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Advances in Calculus of Variations

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Regularity theorems for degenerate quasiconvex energies with (p, q)-growth

Thomas Schmidt1

1Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr.1, 40225 Düsseldorf, Germany. E-mail:

Citation Information: Advances in Calculus of Variations. Volume 1, Issue 3, Pages 241–270, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: 10.1515/ACV.2008.010, November 2008

Publication History

Received:
2007-08-09
Revised:
2008-04-30
Published Online:
2008-11-25

Abstract

We study autonomous integrals

F[u] := ∫Ω ƒ(Du) dx for u : ℝn ⊃ Ω → ℝN

in the multidimensional calculus of variations, where the integrand ƒ is a strictly quasiconvex function satisfying the (p, q)-growth conditions

γ|ξ|p ≤ ƒ(ξ) ≤ Γ(1 + |ξ|q)

with exponents . Imposing the additional assumption that ƒ resembles the degenerate behavior of the p-energy density, we establish a partial C 1,α-regularity theorem for F-minimizers and a similar theorem for minimizers of a relaxed functional.

Our results cover the model case of polyconvex integrands

,

where h is a smooth convex function with -growth

Keywords.: Calculus of variations; partial regularity; quasiconvexity; polyconvexity; nonstandard growth; degeneration; relaxation

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