UNIFORM POINCARE-SOBOLEV AND ISOPERIMETRIC INEQUALITIES FOR CLASSES OF DOMAINS

The aim of this paper is to prove an isoperimetric inequality relative to a convex domain Omega subset of R-d intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius r > 0 and the position y is an element of (Omega) over bar of the center of the ball. For this, uniform Sobolev, Poincare and Poincare-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 is an element of [1; d).