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# 7.3.3: Boundary Value Problems: Mixed Boundary Value Problem

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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

• Integrated by Justin Marshall.