Compactness versus regularity in the calculus of variations

Daniel Faraco; Jan Kristensen

doi:10.3934/dcdsb.2012.17.473

Article Contents

2012, Volume 17, Issue 2: 473-485. Doi: 10.3934/dcdsb.2012.17.473

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Compactness versus regularity in the calculus of variations

Daniel Faraco^1, and
Jan Kristensen^2,

1.
Department of Mathematics, Universidad Autónoma de Madrid, and Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, Madrid, 28049
2.
Mathematical Institute, 24–29 St Giles’, University of Oxford, OX1 3LB Oxford

Received: September 2010

Revised: February 2011

Published: March 2012

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Abstract

In this note we take the view that compactness in $L^p$ can be seen quantitatively on a scale of fractional Sobolev type spaces. To accommodate this viewpoint one must work on a scale of spaces, where the degree of differentiability is measured, not by a power function, but by an arbitrary function that decays to zero with its argument. In this context we provide new $L^p$ compactness criteria that were motivated by recent regularity results for minimizers of quasiconvex integrals. We also show how rigidity results for approximate solutions to certain differential inclusions follow from the Riesz--Kolmogorov compactness criteria.

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Mathematics Subject Classification: Primary: 49J45; Secondary: 46E35, 46E30.

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