Calculus of Variations and Partial Differential Equations

, Volume 32, Issue 1, pp 1–24

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

Article

DOI: 10.1007/s00526-007-0126-5

Cite this article as:
Schmidt, T. Calc. Var. (2008) 32: 1. doi:10.1007/s00526-007-0126-5

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 W1,p-quasiconvex C2-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 C1,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.

Mathematics Subject Classification (2000)

49N6049J4535J50

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematisches InstitutHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany