cdot,t)\Vert_\infty <\epsilon$

where $\epsilon>0$ is a small constant. Prove that for fixed $T>0$, $u-\widetilde{u}\rightarrow 0$ uniformly in $x$ on $[0,T]$.

Problem 3

Derive the solution explicitly of the initial-boundary value problem
$$u_{tt}-c^2u_{xx}=0,\quad 0<x<1,\,t>0$$
with initial conditions
$$\left\{\begin{array}{ll} u(x,0)=\sin \pi x,& 0<x<1\\ u_t(x,0)=0,& 0<x<1 \end{array} \right.$$
and boundary conditions
$$\left\{\begin{array}{ll} u(0,t)=0,& t>0\\ u(1,t)=0,& t>0. \end{array} \right.$$

Problem 4

(a) Show that for $n=3$ the general solution of $$\square u=u_{tt}-c^2\Delta u=0,$$
with spherical symmetry about the origin has the form
$$u=\frac{F(r+ct)+G(r-ct)}{r},\quad r=\vert x\vert$$
with suitable $F, G$.

(b) Show that the solution with initial data of the form $u=0$, $u_t=g(r)$; ($g$ is an even function of $r$) is given by
$$u(r,t)=\frac{1}{2cr}\int_{r-ct}^{r+ct}\rho g(\rho)\,d\rho.$$

Assignment 5 ( Done April 23, 2020 )

Problem 1

Show that
$$\mathcal{F}^{-1}\left(e^{-s^{2}t}\right)=\frac{1}{\sqrt{2t}}e^{-x^{2}/4t}.$$

Problem 2

(i) Derive
$$\begin{aligned} u(x,t) &=\int_{\mathbb{R}^{n}}K(x,y,t)\phi (y)\,dy \\ &=\int_{\mathbb{R}^{n}}(4k\pi t)^{-n/2}e^{-\left \vert x-y\right \vert ^{2}/4kt}\phi (y)\,dy \end{aligned}$$
by Fourier transform 

$$\hat{f}(\xi )=\mathcal{F}(f)=(2\pi )^{-n/2}\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi }f(x)\,dx.$$

(ii) Show that

• $$K(x,y,t)\in C^{\infty }\,\,\text{ for }\,\,x\in \mathbb{R}^{n},\text{ }y\in \mathbb{R}^{n},\text{ }t>0.$$
• $$\left( \frac{\partial }{\partial t}-k\Delta _{x}\right) K(x,y,t)=0\,\,\text{ for }\,\,t>0.$$
• $$K(x,y,t)>0\,\,\text{for}\,\,t>0.$$
• $$\int_{\mathbb{R}^{n}}K(x,y,t)\,dy=1\,\,\text{for }\,\,x\in \mathbb{R}^{n},\text{ }%t>0.$$
•  For any $\delta >0$ we have $$\lim \limits_{t\rightarrow 0^{+}}\int_{\left \vert y-x\right \vert >\delta }K(x,y,t)\,dy=0.$$
  uniformly for $x\in \mathbb{R}^{n}$. That is, $$\lim \limits_{t\rightarrow 0^{+}}\int_{\left \vert y-x\right \vert \leq \delta}K(x,y,t)\,dy=1.$$

Problem 3

Construct a solution of the nonhomogeneous 1-dimensional heat equation 

$$\left\{\begin{array}{ll} u_{t}=u_{xx}+f(x,t), & \text{for }-\infty <x<\infty ,\text{ }t>0, \\ u(x,0)=\phi (x), & \text{for }-\infty <x<\infty , \\ u(x,t)\rightarrow 0, & \text{as }\left \vert x\right \vert \rightarrow \infty , \,\,t>0, \end{array} \right.$$

by Fourier transform.



Assignment 6  ( Done May 28, 2020 )

Problem 1
Let $\Omega$ denote the unbounded set $\left \vert x\right \vert >1$.

