Regularity for higher order quasiconvex problems with linear growth from below

Abstract

We announce new existence and $\epsilon$-regularity results for minimisers of the relaxation of strongly quasiconvex integrals that on smooth maps $u:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^N$ are defined by $$u\mapsto {\int }_{\Omega }F({\nabla }^{k}u)dx.$$ The results cover the case of integrands $F$ with $(1,q)$-growth in the full range of exponents $1<q<\frac{n}{n-1}$ for which a measure representation of the relaxed functional is possible and the minimizers belong to the space $B{V}^k$ of maps whose $k$-th order derivatives are measures.

Announcement, 5 pages