Non-uniformly elliptic variational problems on BV

TL;DR

This paper proves regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on BV spaces, even with non-uniform ellipticity and only linear growth from below, extending previous results.

Contribution

It establishes new regularity results for BV minimizers under non-uniform elliptic conditions with linear growth, broadening the scope of classical theories.

Findings

W^{1,1}-regularity for relaxed minimizers

Higher gradient integrability results

Extension of bounds to non-uniformly degenerate elliptic cases

Abstract

We establish $\mathrm{W}^{1,1}$-regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on $\mathrm{BV}$. Unlike classical examples such as the minimal surface integrand, we only require linear growth from below but not necessarily from above. This typically comes with a non-uniformly degenerate elliptic behaviour, for which our results extend the presently available bounds from the superlinear growth case in a sharp way.