(a) Let $u\in C^{2}(\overline{\Omega })$, $\Delta u=0$ in $\Omega$ and $\lim_{x\rightarrow \infty }u(x)=0$. Show that
$$\begin{aligned} \max_{\overline{\Omega }}\left \vert u\right \vert =\max_{\partial \Omega }\left \vert u\right \vert . \end{aligned}$$

(b) If the condition $\lim_{x\rightarrow \infty }u(x)=0$ in part (a) is deleted, does the result still hold? Give a counterexample or a proof.

Problem 2
Show that when $n=2$, $x=(r\cos \theta ,r\sin\theta )$, $y=(R\cos \phi ,R\sin \phi )$, then
$$u(x)=\left \{ \begin{array}{ll} \displaystyle\frac{R^{2}-\left \vert x\right \vert ^{2}}{\omega _{n}R}\int_{\partial B}\frac{\varphi (y)}{\left \vert x-y\right \vert ^{n}}\,ds_{y} & \text{for}\, x\in B \\ \varphi (x) & \text{for}\, x\in \partial B \end{array} \right.$$
reduces to
$$\begin{aligned} u(r,\theta )=\frac{1}{2\pi }\int_{0}^{2\pi }\frac{(R^{2}-r^{2})\varphi (\phi)}{R^{2}+r^{2}-2Rr\cos (\theta -\phi )}d\phi \,\text{for }r<R. \end{aligned}$$

Problem 3
Derive
$$\begin{aligned} \frac{\partial G}{\partial \nu }=\frac{\partial G}{\partial \left \vert x\right \vert }=\frac{R^{2}-\left \vert y\right \vert ^{2}}{\omega _{n}R} \left \vert x-y\right \vert ^{-n}\left \{ \begin{array}{l} >0\,\,\text{if}\ y\in B_{R}\text{ }(y\neq x), \\ =0\,\,\text{ if}\ y\in \partial B_{R}\text{ }(y\neq x). \end{array} \right. \end{aligned}$$

Problem 4
Recall that the Liouville theorem says that a harmonic function defined over $\mathbb{R}^{n}$ $(n\geq 2)$ and bounded above is constant.

(a) Suppose $u\in C(\bar{B}_{R})$ is a nonnegative harmonic function in $B_{R}=B_{R}(x_{0})$. Show that
$$\left \vert Du(x_{0})\right \vert \leq \frac{n}{R}u(x_{0}).$$
(b) Prove the Liouville theorem by (a).
(c) Prove Liouville's theorem by the Harnack inequality
$$\frac{R^{n-2}(R-\vert x\vert)}{(R+\vert x\vert)^{n-1}}u(0)\leq u(x)\leq \frac{R^{n-2}(R+\vert x\vert)}{(R-\vert x\vert)^{n-1}}u(0)$$
when $n=2$.

Problem 5
(a) If $n=2$ prove that the Liouville theorem (Corollary 1) in Section 6.5 is valid for subharmonic functions.
(b) If $n>2$ prove that a bounded subharmonic function defined over $\mathbb{R}^{n}$ need not be constant.

Problem 6
Suppose $u$ is a function defined on $\mathbb{R}^{n}$ such that

•  $u$ is bounded and continuous on the half-space $x_{n}\geq 0$ of $\mathbb{R}^{n}$,
• $u=0$ on $x_{n}=0$,
• $u$ is harmonic in $x_{n}>0$.

Prove that $u\equiv 0$ on $x_{n}>0$.

Problem 7
(Rellich's Theorem) Let nontrivial function $u$ satisfy $u\in C^{2}(\mathbb{R}^{n})$, $\Delta u=0$ in $\mathbb{R}^{n}$ ($n\geq 1$). Show
that
$$\int_{\mathbb{R}^{n}}u^{2}\,dx$$
does not exist.

