\documentstyle[amssymb,12pt]{article} \def\baselinestretch{1.2} %\textwidth=18cm %\oddsidemargin=-1cm \begin{document} \begin{center} {\bf ASSIGNED EXERCISES OF MATH544: PDE set 7} {\it April, 2001} \end{center} \vspace{0.4cm} \noindent {\bf THE LAPLACE EQUATION}\\ Ronald B Guenther and John W Lee, Partial Differential Equations of Mathematical Physics and Integral Equations \vspace{0.3cm} \noindent In all types of equations we studied the following: \vspace{0.3cm} \noindent {\bf 1.\, Exact Solutions\\ 2.\, Uniqueness of solutions\\ 3.\, Maximum-Minimum Principles} \vspace{0.2cm} \noindent Here, in the elliptic case, I will include into our study the subject {\bf Green's Function Technique} for the Laplace operator. \vspace{0.2cm} \noindent {\bf 1}.\, {\bf Green's function for the Laplace operator}: Let $D$ be a domain and $B$ its boundary. Let the closure of $D$ be $\bar{D} =D \cap B$. Let $u,v \in C^{2}(D) \cap C^{1}(\bar{D})$. Then the following is an identity \begin{equation} \int_{D}\, [v \nabla^{2}\,u-u \nabla^{2}\, v]\, dV=\int_{B}\, [v {\partial u \over \partial n}- u {\partial v \over \partial n}] \,dS \label{grn} \end{equation} \noindent where ${ \partial u \over \partial n}=$. Here $N$ is the unit normal vector field defined at each point of the surface $B$. Now consider the following Dirichlet problem \begin{eqnarray} \nabla^{2}\,u=h, ~~~ in ~~D,\\ u=f, ~~~ on ~~B. \end{eqnarray} \noindent where $h$ and $f$ some continuous functions in $D$ and on $B$ respectively. Let $v=G(x,y)$ with \begin{eqnarray} \nabla_{y}^{2}\, G(x,y)=\delta(x-y), ~~in ~~D \label{g1}\\ G(x,y)|_{y \in B}=0. \label{g2} \end{eqnarray} \noindent Then we find that \begin{equation} u(x)=\int_{D}\, G(x,y) h(y)\, dv_{y}+\int_{B}\, f(y)\, {\partial G(x,y) \over \partial n_{y}}\, dS_{y} \end{equation} \noindent Hence the solution of a Dirichlet problem reduces to the determination of Green's function $G(x,y)$ satisfying (\ref{g1})-(\ref{g2}). \vspace{0.3cm} \noindent {\bf 2}.\, Find the Green function when $D$ is a ball and $B$ is the sphere with radius $a$. Since ${1 \over |x-y|}$ satisfies $$\nabla^{2} {1 \over |x-y|}=4\pi \delta(x-y)$$ then the following form for $G$ is more suitable $$G(x,y)={1 \over 4\pi}[{1 \over |x-y|}-{q \over |x^{*}-y|}]$$ where $q$ and $x^{*}$ are to be determined thorough the boundary condition. Let $y \in B$ then $$G|_{y \in B}=0$$ or $${1 \over |x-y|}={q \over |x^{*}-y|}$$ This gives \begin{equation} q={a \over r},~~~ x^{*}=q^2\,x. \end{equation} \noindent where $r^2=|x|^2$. We find also $${\partial G(x,y) \over \partial n_{y}}= <{y \over a}, grad_{y} G(x,y)> |_{y \in B}= {r^2-a^2 \over 4\pi\,|x-y|^{3}}$$ Hence the solution of the Dirichlet problem when $h=0$ is $$ u(x)=\int_{B}\, {a^2-r^2 \over 4\pi \, |x-y|^3} \,f(y)\,dS_{y}$$ \noindent {\bf 3}.