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7.3.3: Boundary Value Problems: Mixed Boundary Value Problem

Last updated

Jan 2, 2021
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- 7.3.2: Boundary Value Problems: Neumann Problem
- 7.4: Green's Function for $\Delta$

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The Mixed boundary value problem (third boundary value problem) is to find a solution $u\in C^2(\Omega)\cap C^1(\overline{\Omega})$ of

$\begin{eqnarray} \label{M1}\tag{7.3.3.1} \triangle u&=&0\ \ \mbox{in}\ \Omega\\ \label{M2} \tag{7.3.3.2} \frac{\partial u}{\partial n}+hu&=&\Phi\ \ \mbox{on}\ \partial\Omega, \end{eqnarray}$
where

$\Phi$ and

$h$ are given and continuous on

$\partial\Omega$ .e

$\Phi$ and

$h$ are given and continuous on

$\partial\Omega$ .

Proposition 7.6. Assume $\Omega$ is bounded and sufficiently regular, then a solution to the mixed problem is uniquely determined in the class $u\in C^2(\overline{\Omega})$ provided $h(x)\ge 0$ on $\partial\Omega$ and $h(x)>0$ for at least one point $x\in\partial\Omega$ .

Proof. Exercise. Hint: Multiply the differential equation $\triangle w=0$ by $w$ and integrate the result over $\Omega$ .

Contributors and Attributions

Prof. Dr. Erich Miersemann (Universität Leipzig)
Integrated by Justin Marshall.

Contributors and Attributions

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