Regularity of minimizers of W^1,p-quasiconvex variational integrals with (p,q)-growth

Abstract

We consider autonomous integrals

$$F[u]:=\int_\Omega f(Du)dx \quad{\rm for}\,\,u:{\mathbb{R}}^{n}\supset\Omega\to{\mathbb{R}}^{N} $$

in the multidimensional calculus of variations, where the integrand f is a strictly W ^1,p-quasiconvex C ²-function satisfying the (p,q)-growth conditions

$$ \gamma |A|^p\,\le\,f(A) \le \Gamma(1+|A|^q)\quad {\rm for \quad every}\,A \in \mathbb{R}^{nN}$$

with exponents 1 < p ≤  q < ∞. Under these assumptions we establish an existence result for minimizers of F in $W^{1,p}(\Omega;{\mathbb{R}}^N)$ provided $q\quad < \quad\frac{np}{n-1}$ . We prove a corresponding partial C ^1,α-regularity theorem for $q < p +\frac{{\rm min}\{2,p\}}{2n}$ . This is the first regularity result for autonomous quasiconvex integrals with (p,q)-growth.