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

Abstract

Abstract We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions. Our main result is a sharp lower bound on the L 2 {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 Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity.