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# 7.3.2: Boundary Value Problems: Neumann Problem

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The Neumann problem (second boundary value problem) is to find a solution $$u\in C^2(\Omega)\cap C^1(\overline{\Omega})$$ of
\begin{eqnarray}
\label{N1}\tag{7.3.2.1}
\triangle u&=&0\ \ \mbox{in}\ \Omega\\
\label{N2} \tag{7.3.2.2}
\frac{\partial u}{\partial n}&=&\Phi\ \ \mbox{on}\ \partial\Omega,
\end{eqnarray}
where $$\Phi$$ is given and continuous on $$\partial\Omega$$.

Proposition 7.5. Assume $$\Omega$$ is bounded, then a solution to the Dirichlet problem is in the class $$u\in C^2(\overline{\Omega})$$ uniquely determined up to a constant.

Proof. Exercise. Hint: Multiply the differential equation $$\triangle w=0$$ by $$w$$ and integrate the result over $$\Omega$$.
Another proof under the weaker assumption $$u\in C^1(\overline{\Omega})\cap C^2(\Omega)$$ follows from the Hopf boundary point lemma, see Lecture Notes: Linear Elliptic Equations of Second Order, for instance.

### Contributors

• Integrated by Justin Marshall.