Let \(\Omega\subseteq \mathbb{R}^d\) be a domain with boundary and \(f\in C(\partial \Omega)\). Then the boundary value problem
\[ \left\{ \begin{aligned} -\Delta u &= 0 &&\text{ in } \Omega\\ u&=g &&\text{ on } \partial \Omega, \end{aligned} \right. \]is called Dirichlet problem for Laplace’s equation.
Remarks
Regarding the solution Link to heading
- A solution is unique.
- If \(\Omega\) is bounded and has \(C^1\) boundary the solution is given by \[u(x) = - \int_{\partial U} g(y) \frac{\partial G}{\partial \nu}(x, y) \, dS(y)\] where \(G\) denotes the Greens function of the Laplacian (cf. Poisson’s equation case ). The kernel \(\frac{\partial G}{\partial \nu}(x,y)\) is often called Poisson’s kernel.
- This problem is solvable if and only if all boundary points are regular.