Let \(\Omega\subseteq \mathbb{R}^d\) be a domain with boundary , \(f\in C(\Omega)\) and \(g\in C(\partial \Omega)\). Then the boundary value problem
\[ \left\{ \begin{aligned} -\Delta u &= f &&\text{ in } \Omega\\ u&=g &&\text{ on } \partial \Omega, \end{aligned} \right. \]is called Dirichlet problem for Poisson’s equation.
Remarks
- A solution is unique.
- If \(\Omega\) is bounded and has \(C^1\) boundary the solution is given by \[u(x) = - \int_{\partial U} g(y) \frac{\partial G}{\partial \nu}(x, y) \, dS(y) + \int_U f(y) G(x, y) \, dy,\] where \(G\) denotes the Greens function of the Laplacian and \[\frac{\partial G}{\partial \nu}(x, y) = D_\nu G(x, y) \cdot \nu(y).\] [1, 2.2 Theorem 12]
See also Link to heading
References Link to heading
- L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.