\, Using the Green's function technique solve the following Dirichlet problem. \begin{eqnarray} \nabla^{2}\,u=0, ~~~ -\infty < x,y < \infty,~~ z > 0,\\ u(x,y,0)=f(x,y),~~~ -\infty 0$ so small that $K_{\varepsilon}(x), K_{\varepsilon}(x^{\prime}) \subset D$ and $|x-x^{\prime}| > 2 \varepsilon$. Apply (\ref{grn}) to the domain $$D_{\varepsilon}=D-(K_{\varepsilon}(x) \cup K_{\varepsilon}(x^{\prime}))$$ for $u(y)=G(x,y)$, and $v(y)=G(x^{\prime},y)$ \vspace{0.3cm} \noindent {\bf 6}.\, In two dimensions let $D$ be a disc with radius $a$ and $B$ is the circle bounding $D$ with radius $a$. Consider \begin{eqnarray} \nabla^{2}\,u=0, ~~~ r < a, \\ u=f~~~ on~~ B. \end{eqnarray} \noindent {\bf a}.\, Find the formal solution.\\ {\bf b}.\, Find reasonable restrictions on $f$ so that the formal solution is a solution. \vspace{0.3cm} \noindent {\bf 7}.\, Let $D=\{(x,y,z) \in R^3 | z > 0\}$ and $B= \{ (x,y,z) | z=0 \}$. Solve the following Dirichlet problem \begin{eqnarray} \nabla^2 \, u=0, ~~~ in~~ D, u=f(x,y)~~~ on ~~ B. \end{eqnarray} \vspace{0.3cm} \noindent {\it Remark 1}. The last two problems were solved in class. They respectively represent discrete and continuous spectrum in the method of separation of variables. \vspace{0.3cm} \noindent {\bf 8}.\, In two dimensions consider \begin{eqnarray} \nabla^2\, u=0, ~~~ 0 < x < a, ~~ 0 < y < b,\\ u(x,0)=f(x),~~u(x,b)=0,~~ 0 \le x \le a,\\ u(0,y)=u(a,y)=0, ~~ 0 \le y \le b. \end{eqnarray} \noindent {\bf a}.\, Find the formal solution,\\ {\bf b}.\, Find reasonable restrictions on $f(x)$ so that the formal solution is a solution. \vspace{0.3cm} \newpage \noindent {\bf 9}.\, Consider \begin{eqnarray} \nabla^2 \, u(r,\theta)=0, ~~~ 0 \le r < a,\\ {\partial u(a,\theta) \over \partial r}+\alpha \, u(a,\theta)=f(\theta), \end{eqnarray} \noindent with $f$ is $2\pi$ periodic continuous function and $\alpha > 0$ a constant. \noindent {\bf a}.\, Solve formally by separation of variables.\\ {\bf b}.\, Find reasonable conditions on $f$ so that the formal solution is a solution. \vspace{0.3cm} \noindent {\bf 10}.\,Suppose that $k > 0$ is a constant and $\nabla^2 u-k^2 u=0$. Formulate Dirichlet, Neumann , and Robin problems of this operator $L=\nabla^2-k^2$ and prove that each problem has at most one solution. \vspace{0.3cm} \noindent {\bf 11}.\, Prove the following theorems. \vspace{0.2cm} \noindent {\bf Theorem 1}.\, {\it Let $ u \in C^{2}(D) \cap C^{1}(\bar{D})$, where $D$ is abounded domain. \vspace{0.2cm} \noindent (i)~ If $ \nabla^2 \,u \le 0$ in $D$ and $u \ge 0$ on $B$. then $u \ge 0$ in $D$,\\ {ii)~ If $\nabla^2 \,u \ge 0$ in $D$ and $u \le 0$ on , $B$ then $u \le 0$ in $D$. } \vspace{0.2cm} \noindent {\bf Theorem 2}.