Concavity of Condenser Energy Under Boundary Variations

Research output: Contribution to journalArticlepeer-review

Abstract

Let D, D1 be two bounded domains in Rn, n≥ 2 , such that D¯ ⊂ D1 and ∂D and ∂D1 are closed surfaces. Consider a variation of D to D1 via a family of smooth domains Dt, t∈ (0 , 1) , whose boundaries ∂Dt are level sets of a C2 function V on D1\ D. Let K be an arbitrary compact subset of D and let I(Dt, K) be the equilibrium energy of the condenser (Dt, K). We show that the function f(t) : = I(Dt, K) is continuously differentiable. In addition, we show that, if V is subharmonic, then f is a concave function. We characterize the cases where f is affine by showing that this occurs if and only if ∂Dt are level sets of the equilibrium potential of the condenser (D1, K). This is a generalization of a result obtained by R. Laugesen [14] when the domains Dt are concentric balls.

Original languageEnglish
JournalJournal of Geometric Analysis
DOIs
StateAccepted/In press - 2020

Keywords

  • Capacity constant
  • Condenser energy
  • Harmonic radius
  • Parametric deformation

Fingerprint Dive into the research topics of 'Concavity of Condenser Energy Under Boundary Variations'. Together they form a unique fingerprint.

Cite this