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On the L∞-maximization of the solution of Poisson's equation: Brezis–Gallouet–Wainger type inequalities and applications

Part of: Elliptic equations and systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  20 February 2020

Davit Harutyunyan  and
Hayk Mikayelyan
Davit Harutyunyan
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA93106-3080, USA (harutyunyan@ucsb.edu)
Hayk Mikayelyan
Affiliation:
School of Mathematical Sciences, University of Nottingham Ningbo, 199 Taikang East Road, Ningbo315100, PR China (Hayk.Mikayelyan@nottingham.edu.cn)
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Abstract

For the solution of the Poisson problem with an L∞ right hand side

\begin{cases} -\Delta u(x) = f (x) & {\rm in}\ D, \\ u=0 & {\rm on}\ \partial D \end{cases}
we derive an optimal estimate of the form
\|u\|_\infty\leq \|f\|_\infty \sigma_D(\|f\|_1/\|f\|_\infty),
 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 $\|f\|_1$ and $\|f\|_\infty .$ We also show that
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],
 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.

Keywords

Optimal rearrangementsBrezis–Gallouet–Wainger inequalityBathtub principleLaplace eigenfunctions

MSC classification

Primary: 35R11: Fractional partial differential equations
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 79 - 92
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Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

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