Regularity for higher order quasiconvex problems with linear growth from below

TL;DR

This paper establishes new existence and regularity results for minimizers of higher-order quasiconvex variational problems with linear growth, covering a broad range of growth conditions and measure-based minimizers.

Contribution

It introduces novel existence and ε-regularity results for relaxed strongly quasiconvex integrals involving higher derivatives with linear growth.

Findings

Existence of minimizers in BV^k space for specified growth conditions.

Regularity results under (1,q)-growth with 1<q<n/(n-1).

Measure representation of the relaxed functional.

Abstract

We announce new existence and $\varepsilon$-regularity results for minimisers of the relaxation of strongly quasiconvex integrals that on smooth maps $u\colon\Omega\subset\mathbb{R}^{n}\to\mathbb{R}^{N}$ 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 $BV^k$ of maps whose $k$-th order derivatives are measures.