Given the Laplace equation with Dirichlet boundary data
\[ \left\{ \begin{aligned} -\Delta u &= 0 &&\text{ in } B_r(0)\\ u&=g &&\text{ on } \partial B_r(0), \end{aligned} \right. \]with \(g \in C(\partial B_r(0))\).
The Green’s function for the unit ball is given by
\[G(x, y) = \phi(y - x) - \phi(|x|y - \tilde{x})\]where \(\tilde{x} = \frac{x}{|x|^2}\) is the dual point of \(x\) and \(\phi\) denotes the fundamental solution .
With a scaling argument we obtain
\[u(x) = \int_{\partial \mathbb{R}^d_+} K(x, y) g(y) \, dS(y)\]with
\[K(x, y) := \frac{r^2 - |x|^2}{d\omega_d r} \cdot \frac{1}{|x - y|^d},\]where \(x \in B_r(0)\) and \(y \in \partial B_r(0)\). The solution satisfies
- \(u \in C^\infty\),
- \(\Delta u = 0\),
- \(\lim_{x \to x_0} u(x) = g(x_0), \quad x_0 \in \partial\mathbb{R}^d_+\).
See also Link to heading
References Link to heading
- L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.