Sobolev inequalities in arbitrary domains

TL;DR

This paper develops a general theory of Sobolev inequalities applicable to any open set in Euclidean space, removing the need for boundary regularity and providing geometry-independent constants with classical critical exponents.

Contribution

It introduces a novel approach using representation formulas and pointwise estimates to establish Sobolev inequalities in arbitrary domains without boundary regularity.

Findings

Constants are independent of domain geometry.

Inequalities retain classical critical exponents.

Applicable to all open sets in Euclidean space.

Abstract

A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set.