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Preprint (Vorabdruck) zugänglich unter
Lavrentiev phenomenon, relaxation and some regularity results for anisotropic functionals
URN: urn:nbn:de:bsz:291-scidok-44617
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4461/
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Dokument 1.pdf (386 KB)
Institut:
DDC-Sachgruppe:
Mathematik
Dokumentart:
Preprint (Vorabdruck)
Schriftenreihe:
Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Bandnummer:
103
Sprache:
Englisch
Erstellungsjahr:
2004
Publikationsdatum:
10.02.2012
Kurzfassung auf Englisch:
We study local minimizers of anisotropic variational integrals of the form J[u]=\int_{\Omega}f(\cdot,\nabla u)dx with integrand f satisfying a (p,\bar{q})-growth condition w.r.t. \nabla u and with D_{P}f(x,P) satisfying a Lipschitz condition w.r.t. x\in\Omega. If the Lavrentiev gap functional \mathcal{L} relative to J vanishes for all balls B_{R}\Subset\Omega and if \bar{q}<p(1+1/), then (partial) C^{1,\alpha}-regularity holds. Moreover, the bound on the exponents can be replaced by \bar{q}<p+1 provided we study locally bounded minimizers.
We also investigate the relaxation of global minimization problems and discuss the regularity of the corresponding solutions. The importance of the condition \mathcal{L}\equiv0 was recently discovered by Esposito, Leonetti and Mingione in [ELM], where besides other results the higher integrability of the gradient is proved even under weaker assumptions than used here.
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