7.3.2: Boundary Value Problems: Neumann Problem
( \newcommand{\kernel}{\mathrm{null}\,}\)
The Neumann problem (second boundary value problem) is to find a solution u∈C2(Ω)∩C1(¯Ω) of
△u=0 in Ω∂u∂n=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 u∈C2(¯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 u∈C1(¯Ω)∩C2(Ω) follows from the Hopf boundary point lemma, see Lecture Notes: Linear Elliptic Equations of Second Order, for instance.
Contributors and Attributions
Integrated by Justin Marshall.