I read a bit through uniqueness of the solution to Poisson's equation with boundary data and also existence/uniqueness of the variational formulation. But under what conditions can one guarantee the existence of a classical solution for Poissons equation $$-\Delta u =f$$ with given boundary data, say dirichlet, $\Omega \subset \mathbb R^n$?
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I am also happy with reference to literature where the existence of a classical solution is being discussed – Tesla Jun 28 '19 at 08:00
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Pages 365-368 of "Methods of Mathematical Physics" (Vol 1) by Courant and Hilbert might be helpful for the case n=2,3 – Inzinity Mar 29 '23 at 12:13