TY - JOUR

T1 - Concavity of Condenser Energy Under Boundary Variations

AU - Pouliasis, Stamatis

N1 - Publisher Copyright:
© 2020, Mathematica Josephina, Inc.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Capacity constant

KW - Condenser energy

KW - Harmonic radius

KW - Parametric deformation

UR - http://www.scopus.com/inward/record.url?scp=85093922300&partnerID=8YFLogxK

U2 - 10.1007/s12220-020-00547-3

DO - 10.1007/s12220-020-00547-3

M3 - Article

AN - SCOPUS:85093922300

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

ER -