Some remarks on quasiconvexity

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.