Chapter 10.7, Problem 20E

Stability.Use the maximum principle to prove the following theorem on the continuous dependence of the solution on the boundary conditions:

Theorem. Let $f_1$ and $f_2$ be continuous functions on $\partial D$, where $D$ is a bounded domain. For $i=1$ and $$, let $u_i$ be the solution to the Dirichlet problem

$\begin{array}{rll}\Delta u& =& 0\text{ in }D,\\ u& =& f_i\text{ on }\partial \text{D}\end{array}$.

If the boundary values satisfy

$\left|f_1\left(x,y\right)-f_2\left(x,y\right)\right|\le \epsilon \text{ for all }\left(x,y\right)\text{ on }\partial D$,

Where $\epsilon >0$ in some constant, then

$\left|u_1\left(x,y\right)-u_2\left(x,y\right)\right|\le \epsilon \text{ for all }\left(x,y\right)\text{ on }D$.