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# 7.3.1: Boundary Value Problems: Dirichlet Problem

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The Dirichlet problem (first boundary value problem) is to find a solution $$u\in C^2(\Omega)\cap C(\overline{\Omega})$$ of
\begin{eqnarray}
\label{D1}\tag{7.3.1.1}
\triangle u&=&0\ \ \mbox{in}\ \Omega\\
\label{D2}\tag{7.3.1.2}
u&=&\Phi\ \ \mbox{on}\ \partial\Omega,
\end{eqnarray}
where $$\Phi$$ is given and continuous on $$\partial\Omega$$.

Proposition 7.4. Assume $$\Omega$$ is bounded, then a solution to the Dirichlet problem is uniquely determined.

Proof. Maximum principle.

Remark. The previous result fails if we take away in the boundary condition (\ref{D2}) one point from the the boundary as the following example shows. Let $$\Omega\subset\mathbb{R}^2$$ be the domain
$$\Omega=\{x\in B_1(0):\ x_2>0\},$$

Figure 7.3.1.1: Counterexample

Assume $$u\in C^2(\Omega)\cap C(\overline{\Omega}\setminus\{0\})$$ is a solution of
\begin{eqnarray*}
\triangle u&=&0\ \ \mbox{in}\ \Omega\\
u&=&0\ \ \mbox{on}\ \partial\Omega\setminus\{0\}.
\end{eqnarray*}
This problem has  solutions $$u\equiv 0$$ and $$u=\mbox{Im}(z+z^{-1})$$, where $$z=x_1+ix_2$$. Concerning another example see an exercise.

In contrast to this behavior of the Laplace equation, one has uniqueness if $\triangle u=0$ is replaced by the minimal surface equation
$$\frac{\partial}{\partial x_1}\left(\frac{u_{x_1}}{\sqrt{1+|\nabla u|^2}}\right)+ \frac{\partial}{\partial x_2}\left(\frac{u_{x_2}}{\sqrt{1+|\nabla u|^2}}\right)=0.$$

### Contributors

• Integrated by Justin Marshall.