# 7.2: Representation Formula

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In the following we assume that $$\Omega$$, the function $$\phi$$ which appears in the definition of the fundamental solution and the potential function $$u$$ considered are sufficiently regular such that the following calculations make sense, see [6] for generalizations. This is the case if $$\Omega$$ is bounded, $$\partial\Omega$$ is in $$C^1$$, $$\phi\in C^2(\overline{\Omega})$$ for each fixed $$y\in\Omega$$ and $$u\in C^2(\overline{\Omega})$$.

Figure 7.2.1: Notations to Green's identity

Theorem 7.1. Let $$u$$ be a potential function and $$\gamma$$ a fundamental solution, then for each fixed $$y\in\Omega$$
$$u(y)=\int_{\partial\Omega}\left(\gamma (x,y)\frac{\partial u(x)}{\partial n_x}-u(x)\frac{\partial \gamma(x,y)}{\partial n_x}\right)\ dS_x.$$

Proof. Let $$B_\rho(y)\subset\Omega$$ be a ball. Set $$\Omega_\rho(y)=\Omega\setminus B_\rho(y)$$. See Figure 7.2.2 for notations.

Figure 7.2.2: Notations to Theorem 7.1

From Green's formula, for $$u,\ v\in C^2(\overline{\Omega})$$,
$$\int_{\Omega_\rho(y)}\ (v\triangle u-u\triangle v)\ dx=\int_{\partial\Omega_\rho(y)}\ \left(v\frac{\partial u}{\partial n}-u\frac{\partial v}{\partial n}\right)\ dS$$
we obtain, if $$v$$ is a fundamental solution and $$u$$ a potential function,
$$\int_{\partial\Omega_\rho(y)}\ \left(v\frac{\partial u}{\partial n}-u\frac{\partial v}{\partial n}\right)\ dS=0.$$
Thus we have to consider
\begin{eqnarray*}
\int_{\partial\Omega_{\rho}(y)}\ v\frac{\partial u}{\partial n}\ dS&=&\int_{\partial\Omega}\ v\frac{\partial u}{\partial n}\ dS+\int_{\partial B_\rho(y)}\ v\frac{\partial u}{\partial n}\ dS\\
\int_{\partial\Omega_{\rho}(y)}\ u\frac{\partial v}{\partial n}\ dS&=&\int_{\partial\Omega}\ u\frac{\partial v}{\partial n}\ dS+\int_{\partial B_\rho(y)}\ u\frac{\partial v}{\partial n}\ dS.
\end{eqnarray*}
We estimate the integrals over $$\partial B_\rho(y)$$:

(i)
\begin{eqnarray*}
\left|\int_{\partial B_\rho(y)}\ v\frac{\partial u}{\partial n}\ dS\right|&\le&M\int_{\partial B_\rho(y)}\ |v|\ dS\\
&\le&M\left(\int_{\partial B_\rho(y)}\ s(\rho)\ dS+C\omega_n\rho^{n-1}\right),
\end{eqnarray*}
where
\begin{eqnarray*}
M&=&M(y)=\sup_{B_{\rho_0}(y)}|\partial u/\partial n|,\ \ \rho\le\rho_0,\\
C&=&C(y)=\sup_{x\in B_{\rho_0}(y)}|\phi(x,y)|.
\end{eqnarray*}
From the definition of $$s(\rho)$$ we get the estimate as $$\rho\to 0$$

\label{ell1}
O(\rho|\ln\rho|)&n=2\\
O(\rho)&n\ge3.
\end{array}\right.

(ii) Consider the case $$n\ge3$$, then
\begin{eqnarray*}
\int_{\partial B_\rho(y)}\ u\frac{\partial v}{\partial n}\ dS&=&
\frac{1}{\omega_n}\int_{\partial B_\rho(y)}\ u\frac{1}{\rho^{n-1}}\ dS+\int_{\partial B_\rho(y)}\ u\frac{\partial \phi}{\partial n}\ dS\\
&=&\frac{1}{\omega_n\rho^{n-1}}\int_{\partial B_\rho(y)}\ u\ dS+O(\rho^{n-1})\\
&=&\frac{1}{\omega_n\rho^{n-1}}u(x_0)\int_{\partial B_\rho(y)}\ dS+O(\rho^{n-1}),\\ &=&u(x_0)+O(\rho^{n-1}).
\end{eqnarray*}
for an $$x_0\in\partial B_\rho(y)$$.

Combining this estimate and (\ref{ell1}), we obtain the representation formula of the theorem.

$$\Box$$

Corollary. Set $$\phi\equiv 0$$ and $$r=|x-y|$$ in the representation formula of Theorem 7.1, then
\begin{eqnarray}
\label{ell2}
u(y)&=&\frac{1}{2\pi}\int_{\partial\Omega}\ \left(\ln r\frac{\partial u}{\partial n_x}-u\frac{\partial(\ln r)}{\partial n_x}\right)\ dS_x,\ \ n=2,\\
\label{ell3}
u(y)&=&\frac{1}{(n-2)\omega_n}\int_{\partial\Omega}\ \left(\frac{1}{r^{n-2}}\frac{\partial u}{\partial n_x}-u\frac{\partial(r^{2-n})}{\partial n_x}\right)\ dS_x,\ \ n\ge3.
\end{eqnarray}