## How to solve the Dirichlet problem of Possion equation

The problem is to solve

$\displaystyle \begin{cases}\Delta u=f \text{ in }\Omega\\ u=g\text{ on }\partial \Omega\end{cases}$

There are different ways to solve adapting to the assumption of $f$ and $\Omega$. We only consider $\Omega$ is bounded.

(1) If $f$ is bounded and locally Holder continuous, $f\in C^{\alpha}(\Omega)$, then the Newtonian potential

$\displaystyle w=\int_{\Omega}\Gamma(x-y)f(y)dy$

is $w\in C^{2,\alpha}(\Omega)$ and $\Delta w=f$. So we can let $v=u-w$, solve

$\displaystyle \begin{cases}\Delta v=0\text{ in }\Omega\\ v=g-w\text{ on }\partial \Omega\end{cases}$

$v$ is a harmonic function. If $\partial \Omega$ is regular everywhere, we can solve this by the perron method, provided $g$ is continuous on $\partial \Omega$.

(2) If we only know $f\in L^2(\Omega)$ or $f\in H^{-1}(\Omega)$. We can use the Lax-Milgram to obtain the weak solution. First let us solve the problem of $g=0$

$\displaystyle \displaystyle \begin{cases}\Delta u=f \text{ in }\Omega\\ u=0\text{ on }\partial \Omega\end{cases}$

Then define $\displaystyle B[u,v]=\int_{\Omega}\nabla u\nabla vdx$ on $H_0^1(\Omega)$, the weak solution of the above equation is

$\displaystyle B[u,v]=-(f,v)_{L^2}$, $\forall v\in H_0^1(\Omega)$

Then $B$ is bounded and coercity on $H_0^1(\Omega)$, by the Lax-Milgram theorem, $\forall f\in L^2$, we can find $u$ such that solves the Possion equation in weak sense.

For the general case, $u=g$ on $\partial \Omega$. One way is to reduce this case to $u=0$ on $\partial \Omega$ by extending $g$ to a $H^1_0$ function inside $\Omega$, say $\psi$. Solve

$\displaystyle B[u,v]=-(f,v)_{L^2}+(\psi,v)_{H^1_0}$, $\forall v\in H^1_0(\Omega)$

Then $u+\psi$ is the solution we want to find.

(3) If $f\in L^p(\Omega)$, $p\neq 2$. Assume $\partial\Omega$ is smooth and $g$ can be extended to be a $W^{2,p}(\Omega)$ function. Then we only need to consider the homogeneous boundary condition. Since the estimate

$\displaystyle ||u||_{W^{2,p}(\Omega)}\leq C||f||_p$

approximate $f$ by smooth functions $f_n$, correspondingly you have solution $u_n$

$\displaystyle \begin{cases}\Delta u_n=f_n \text{ in }\Omega\\ u_n=0\text{ on }\partial \Omega\end{cases}$

Then $u_n$ will be a cauchy sequence in $W^{2,p}(\Omega)$. Suppose $u_n\to u$, then $\Delta u=f$ and $u_n=0$ on $\partial\Omega$.