Non-autonomous functionals, borderline cases and related function classes

Abstract

We consider a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and prove some regularity results for related minimisers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with (p, q)-growth. We also discuss similar functionals related to Musielak-Orlicz spaces in which basic properties like density of smooth functions, boundedness of maximal and integral operators, and validity of Sobolev type inequalities naturally relate to the assumptions needed to prove regularity of minima. 1. Almost fifty years of degenerate operators in Russia In 1967 a seminal paper [60] of Ural’tseva appeared, featuring the proof of the C-nature of energy solutions to the degenerate equation (1.1) − div(|Du|p−2Du) = 0 . The one on the left hand side is nowadays very well known as the p-Laplacean operator. This operator is relevant in a large number of situations as for instance in the Calculus of Variations, in Geometric Analysis, in the theory of quasiconformal mappings, in the modelling of non-Newtonian fluids. Ural’tseva herself, by exhibiting a counterexample, showed that the regularity of solutions does not go beyond Holder continuity of the gradient, for some exponent β ∈ (0, 1). The proof the Holder gradient continuity result also appears in the second, yet untranslated, edition of the classical book [41]. Ural’tseva’s fundamental result is at the origin of a huge literature, up to the point that it is nowadays hard to find another single nonlinear operator that has attracted so much attention as long as the elliptic regularity theory is concerned. We quote here the important paper of Uhlenbeck [59], where Ural’tseva’s result has been extended to the vectorial case, and the papers [23, 29, 42, 47], where different proofs and extensions to equations with coefficients have been given. The equation appearing in (1.1) is the Euler-Lagrange equation of the functional (1.2) w 7→ ∫ |Dw| dx , p > 1 and, in fact, several of the results and techniques coming from the analysis of (1.1) have been eventually found to be useful in the Calculus of Variations. Starting from the eighties, new models and functionals related to the one in (1.2) were developed by Zhikov [62, 63, 64, 65, 66], together with a group of Russian mathematicians, in order to describe the behaviour of strongly anisotropic materials in the context of homogenisation and nonlinear elasticity. These functionals revealed to be important 1 2 BARONI, COLOMBO, AND MINGIONE also in the study of duality theory and in the context of the Lavrentiev phenomenon. They are non-autonomous functionals of the form