Everywhere regularity of vectorial minimizers of some non-convex functionals

The convexity of the integrand of a functional of the calculus of variations is equivalent to the lower semicontinuity of the functional in the scalar case, but it is only a sufficient condition in the vectorial case. So, it is not satisfied by many interesting examples to which the existence theorems apply. Moreover, the convexity of the integrand turns out to be a too strong and unrealistic assumption in applications, as for instance in mathematical models in nonlinear elasticity (Ball 1977). In the vectorial framework more appropriate and weaker conditions than the convexity are the polyconvexity and the quasiconvexity. Under these assumptions, many results were proved concerning the partial regularity of minimizers (regularity on open sets of full measure), but the results concerning the everywhere regularity are very few and mainly in low dimensions (n=N=2). We will discuss recent everywhere regularity results of vectorial minimizers for some classes of polyconvex and quasiconvex functionals (n,N >2) obtained in collaboration with F. Leonetti and E. Mascolo (local boundedness) and with them and M. Focardi (Holder continuity). The proofs rely on the power and elegant (typically scalar) method by De Giorgi (1957).