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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board Member: Armstrong, Scott N. / Astala, Kari / Colding, Tobias / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Finster, Felix / Gianazza, Ugo / Gursky, Matthew / Hardt, Robert / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / McCann, Robert / Mingione, Giuseppe / Nystrom, Kaj / Pacard, Frank / Preiss, David / Riviére, Tristan / Schaetzle, Reiner / Silvestre, Luis

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IMPACT FACTOR 2016: 1.182

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1864-8266
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Volume 1, Issue 3 (Jan 2008)

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

Thomas Schmidt
  • Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr.1, 40225 Düsseldorf, Germany. E-mail:
  • Other articles by this author:
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Published Online: 2008-11-25 | DOI: https://doi.org/10.1515/ACV.2008.010

Abstract

We study autonomous integrals

F[u] := ∫Ω ƒ(Du) dx for u : ℝ n ⊃ Ω → ℝ N

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

About the article

Received: 2007-08-09

Revised: 2008-04-30

Published Online: 2008-11-25

Published in Print: 2008-10-01


Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/ACV.2008.010.

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