Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

Marita Thomas

doi:10.3934/dcds.2015.35.2741

Article Contents

2015, Volume 35, Issue 6: 2741-2761. Doi: 10.3934/dcds.2015.35.2741

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Uniform Poincaré-Sobolev and isoperimetric inequalities for classes of domains

Marita Thomas^1,

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin

Received: December 2013

Revised: April 2014

Published: May 2017

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Abstract

The aim of this paper is to prove an isoperimetric inequality relative to a convex domain $\Omega\subset\mathbb{R}^d$ intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius $r>0$ and the position $y\in\overline{\Omega}$ of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension $d,$ the diameter of the domain and the integrability exponent $p\in[1,d)$.

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Mathematics Subject Classification: Primary: 46E35, 26D10; Secondary: 52A38.

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