An inequality for the normal derivative of the Lane-Emden ground state

TL;DR

This paper establishes a sharp lower bound on the normal derivative of Lane-Emden ground states, leading to an isoperimetric inequality applicable to arbitrary Lipschitz domains, extending previous results beyond convex sets.

Contribution

It provides the first sharp inequality for the normal derivative of Lane-Emden ground states on general Lipschitz domains, broadening the scope of isoperimetric inequalities in this context.

Findings

Derived a sharp lower bound on the normal derivative in terms of energy.

Extended isoperimetric inequalities to non-convex Lipschitz domains.

Applicable for polytropic index 0 ≤ q-1 ≤ 1.

Abstract

We consider Lane-Emden ground states with polytropic index $0\leq q-1\leq 1$, that is, minimizers of the Dirichlet integral among $L^q$-normalized functions. Our main result is a sharp lower bound on the $L^2$-norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets $\Omega\subset\mathbb{R}^d$, without assuming convexity.