Assignment 7  ( Done June 4, 2020 )

Problem 1

Consider
$$\begin{array}{ll} y^{\prime \prime }\left( x\right) +\lambda y\left( x\right) =0, & 0<x<1, \\ y\left( 0\right) =0, & \\ y\left( 1\right) +ky^{\prime }\left( 1\right) =0, & k>0\text{ is a constant.} \end{array}$$

(a) Show that the above boundary value problem has infinitely
many positive eigenvalues
$$\lambda _{0}<\lambda _{1}<\lambda _{2}<\cdots <\lambda _{n}<\cdots$$
with $\lim_{n\rightarrow \infty }\lambda _{n}=\infty$.
(b) Show that for each $n$ the corresponding eigenfunction $\phi_{n}$ has exactly $n$ zeros in the interval $\left( 0,1\right)$.

Problem 2
Consider the periodic Sturm-Liouville eigenvalue problem%
$$\left \{ \begin{array}{l} \left. y^{\prime \prime }+\lambda y=0,\;-\pi \leq x\leq \pi ,\right. \\ y(-\pi )=y(\pi ),\,y^{\prime }(-\pi )=y^{\prime }(\pi )\text{ (periodic end conditions).} \end{array} \right.$$

(a) Show that the eigenvalues of the periodic Sturm-Liouville eigenvalue problem are $0,\{n^{2}\}$, and the corresponding eigenfunctions are $1,$ $\{ \cos nx\},$ $\{ \sin nx\}$, where $n$ is a positive integer.
$$\begin{array}{cc} \text{eigenvalue} & \text{eigenfunction(s)} \\ \lambda =0 & 1 \\ \lambda =n^{2} & \cos nx,\; \sin nx \end{array}$$
(b) Compare it with Theorem 7.1.7.

Problem 3
Consider the regular Sturm-Liouville eigenvalue problem:
$$\frac{d}{dx}\left( p\frac{dy}{dx}\right) +(q+\lambda s)y=0\text{,}$$
$$a_{1}y(a)+a_{2}y^{\prime }(a)=0,\; \ b_{1}y(b)+b_{2}y^{\prime }(b)=0$$
in Section 7.1.2., where
(1) $p\in C^{1}[a,b],$ $q\in C[a,b],$ $s\in C[a,b]$,
(2) $p(x)>0$ and $s(x)>0$ on $[a,b]$,
(3) $a_{1}^{2}+a_{2}^{2}\neq 0$,
(4) $b_{1}^{2}+b_{2}^{2}\neq 0$

(a) Suppose that $a_{1}a_{2}\leq 0,$ $b_{1}b_{2}\geq 0,$ and $q(x)\leq 0$. Show that $\lambda =0$ is an eigenvalue implies $a_{1}=b_{1}=0$.
(b) Suppose that $a_{1}a_{2}\leq 0,$ $b_{1}b_{2}\geq 0,$ and $q(x)\equiv 0.$ Show that $a_{1}=b_{1}=0$ implies $\lambda =0$ is an eigenvalue.

Problem 4
Show that any second-order equation
$$a_{0}(x)y^{\prime \prime }+a_{1}(x)y^{\prime }+a_{2}(x)y=0,\;(a_{0}\neq 0)$$
can be transformed into normal form
$$u^{\prime \prime }+q(x)u=0$$
by the change of variables
$$y=u\exp \left( -\frac{1}{2}\int^{x}\frac{a_{1}}{a_{0}}dt\right) ,$$
where
$$q(x)=\frac{a_{2}}{a_{0}}-\frac{1}{4}\frac{a_{1}^{2}}{a_{0}^{2}}-\frac{1}{2} \frac{a_{1}^{\prime }a_{0}-a_{1}a_{0}^{\prime }}{a_{0}^{2}}.$$

Problem 5
Suppose that $q\in C[x_{1},x_{2}]$ and $q\leq 0$ in $(x_{1},x_{2}).$ Show that any nontrivial solution $\phi (x)$ of  $y^{\prime \prime }+q(x)y=0$ has at most one zero in $(x_{1},x_{2})$.

Problem 6
Show that Airy's equation $y^{\prime \prime}(x)+xy(x)=0$ is oscillatory.