Given the Laplace equation with Dirichlet boundary data
\[ \left\{ \begin{aligned} -\Delta u &= 0 &&\text{ in } \mathbb{R}^d_+\\ u&=g &&\text{ on } \partial \mathbb{R}^d_+, \end{aligned} \right. \]with \(g \in C(\mathbb{R}^{d-1}) \cap L^\infty(\mathbb{R}^{d-1})\).
The Green’s function is given by
\[G(x, y) = \phi(y - x) - \phi(y - \tilde{x}),\]where \(\tilde{x} = (x_1,\ldots , x_{d-1}, -x_d)\) is the reflection point of \(x\) along the \(x_n\) axis and \(\phi\) denotes the fundamental solution .
Then
\[u(x) = \int_{\partial \mathbb{R}^d_+} K(x, y) g(y) \, dS(y)\]with
\[K(x, y) = \frac{2x_d}{d\omega_d} \frac{1}{|x - y|^d}\]satisfies
- \(u \in C^\infty \cap L^\infty\),
- \(\Delta u = 0\),
- \(\lim_{x \to x_0} u(x) = g(x_0), \quad x_0 \in \partial\mathbb{R}^d_+\).
See als Link to heading
References Link to heading
- L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.