Equilibrium measures and equilibrium potentials in the Born-Infeld model

Abstract

In this paper, we consider the electrostatic Born-Infeld model(BI){−div(∇ϕ1−|∇ϕ|2)=ρin RN,lim|x|→∞⁡ϕ(x)=0 where ρ is a charge distribution on the boundary of a bounded domain Ω⊂RN, with N⩾3. We are interested in its equilibrium measures, i.e. charge distributions which minimize the electrostatic energy of the corresponding potential among all possible distributions with fixed total charge. We prove existence of equilibrium measures and we show that the corresponding equilibrium potential is unique and constant in Ω‾. Furthermore, for smooth domains, we obtain the uniqueness of the equilibrium measure, we give its precise expression, and we verify that the equilibrium potential solves (BI). Finally we characterize balls in RN as the unique sets among all bounded C2,α-domains Ω for which the equilibrium distribution is a constant multiple of the surface measure on ∂Ω. The same results are obtained also for Taylor approximations of the electrostatic energy.