Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth

摘要

We consider autonomous integrals F[u] := integral(Omega) f(Du) dx for u : R-n superset of Omega -> R-N in the multidimensional calculus of variations, where the integrand f is a strictly quasiconvex C-2-function satisfying the (p, q)-growth conditions. gamma|A|(p) <= f (A) <= Gamma(1 + |A|(q)) for every A is an element of R-nN with exponents 1 < p <= q < infinity. We examine the Lebesgue - Serrin extension F-loc[u] := inf{lim inf(k-->infinity) F[u(k)] : W-loc(1,q) (sic) u(k) -> (k ->infinity) u weakly in W-1,W- p} of F and establish an existence result for minimizers of F-loc. Furthermore, we prove a corresponding partial C-1,C-alpha-regularity theorem for q < p+min{2, p}/2n, which is the first regularity result for this class of integrands.