Non-uniformly elliptic variational problems on BV

Abstract We establish $$\operatorname {W}^{1,1}$$ W 1 , 1 -regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on $$\operatorname {BV}$$ 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.