# Sobolev inequalities for the symmetric gradient in arbitrary domains

## TL;DR

This paper introduces new Sobolev inequalities for the symmetric gradient of vector functions applicable to arbitrary domains, replacing boundary regularity with boundary trace information, and providing constants independent of domain geometry.

## Contribution

It develops Sobolev inequalities for the symmetric gradient that hold in arbitrary domains, with boundary trace conditions replacing boundary regularity, and features domain-independent constants.

## Key findings

- Inequalities match classical exponents in regular domains.
- Constants are independent of domain geometry.
- Applicable to arbitrary ground domains in R^n.

## Abstract

A form of Sobolev inequalities for the symmetric gradient of vector-valued functions is proposed, which allows for arbitrary ground domains in $\mathbb R ^n$. In the relevant inequalities, boundary regularity of domains is replaced with information on boundary traces of trial functions. The inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of Sobolev functions, in regular domains. Furthermore, they involve constants independent of the geometry of the domain, and hence yield novel results yet for smooth domains. Our approach relies upon a pointwise estimate for the functions in question via a Riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace.