On the torsion function with Robin or Dirichlet boundary conditions

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 $\Om$.