= Solution
Impose the homogeneous <Dirichlet boundary condition> on the boundary variable:
$$
\boxed{G(\mathbf x,\mathbf x')=0\quad\text{for }\mathbf x\in S,\ \mathbf x'\in V.}
$$
This removes the term containing the unknown normal derivative of $\phi$. The prescribed boundary values alone then determine the solution through
$$
\phi(\mathbf x')=-\int_S f(\mathbf x)\,\partial_{n_{\mathbf x}}G(\mathbf x,\mathbf x')\,dS.
$$
For an unbounded domain one also imposes the appropriate decay condition at infinity. This defines the required <Dirichlet Green function>.
Back to article page
