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Mathematics LibreTexts

7.3.2: Boundary Value Problems: Neumann Problem

( \newcommand{\kernel}{\mathrm{null}\,}\)

The Neumann problem (second boundary value problem) is to find a solution uC2(Ω)C1(¯Ω) of
u=0  in Ωun=Φ  on Ω,


where Φ is given and continuous on Ω.

Proposition 7.5. Assume Ω is bounded, then a solution to the Dirichlet problem is in the class uC2(¯Ω) uniquely determined up to a constant.

Proof. Exercise. Hint: Multiply the differential equation w=0 by w and integrate the result over Ω.
Another proof under the weaker assumption uC1(¯Ω)C2(Ω) follows from the Hopf boundary point lemma, see Lecture Notes: Linear Elliptic Equations of Second Order, for instance.

Contributors and Attributions


This page titled 7.3.2: Boundary Value Problems: Neumann Problem is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

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