REGULARITY RESULTS FOR A CLASS OF QUASICONVEX FUNCTIONALS WITH NONSTANDARD GROWTH (original) (raw)

Quasiconvex functionals of (p,q)(p,q)(p,q)-growth and the partial regularity of relaxed minimizers

arXiv (Cornell University), 2022

We establish C ∞-partial regularity results for relaxed minimizers of strongly quasiconvex functionals F [u; Ω] :=ˆΩ F (∇u) dx, u : Ω → R N , subject to a q-growth condition |F (z)| c(1+|z| q), z ∈ R N×n , and natural p-mean coercivity conditions on F ∈ C ∞ (R N×n) for the basically optimal exponent range 1 p q < min{ np n−1 , p + 1}. With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F , our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from SCHMIDT's foundational work [124] for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for p = 1. As further key novelties, our results apply to the canonical class of signed integrands and do not rely in any way on measure representationsà la FONSECA & MALÝ [65]. CONTENTS 1. Introduction 1 2. Main results 7 3. Preliminaries 13 4. Trace-preserving operators and Fubini-type theorems 20 5. The good generation theorem 27 6. Mean coercivity and properties of the relaxed functional 38 7. Estimates for linearisations 48 8. A Mazur-type lemma and the Euler-Lagrange system 49 9. The Caccioppoli inequality of the second kind 56 10. Proof of Theorems 2.1 and 2.2 64 11. Appendix 76 References 86

A regularity theorem for minimisers of quasiconvex integrals: the case 1

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1996

We prove partial regularity for minimisers of quasiconvex integrals of the form ∫Ωf(Du(x))dx. More precisely, we consider an integrandf(ξ) having subquadratic growth, i.e. |f(ξ)|≦L(1+|ξ|p) withp< 2. The case of a general integrand depending also onxanduis also considered.

A regularity theorem for minimizers of quasiconvex integrals

Archive for Rational Mechanics and Analysis, 1987

We prove C 1'~ partial regularity for minimizers of functionals with quasiconvex integrand f(x, u, Du) depending on vector-valued functions u. The integrand is required to be twice continuously differentiable in Du, and no assumption on the growth of the derivatives of f is made: a polynomial growth is required only on f itself.