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.

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