Talenti comparison results for solutions to $p$-Laplace equation on multiply connected domains

TL;DR

This work extends Talenti-type comparison principles to $p$-Laplace problems on multiply connected domains with Robin exterior boundary and non-homogeneous interior Dirichlet data. It develops a Lorentz-space framework for sharp pointwise and norm inequalities by adapting Schwarz rearrangement to holes via constant extensions, using level-set, coarea, and isoperimetric tools plus Gronwall-type arguments. The authors obtain $L^{k,1}$ and $L^{pk,p}$ comparisons between the original and symmetrized problems, with additional pointwise and $L^1,L^p$ results in the linear and low-exponent regimes; in particular, for $f\equiv1$ they obtain explicit conditions on $p$ for pointwise comparison and demonstrate an annulus as extremal for $p$-torsion and Robin eigenvalues. These findings yield Bossel–Daners and Saint-Venant-type inequalities for multiply connected domains and contribute to shape-optimization results involving holes.

Abstract

In the last years comparison results of Talenti type for Elliptic Problems have been widely investigated. In this paper we obtain a comparison result for the $p$-Laplace operator in multiply connected domains with Robin boundary condition on the exterior boundary and non-homogeneous Dirichlet boundary conditions on the interior one, generalizing the results obtained in \cite{ANT, AGM} to this type of domains. This will be a generalization to Robin boundary condition of the results obtained in \cite{B, B2}, with an improvement of the $L^2$ comparison in the case $p=2$. As a consequence, we obtain a Bossel-Daners and Saint-Venant type inequalities for multiply connected domains.

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.1: Coarea formula
• Lemma 2.2
• Proposition 3.1
• proof