On the torsion function with Robin or Dirichlet boundary conditions

TL;DR

This paper derives bounds for the $p$-torsion function with Robin or Dirichlet boundary conditions, linking the $L^ fty$ norm to spectral properties, domain measure, and boundary parameters in various cases.

Contribution

It provides new bounds for the $L^ fty$ norm of the $p$-torsion function based on spectral data, domain measure, and boundary conditions, extending previous results to nonlinear and Robin cases.

Findings

Bounds depend on the bottom of the spectrum, boundary parameter, and domain measure.

Explicit bounds are obtained for the linear case ($p=2$) with Robin boundary conditions.

Bounds for the nonlinear case ($p eq 2$) with Dirichlet boundary conditions involve domain measure.

Abstract

For $p\in (1,+\infty)$ and $b \in (0, +\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set $\Om \subset \R^m$ satisfies formally the equation $-\Delta_p =1$ in $\Om$ and $|\nabla u|^{p-2} \frac{\partial u}{\partial n} + b|u|^{p-2} u =0$ on $\partial \Om$. We obtain bounds of the $L^\infty$ norm of $u$ {\it only} in terms of the bottom of the spectrum (of the Robin $p$-Laplacian), $b$ and the dimension of the space in the following two extremal cases: the linear framework (corresponding to $p=2$) and arbitrary $b>0$, and the non-linear framework (corresponding to arbitrary $p>1$) and Dirichlet boundary conditions ($b=+\infty$). In the general case, $p\not=2, p \in (1, +\infty)$ and $b>0$ our bounds involve also the Lebesgue measure of $\Om$.