Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
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
\triangle u&=&0\ \ \mbox{in}\ \Omega\\
\label{N2} \tag{}
\frac{\partial u}{\partial n}&=&\Phi\ \ \mbox{on}\ \partial\Omega,
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.