\,{\it Let $ u \in C^{2}(D) \cap C^{1}(\bar{D})$, where $D$ is a bounded domain. \vspace{0.2cm} \noindent (i)~~~ If $\nabla^2 \, u \ge 0$ in $D$, then $u(x) \le M=max_{y \in B}\, u(y)$ for all $x \in D$.\\ (ii)~~ If $\nabla^2\, u \le 0$ in $D$, then $u(x) \ge m=\min_{y \in B}\, u(y)$ for all $x \in D$.\\ (iii)~ If$ \nabla^2\, u=0$ in $D$, then $ m \le u(x) \le M$ for all $x \in D$. } \vspace{0.3cm} \noindent {\bf Theorem 3}.\, {\it Let $D$ be a bounded domain and $B$ its boundary. Let $u \in C^{2}(D) \cap C^{1}(\bar{D})$. Then there is at most one solution of the problem } \begin{eqnarray} \nabla^{2}\, u=h, ~~~ in ~~D,\\ u=f, ~~~ on~~ B. \end{eqnarray} \vspace{0.2cm} \noindent {\bf Theorem 4}.\,{\it Suppose that $u \in C^{2}(D) \cap C^{1}(\bar{D})$ and satisfies the Dirichlet problem as in the previous theorem (Thm. 3). Suppose that $h(x)$ is bounded by a constant $H$ in $D$ and $f$ is bounded by $F$ on $B$. Then there is a constant $C$ depending only on the domain $D$ such that $|u(x)| \le F+CH$ } \vspace{0.2cm} \noindent {\bf Proof}. Find a function $w$ such that \begin{equation} \nabla^ 2 \, w \le -H~~ in ~~D~ and~~ w \ge \pm u~~ on~ B. \label{eqw} \end{equation} \noindent Given such a function we have \noindent $\nabla^2\, (w \pm u) \le - H \pm h \le 0$ in $D$ and $w \pm u \le 0$ on $B$ \noindent Using Thm. 1 above we must have $$w \pm u \ge 0$$ or $w \ge |u|$ in $D$. Hence we have to find an $w$ satisfying (\ref{eqw}). Let \begin{equation} w(x)=F+H[e^{d}-e{x_{1}-c_{c}}], ~~ x \in \bar{D} \end{equation} \noindent where $D$ lies in between the planes $x_{1}=c_{1}$ and $x_{1}=c_{1}+d$ and $C=e^{d}$. {\it Prove that this $w$ satisfies all requirements of (\ref{eqw}). \vspace{0.2cm} \noindent {\bf Theorem 5}.\, {\it Suppose that the functions $u_{i} \in C^{2}(D) \cap C^{1}(\bar{D})$, $i=1,2$, satisfies $\nabla^2 \, u_{i}=h_{i}$ in $D$ and $u_{i}=f_{i}$ on $B$, where $h_{i}$ and $f_{i}$ are given. If $$|h_{1}(x)-h_{2}(x)| \le \varepsilon$$ for all $x \in D$ and $$|f_{1}(x)-f_{2}(x)| \le \varepsilon $$ for all $x \in B$. then $$|u_{1}(x)-u_{2}(x)| \le (1+C) \varepsilon$$ for all $x \in D$, where $C$ is a constant determined by the domain $D$. \vspace{0.2cm} \noindent {\it Remark 2}.\, First three theorems were proved in the class. \newpage \vspace{0.3cm} \noindent {\bf 12}.\, Prove the following theorem. \vspace{0.2cm} \noindent {\bf Theorem 6}.\, {\it Let $u \in C^{2}(D) \cap C^{1}(\bar{D})$ for some bounded domain $D$. \vspace{0.2cm} \noindent (1)~~ If $\nabla^2 \,u < 0 $ in $D$ and $u \ge 0$ on $B$ , then $ u \ge 0$ in $D$.\\ (ii)~~ If $\nabla^2 \, u > 0$ in $D$ and $u \le 0$ on $B$, then $u \le 0$ in $D$.} \end{document}