Seminario di analisi matematica
ore
16:00
presso Aula Vitali
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).