In order for the Green's function of the laplacian to be well defined, it is usually asked for the domain to have $\mathcal C^1$ boundary (see e.g. Evans book on PDEs, section 2.2.4). My question is, at least in 2 dimensions, are we allowed to just have piecewise differentiability? In particular, is the Green's function well defined in a squared domain of $\mathbb R^2$? Physicists usually solve this kind of problems on rectangular domains, but I need to know whether it is mathematically well defined. Any references where I can see this?
2026-03-29 04:47:25.1774759645
Is the Green's function for the laplacian in 2D well defined in a square?
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