Caveat: I don't really know anything about PDEs, so this question might not make sense.

In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace equation on bounded domains with nice (smooth) boundary. My sketchy understanding of the history of this problem (gleaned from Wikipedia) is that in the 19th century everybody "knew" that the problem had to have a unique solution, because of physics. Specifically, if I give you a distribution of charge along the boundary, it has to determine an electric potential in the domain, which turns out to be harmonic. But Dirichlet's proof was wrong, and it wasn't until around 1900 that Hilbert found a correct argument for the existence and uniqueness of the solution, given reasonable conditions (the boundary function must be continuous, and the boundary really has to be sufficiently smooth).

Is the physical heuristic really totally meaningless from a mathematical point of view? Or is there some way to translate it into an actual proof?

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