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{\bf ASSIGNED EXERCISES OF MATH544: PDE set  7} 
{\it April, 2001}
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\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 ${ \partial u \over \partial n}=<N, grad \,u>$. 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 <x,y < \infty .
\end{eqnarray}

\noindent
where $f(x,y)$ is a continuous function in $R^{2}$.


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\noindent
{\bf 4}.\, Solve the above problem by use of separation of variables

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\noindent
{\bf 5}.\, Prove the following theorem:

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\noindent
{\bf Theorem}.\, Let $D$ be a bounded normal domain, and suppose that
the Green's function $G(x,y)$ for the Dirichlet problem exists. Then
$$G(x,y)=G(y,x)$$
for $x,y \in D$. Consequently , in addition to $\nabla_{y}^{2} \, G(x,y)=0$
for $y \ne x$, we also have
$\nabla_{x}^{2}\, G(x,y)=0$ for $ x \ne y$.

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\noindent
{\bf Solution}.\, {\it First way}. Let $u=G(z,x)$ and $v=G(z,y)$ in
 (\ref{grn}). Using (\ref{g1}) and (\ref{g2}) and using the property 
$$\int_{D} \, f(y)\, \delta (y-x) \, d^{3} y=f(x)$$
of the delta-distribution one obtains $G(x,y)=G(y,x)$. {\it Second way}.
Let $K_{a}(c)$ represent a ball with radius $a$ centered at $c$ and $S_{a}(c)$
is the sphere bounding this ball. 
$$K_{a}(a)=\{ x \in R^3 | |x-c| < a \},~~~ S_{a}(c)=\{x \in R^3 | |x-c|=a \}$$
Fix $x, x^{\prime} \in D$. Fix $\varepsilon > 0$ 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 $v(y)=G(x^{\prime},y)$


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\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.

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\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}

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\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.

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\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.

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\noindent
{\bf Theorem 1}.\, {\it Let $ u \in C^{2}(D) \cap C^{1}(\bar{D})$, 
where $D$ is abounded domain.

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\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 $D$.
}

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\noindent
{\bf Theorem 2}.\,{\it  Let $ u \in C^{2}(D) \cap C^{1}(\bar{D})$, where
$D$ is a bounded domain.

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\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 $x \in D$.
}

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\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} 


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\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 $|u(x)| \le F+CH$
}

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\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 $D$.

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\noindent
{\it Remark 2}.\, First three theorems were proved in the class.

\newpage

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\noindent
{\bf 12}.\, Prove the following theorem.

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\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 $D$.}




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