Regularity for higher order quasiconvex problems with linear growth from below

Abstract

We announce new existence and ε-regularity results for minimisers of the relaxation of strongly quasiconvex integrals that on smooth maps u :The results cover the case of integrands F with (1, q) growth in the full range of exponents 1 < q < 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.

Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth

“…In light of the partial regularity results in the linear growth setting [37], a natural question is whether the ∇ 2 -condition is really necessary. In fact the results announced in [38] implies that partial regularity holds if we have merely the growth condition ϕ(t) ≤ Ct q with q < n n−1 , however the general case remains open which we wish to address in future work.…”