
# 7.4: Green's Function for $$\Delta$$

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Theorem 7.1 says that each harmonic function satisfies

\label{green1}
u(x)=\int_{\partial\Omega}\left(\gamma (y,x)\frac{\partial u(y)}{\partial n_y}-u(y)\frac{\partial \gamma(y,x)}{\partial n_y}\right)\ dS_y,

where $$\gamma(y,x)$$ is a fundamental solution. In general, $$u$$ does not satisfies the boundary condition in the above boundary value problems. Since $$\gamma=s+\phi$$, see Section 7.2, where $$\phi$$ is an arbitrary harmonic function for each fixed $$x$$, we try to find a $$\phi$$ such that $$u$$ satisfies also the boundary condition.

Consider the Dirichlet problem, then we look for a $$\phi$$ such that

\label{green2}
\gamma(y,x)=0,\ \ y\in\partial\Omega,\ x\in\Omega.

Then
$$u(x)=-\int_{\partial\Omega}\ \frac{\partial \gamma(y,x)}{\partial n_y}u(y)\ dS_y,\ \ x\in\Omega.$$
Suppose that $$u$$ achieves its boundary values $$\Phi$$ of the Dirichlet problem, then

\label{green3}
u(x)=-\int_{\partial\Omega}\ \frac{\partial \gamma(y,x)}{\partial n_y}\Phi(y)\ dS_y,

We claim that this function solves the Dirichlet problem (7.3.1.1), (7.3.1.2).

A function $$\gamma(y,x)$$ which satisfies (\ref{green2}), and some additional assumptions, is called Green's function. More precisely, we define a Green function as follows.

Definition. A function $$G(y,x)$$, $$y,\ x\in\overline{\Omega}$$, $$x\not= y$$, is called Green function associated to $$\Omega$$  and to the Dirichlet problem (7.3.1.1), (7.3.1.2) if for fixed $$x\in\Omega$$, that is we consider $$G(y,x)$$ as a function of $$y$$, the following properties hold:

(i) $$G(y,x)\in C^2(\Omega\setminus\{x\})\cap C(\overline{\Omega}\setminus\{x\})$$,     $$\triangle_yG(y,x)=0,\ \ x\not=y$$.

(ii) $$G(y,x)-s(|x-y|)\in C^2(\Omega)\cap C(\overline{\Omega})$$.

(iii) $$G(y,x)=0$$ if $$y\in\partial\Omega$$, $$x\not=y$$.

Remark. We will see in the next section that a Green function exists at least for some domains of simple geometry. Concerning the existence of a Green function for more general domains see [13].

It is an interesting fact that we get from (i)-(iii) of the above definition  two further important properties, provided  $$\Omega$$ is bounded, sufficiently regular and connected.

Proposition 7.7. A Green function has the following properties. In the case $$n=2$$ we assume {\rm diam} $$\Omega<1$$.

(A) $$G(x,y)=G(y,x)$$\ \ (symmetry).

(B) $$0<G(x,y)<s(|x-y|), \ \ x,\ y\in\Omega,\ x\not=y$$.

Proof. (A) Let $$x^{(1)},\ x^{(2)}\in\Omega$$. Set $$B_i=B_\rho(x^{(i)})$$, $$i=1,\ 2$$. We assume $$\overline{B_i}\subset\Omega$$ and $$B_1\cap B_2=\emptyset$$. Since $$G(y,x^{(1)})$$ and $$G(y,x^{(2)})$$ are harmonic in $$\Omega\setminus\left(\overline{B_1}\cup\overline{B_2}\right)$$ we obtain from Green's identity, see Figure 7.4.1 for notations,

Figure 7.4.1: Proof of Proposition 7.7

\begin{eqnarray*}
0&=&\int_{\partial\left(\Omega\setminus(\overline{B_1}\cup\overline{B_2})\right)}
\bigg(G(y,x^{(1)})\frac{\partial}{\partial n_y}G(y,x^{(2)})\\
&=&\int_{\partial\Omega}
\left(G(y,x^{(1)})\frac{\partial}{\partial n_y}G(y,x^{(2)})-G(y,x^{(2)})\frac{\partial}{\partial n_y}G(y,x^{(1)})\right) dS_y\\
&+&\int_{\partial B_1}
\left(G(y,x^{(1)})\frac{\partial}{\partial n_y}G(y,x^{(2)})-G(y,x^{(2)})\frac{\partial}{\partial n_y}G(y,x^{(1)})\right) dS_y\\
&+&\int_{\partial B_2}
\left(G(y,x^{(1)})\frac{\partial}{\partial n_y}G(y,x^{(2)})-G(y,x^{(2)})\frac{\partial}{\partial n_y}G(y,x^{(1)})\right) dS_y.
\end{eqnarray*}
The integral over $$\partial\Omega$$ is zero because of property (iii) of a Green function, and
\begin{eqnarray*}
\int_{\partial B_1}\
\bigg(G(y,x^{(1)})\frac{\partial}{\partial n_y}G(y,x^{(2)})&-&G(y,x^{(2)})\frac{\partial}{\partial n_y}G(y,x^{(1)})\bigg) dS_y\\
&\to& G(x^{(1)},x^{(2)}),\\
\int_{\partial B_2}\
\bigg(G(y,x^{(1)})\frac{\partial}{\partial n_y}G(y,x^{(2)})&-&G(y,x^{(2)})\frac{\partial}{\partial n_y}G(y,x^{(1)})\bigg)\ dS_y\\
&\to&
-G(x^{(2)},x^{(1)})
\end{eqnarray*}
as $$\rho\to 0$$.
This follows by considerations as in the proof of Theorem 7.1.

(B) Since
$$G(y,x)=s(|x-y|)+\phi(y,x)$$
and $$G(y,x)=0$$ if $$y\in\partial\Omega$$ and $$x\in\Omega$$ we have for $$y\in\partial\Omega$$
$$\phi(y,x)=-s(|x-y|).$$
From the definition of $$s(|x-y|)$$ it follows that $$\phi(y,x)< 0$$ if $$y\in\partial\Omega$$. Thus, since $$\triangle_y\phi=0$$ in $$\Omega$$, the maximum-minimum principle implies that $$\phi(y,x)<0$$
for all $$y,~x\in\Omega$$. Consequently
$$G(y,x)<s(|x-y|),\ \ x,\ y\in\Omega,\ x\not=y.$$
It remains to show that
$$G(y,x)>0,\ \ x,\ y\in\Omega,\ x\not=y.$$
Fix $$x\in\Omega$$ and let $$B_\rho(x)$$ be a ball such that $$B_\rho(x)\subset\Omega$$ for all $$0<\rho<\rho_0$$. There is a sufficiently small $$\rho_0>0$$ such that for each $$\rho$$, $$0<\rho<\rho_0$$,
$$G(y,x)>0\ \ \mbox{for all}\ y\in\overline{B_\rho(x)},\ x\not=y,$$
see property (iii) of a Green function. Since
\begin{eqnarray*}
\triangle_y G(y,x)&=&0\ \ \mbox{in}\ \Omega\setminus\overline{B_\rho(x)}\\
G(y,x)&>&0\ \ \mbox{if}\ y\in\partial B_\rho(x)\\
G(y,x)&=&0\ \ \mbox{if}\ y\in\partial\Omega
\end{eqnarray*}
it follows from the maximum-minimum principle that
$$G(y,x)>0\ \ \mbox{on}\ \Omega\setminus\overline{B_\rho(x)}.$$

$$\Box$$