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Mathematics LibreTexts

7.3.2: Boundary Value Problems: Neumann Problem

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.

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