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Mathematics LibreTexts

7.3.1: Boundary Value Problems: Dirichlet Problem

  • Page ID
    2186
  • [ "article:topic" ]

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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
    $$
    \Omega=\{x\in B_1(0):\ x_2>0\},
    $$

    Counterexample
    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
    $$
    \frac{\partial}{\partial x_1}\left(\frac{u_{x_1}}{\sqrt{1+|\nabla u|^2}}\right)+
    \frac{\partial}{\partial x_2}\left(\frac{u_{x_2}}{\sqrt{1+|\nabla u|^2}}\right)=0.
    $$

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