Consider the equation $-\Delta u = f$ in a bounded domain $\Omega\subset\mathbb R^3$ with BC $u|_\Gamma=0$.
Let $f \in L^\infty$. Is it true that the weak solution $u \in L^\infty$?
2025-01-13 00:01:05.1736726465
Boundedness of a solution of Poisson equation
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Yes. The proof is based on a version of the weak maximum principle that is valid for weak solutions. See, for example, Theorem 8.1 in Gilbarg and Trudinger.