$$\Delta f=h, x\in\Omega$$ $$f=F, x\in\partial \Omega$$
where $h$ is $C^1$ function s.t. $0\le h$, $0\le h'$ and the domain is in 3D.
then, I want to show that $f$ has its max on boundary. please give me a proof or hint.
2026-03-26 17:52:27.1774547547
How to show that the Poisson equation has its maximum on the boundary?
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HINT:
Try searching the maximum principle. For your particular problem I would focus on the maximum principle for elliptic PDEs.
Basically, this principle states that the solution of elliptic (or parabolic) differential equation reaches its minimum and maximum on the boundary of the domain.
As a reference I would propose this book, although you can find plenty of information on maximum principle on the internet.
Hope my hint and reference are helpful.