Concavity of Condenser Energy Under Boundary Variations
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Concavity of Condenser Energy Under Boundary Variations Stamatis Pouliasis1 Received: 26 February 2019 / Accepted: 15 October 2020 © Mathematica Josephina, Inc. 2020
Abstract Let D0 , D1 be two bounded domains in Rn , n ≥ 2, such that D0 ⊂ D1 and ∂ D0 and ∂ D1 are closed surfaces. Consider a variation of D0 to D1 via a family of smooth domains Dt , t ∈ (0, 1), whose boundaries ∂ Dt are level sets of a C 2 function V on D1 \D0 . Let K be an arbitrary compact subset of D0 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. Keywords Capacity constant · Condenser energy · Harmonic radius · Parametric deformation
1 Introduction Let D be a bounded domain in Rn , n ≥ 2. An old and classical object of study in Analysis and Geometry is the behavior of functionals (e.g. capacity, energy, eigenvalues, torsional rigidity) or characteristic functions (e.g. Green function, harmonic measure, eigenfunctions, equilibrium potential) related to linear elliptic partial differential equations and depending on the domain D, when its boundary ∂ D varies under different types of perturbations. Different types of perturbations can be obtained by vector fields or by the level sets of functions defined in a neighborhood of ∂ D. Applications of this method can be found in conformal mapping, potential theory, spectral geometry and fluid mechanics; see e.g. [3–6,17] and references therein. A systematic treatment of this subject is due to Garabedian and Schiffer in their famous article [8], where they developed a rigorous theory of domain variations, unify-
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Stamatis Pouliasis [email protected] Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA
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S. Pouliasis
ing and generalizing earlier methods, and proved asymptotic formulas for the first and second variation of several functionals and physical quantities including Hadamard’s variational formula for the Green function. Applying the formula for the second variation, they obtained convexity theorems for several domain functionals. In particular, they showed that the equilibrium energy of D is a concave function under a variation obtained by the level sets of a subharmonic function [8, pp. 333–334]. Schiffer [17, pp. 314–318] also indicated the possibility of applying the second variation formula to obtain convexity results in potential theory and conformal mapping. He showed that the electrostatic capacity of D is a convex function under a variation obtained by the level sets of a harmonic function and mentioned that the dependence is actually affine when the variation is obtained by the level s
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