Isoperimetric inequalities for p-conductance

Abstract

An n-dimensional domain K is considered with boundary ∂K = k₀ tu K₁ ∪ K₂ such that the closure ¯K is the image of a cylinder B=Sx[0, 1] (S is a closed (n−1)-dimensional cell) under a one-one Lipschitz map. For the p-conductance of the domain K, defined by the equation

$$c_p (K) = \mathop {\inf }\limits_{U(K)} \int K |\nabla f|^p dx (p > 1),$$

, where∪ (K) = f (x):f ∈ W ¹_p (K)∩C (¯K),f = 1 on k₁,f = 0 on K₀, the isoperimetric inequality c_p(K) ≤ V/r^P is established. Here V is the n-dimensional volume of the domain K, r is the shortest distance between k₀ and K₁, measured in K. Equality is achieved on the right cylinder.