Regularity for the Dirichlet problem on BD

TL;DR

This paper proves that the Dirichlet problem for convex linear growth functionals on functions of bounded deformation (BD) shares the same regularity properties as the well-studied BV case, despite the differences in the underlying spaces.

Contribution

It extends Sobolev and partial regularity results from BV to BD, overcoming the challenge posed by Ornstein's Non-Inequality and applying to all generalized minima.

Findings

Establishes Sobolev regularity for BD-based Dirichlet problems.

Proves partial $C^{1,eta}$-regularity for solutions.

Extends previous BV results to the BD setting in an optimal way.

Abstract

We establish that the Dirichlet problem for convex linear growth functionals on $BD$, the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial $C^{1,\alpha}$-regularity theory as presently available for the full gradient Dirichlet problem on $BV$. By Ornstein's Non-Inequality, $BV$ is a proper subspace of $BD$, and full gradient techniques known from the $BV$-situation do not apply here. In particular, applying to all generalised minima (i.e., minima of a suitably relaxed problem) despite their non-uniqueness and reaching the ellipticity ranges known from the $BV$-case, this paper extends previous results by Kristensen and the author (Gmeineder, F.; Kristensen, J.: Sobolev regularity for convex functionals on BD. J. Calc. Var. (2019) 58:56) in an optimal way.