Date: 03 Aug 2007

Regularity of minimizers of W1,p -quasiconvex variational integrals with (p,q)-growth

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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.