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On the L∞-maximization of the solution of Poisson's equation: Brezis–Gallouet–Wainger type inequalities and applications
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 1 / February 2021
- Published online by Cambridge University Press:
- 20 February 2020, pp. 79-92
- Print publication:
- February 2021
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- Article
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For the solution of the Poisson problem with an L∞ right hand side
we derive an optimal estimate of the form
\begin{cases} -\Delta u(x) = f (x) & {\rm in}\ D, \\ u=0 & {\rm on}\ \partial D \end{cases}where σD is a modulus of continuity defined in the interval [0, |D|] and depends only on the domain D. The inequality is optimal for any domain D and for any values of
\|u\|_\infty\leq \|f\|_\infty \sigma_D(\|f\|_1/\|f\|_\infty),
$\|f\|_1$ and 
$\|f\|_\infty .$ We also show that
where B is a ball and |B| = |D|. Using this optimality property of σD, we derive Brezis–Galloute–Wainger type inequalities on the L∞ norm of u in terms of the L1 and L∞ norms of f. As an application we derive L∞ − L1 estimates on the k-th Laplace eigenfunction of the domain D.
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],
