Isoperimetric profile and uniqueness for Neumann problems

Abstract

Given a connected compact Riemannian surface (M,g), f an absolutely continuous function satisfying f⩾f′>0 and a real parameter α, we deal with classical solutions of{−Δgu=f(u)−αin M,∂u∂n=0on ∂M. We prove that any non-constant solution of the above problem satisfies∫Mf(u)⩾8πinfs∈(0,vol(M)){IM2(s)ISM2(s)}, where IM and ISM denote respectively the isoperimetric profile of M and of the standard two-dimensional sphere having same measure than M (see Definition 2.1 below). This inequality is applied to derive new uniqueness results for mean field type equations. A similar result for linear problems is established and gives lower bounds for the first non-zero Neumann eigenvalue.