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Mathematics LibreTexts

7.3.3: Boundary Value Problems: Mixed Boundary Value Problem

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 \(\Omega\).

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