Regularity under sharp anisotropic general growth conditions (original) (raw)

Variational inequalities for energy functionals with nonstandard growth conditions

Abstract and Applied Analysis, 1998

We consider the obstacle problem minimize I(u) = Ω G(∇u)dx among functions u : Ω → R such that u| ∂Ω = 0 and u ≥ Φ a.e. for a given function Φ ∈ C 2 (Ω), Φ| ∂Ω < 0 and a bounded Lipschitz domain Ω in R n . The growth properties of the convex integrand G are described in terms of a N -function A :

Partial regularity for anisotropic functionals of higher order

ESAIM: Control, Optimisation and Calculus of Variations, 2007

We prove a C k,α partial regularity result for local minimizers of variational integrals of the type I(u) = Ω f (D k u(x))dx, assuming that the integrand f satisfies (p, q) growth conditions. Mathematics Subject Classification. 35G99, 49N60, 49N99.

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