The $L_{p}$-Brunn-Minkowski inequalities for variational functionals with $0\leq p<1$

TL;DR

This paper explores the infinitesimal forms of $L_{p}$-Brunn-Minkowski inequalities for various variational functionals, establishing related Poincaré inequalities and confirming inequalities for torsional rigidity near the unit ball.

Contribution

It introduces new infinitesimal formulations of $L_{p}$-Brunn-Minkowski inequalities for $p eq 1$, and verifies these inequalities for small perturbations of the unit ball.

Findings

Infinitesimal forms lead to Poincaré-type inequalities.

Confirmed $L_{p}$-Brunn-Minkowski inequalities for torsional rigidity near the unit ball.

Extended inequalities to functionals like $q$-capacity and eigenvalues.

Abstract

The infinitesimal forms of the $L_{p}$-Brunn-Minkowski inequalities for variational functionals, such as the $q$-capacity, the torsional rigidity, and the first eigenvalue of the Laplace operator, are investigated for $p \geq 0$. These formulations yield Poincar\'{e}-type inequalities related to these functionals. As an application, the $L_{p}$-Brunn-Minkowski inequalities for torsional rigidity with $0 \leq p < 1$ are confirmed for small $C^{2}$-perturbations of the unit ball.