Paolo Marcellini

As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness results under general assumptions that can be applied to several cases studied in the recent and extensive literature on partial differential equations under general growth. In particular, we propose the class of weak solutions with finite energy in which to search for solutions and in which regularity can be studied and achieved. We emphasize that we are not limited to minimizers of certain integral functionals, as often considered recently in this context of general growth, but to the broader class of weak solutions to Dirichlet problems for general nonlinear elliptic equations in divergence form. When applied to [Formula: see text]problems, a further major novelty is that we consider general [Formula: see text] growth conditions for the principal part of the differential equation, but we do not need an upper bound on the ratio [Formula: see text], we just need nothing more than [Formula: see text].

This article is dedicated to Hedy Attouch, colleague and friend, with whom I shared many studies on Gamma convergence and other types of variational convergence [see the author, Su una convergenza di funzioni convesse, Boll. Unione Mat. Italiana 8 (1973) 137–158], as well described in his book [see H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman, Boston (1984)]. I have a vivid memory of the discussions we had together on these and other topics in calculus of variations. This manuscript deals with some properties of quasiconvex functions and their applications to problems of the calculus of variations, in particular lower semicontinuity, existence and regularity – partial and everywhere regularity – of minimizers of convex and quasiconvex energy integrals under p,q-growth conditions. Motivated by a model energy integral relevant in nonlinear elasticity and by the phenomenon of cavitation, we point out the interest to study lower semicontinuity and relaxation in the context of subcritical growth. Subcritical growth appears to be a context for a real challenge on quasiconvexity.

As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness results under general assumptions that can be applied to several cases studied in the recent and extensive literature on partial differential equations \textit{under general growth}. In particular, we propose the class of \textit{weak solutions with finite energy} in which to search for solutions and in which regularity can be studied and achieved. We emphasize that we are not limited to minimizers of certain integral functionals, as often considered recently in this context of general growth, but to the broader class of weak solutions to Dirichlet problems for general nonlinear elliptic equations in divergence form.

This paper deals with some properties of convex functions and their applications to problems of the calculus of variations, both in the convex case, where Rockafellar is one of the main experts in the world, and also in the quasiconvex case, a field which still is not completely understood.