Equality cases for condenser capacity inequalities under symmetrization

Dimitrios Betsakos,Stamatis Pouliasis

doi:10.2478/v10062-012-0009-x

Abstract

It is well known that certain transformations decrease the capacity of a condenser. We prove equality statements for the condenser capacity inequalities under symmetrization and polarization without connectivity restrictions on the condenser and without regularity assumptions on the boundary of the condenser.

Highlights

A condenser in Rn, n ≥ 2, is a pair (D, K), where D is an open subset of Rn and K is a nonempty compact subset of D
Let ACL2(G) be the set of continuous functions φ : G → R on the open set G ⊂ Rn, which are absolutely continuous on lines and their partial derivatives are in L2loc(G)
Equality statements for Schwarz symmetrization have been proved under regularity conditions on the boundary of the condenser; see [2, p. 57], [12, p. 17], [17, pp. 71–72]

Summary

Introduction

A condenser in Rn, n ≥ 2, is a pair (D, K), where D is an open subset of Rn and K is a nonempty compact subset of D. Shlyk [22] proved an equality statement for circular symmetrization of connected condensers, without the admissibility condition. Equality statements for Schwarz symmetrization have been proved under regularity (or smoothness) conditions on the boundary of the condenser; see [2, p. Fusco [8] proved equality statements in Steiner symmetrization inequalities for Dirichlet-type integrals under connectedness, boundedness and boundary conditions. Our purpose is to prove equality statements for condensers in Rn under Steiner and Schwarz symmetrizations and polarization, without any connectivity or regularity assumptions. Let G be an open subset of Rn. We shall denote by I(G) the set of irregular boundary points of G for the Dirichlet problem

Consider the decomposition

Suppose that holds for every oriented horizontal plane

Ωi i

Therefore and