Green’s Function and Its Properties
In this section, we would like to discuss Dirichlet boundary-value problem. The method is to use Green’s identity and Green’s second formula to transform the problem to another specialized Dirichlet boundary-value problem. In the process, we naturally derive Green’s function.
Generally speaking, There are two class of Dirichlet boundary-value problem for elliptic partial equations, i.e., the Dirichlet problem of harmonic equation
\begin{equation}
\begin{cases}
\Delta u=0, &\text{in }\Omega\\
u=\phi,&\text{on }\pt\Omega,
\end{cases}
\end{equation}
and the Dirichlet problem of Poisson equation
\begin{equation}\label{eq:2}
\begin{cases}
\Delta u=f, &\text{in }\Omega\\
u=\phi,&\text{on }\pt\Omega,
\end{cases}
\end{equation}
where $f$ is a continuous function in $\bar\Omega$ and $\phi$ is a continuous function on $\pt\Omega$. Of course, the first problem can be viewed as a special case of the second one.
The elementary mathematics concern about the qualitative theory, i.e., the existence and uniqueness of the solution; whereas the applied mathematics concern more about the property of solution.Let $u\in C^1(\bar\Omega)\cap C^2(\Omega)$ be a solution of \eqref{eq:2}, by Green’s representationThe elliptic equation always have classic solutions (compared with weak solution), we can say that it is complete. Thus, the elliptic equation is quite important in the constructing of general theory of PDE.
\begin{equation}\label{eq:3}
u(x)=\int_\Omega\Gamma\Delta u\rd y
-\int_{\pt\Omega}\left(
\Gamma\frac{\pt u}{\pt\n}-u\frac{\pt\Gamma}{\pt\n}
\right)\rd S\end{equation}
thus, the only unknown term is $\frac{\pt u}{\pt\n}$ on $\pt\Omega$, we should note that we cannot prescribe $u$ and $\frac{\pt u}{\pt\n}$ on $\pt\Omega$ simultaneously. In fact, we will eliminate this term by adjusting $\Gamma$.
Our observation is that the Green’s representation is mostly similar to the Green’s second formula. Consider a harmonic function $\Phi\in C^1(\bar\Omega)\cap C^2(\Omega)$, by Green’s second formula, we have
\[
\int_\Omega \Phi\Delta u-u \Delta\Phi\rd y
=\int_{\pt\Omega}\left(\Phi\frac{\pt u}{\pt\n}-u \frac{\pt\Phi}{\pt\n}\right)\rd S,
\]
i.e.,
\begin{equation}\label{eq:4}
0=\int_\Omega\Phi\Delta u\rd y-\int_{\pt\Omega}\left(\Phi\frac{\pt u}{\pt\n}-u \frac{\pt\Phi}{\pt\n}\right)\rd S.
\end{equation}
Set $G=\Gamma+\Phi$, and if we can find $\Phi$ (adjusting!), such that $G\equiv0$ on $\pt\Omega$, then, by \eqref{eq:3} and \eqref{eq:4}
\begin{equation}\label{eq:5_1}
u(x)=\int_\Omega G\Delta u\rd y
+\int_{\pt\Omega}u\frac{\pt G}{\pt\n}\rd S.
\end{equation}
Thus, we obtain the solution for the Dirichlet problem \eqref{eq:2}
\begin{equation}\label{eq:5}
u(x)=\int_\Omega G(x,y) f(y)\rd y
+\int_{\pt\Omega}\phi(y)\frac{\pt G}{\pt\n_y}(x,y)\rd S.
\end{equation}
From the above discussion, we have transform the problem of solve \eqref{eq:2} to the following Dirichlet problem
\begin{equation}\label{eq:6}
\begin{cases}
\Delta_y\Phi(x,y)=0, &y\in\Omega\\
\Phi(x,y)=-\Gamma(x-y),&y\in\pt\Omega,
\end{cases}
\end{equation}
where $x\in\Omega$ is a fixed point, and $\Phi(x,\cdot)\in C^1(\bar\Omega)\cap C^2(\Omega)$.
- If we obtain the Green’s function for some domain $\Omega$, then the existence of Dirichlet problem \eqref{eq:2} have solved and its solution can be represented by \eqref{eq:6};
- Use the solution \eqref{eq:6} we can discussion the property of solution;
- For some special domain, such as ball, half-space, the first octant, we can find Green’s function by elementary method, and the Dirichlet problem of these domain are important in the theory of PDE.
Now we turn to the properties of Green’s function.
- $\Gamma(x-y) < G(x,y) < 0$, for any $y\in\Omega$ and $y\neq x$;
- $G(x,y)$ as a function of $y$ is a harmonic function on $\Omega\setminus\set{x}$, and $\lim\limits_{y\to x}G(x,y)=\infty$, furthermore, if $n\geq3$, then $G(x,y)=O(1/|x-y|^{n-2})$;
- $G(x,y)=G(y,x)$, for any $x,y\in\Omega$ and $x\neq y$;
- \[ \int_{\pt\Omega}\frac{\pt G}{\pt\n}\rd S=1. \]
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