18.3.1.2 Solution ppc-b

Question:

Assuming that the Dirichlet boundary-value problem for the Laplace equation on a finite domain,

$$\nabla^2 u = 0 \quad\mbox{on}\quad\Omega\qquad u = f \quad\mbox{on}\quad\delta\Omega$$

is solvable, show that it depends continuously on the data.

Answer:

Assume that data $f_1$ produce a solution $u_1$ and $f_2$ a solution $u_2$. Define $v$ as the difference between the two solutions, and $g$ as the difference between the two data.

Show that $v$ satisfies the Laplace equation. Show that on the boundary $v$ equals the change in the data $g$. Then, using the maximum and minimum properties of the Laplace equation, argue that the (maximum) change in the solution is no larger than the (maximum) change in the data.