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Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint.

Consider the problem

$$\Delta u = f \quad\text{in $\Omega$}$$ $$u = g \quad\text{on $\Gamma_1$}$$ $$\partial_\nu u = h \quad\text{on $\Gamma_2$}$$ where $f, g, h$ are given data such that $f,g,h \in L^\infty\cap H^1$ on the respective domains and $\nu$ is the outward-pointing unit normal vector. The data can be taken to be more regular Sobolev functions if needed.

Does anyone know a reference or method of proof to obtain an $L^\infty(\Omega)$-estimate on the weak solution $u$, in terms of the data?

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Too long for comment. take for instance $f=0, g=0$. Then the mapping $h\mapsto u$ is a pseudo-differential operator which will have some Sobolev continuity properties for spaces $W^{s,p}$ with $p\in (1,+\infty)$, but no continuity properties on $W^{s,+\infty}$.

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  • $\begingroup$ If say $\Omega$ is annular region, can I not expect that $u$ is in $L^\infty$? $\endgroup$
    – BBB
    Commented Mar 29 at 10:26

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