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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the continuity of the Luxemburg norm of the gradient in $ L^{p(\cdot)}$ with respect to $ p(\cdot)$


Authors: Marian Bocea and Mihai Mihăilescu
Journal: Proc. Amer. Math. Soc. 142 (2014), 507-517
MSC (2010): Primary 35D30, 46E30, 49J40, 49J45
Published electronically: October 11, 2013
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Abstract: The asymptotic behavior of a sequence of functionals involving the Luxemburg norm of the gradient in variable exponent Lebesgue spaces is studied in the framework of $ \Gamma $-convergence. As a consequence, we prove the convergence of minima for closely related functionals to a corresponding quantity associated to the $ \Gamma $-limit.


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Additional Information

Marian Bocea
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, Illinois 60660
Email: mbocea@luc.edu

Mihai Mihăilescu
Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania – and – “Simion Stoilow” Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania – and – School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Email: mmihailes@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-12017-4
PII: S 0002-9939(2013)12017-4
Keywords: $\Gamma$-convergence, Luxemburg norm, power-law functionals, variable exponent Lebesgue and Sobolev spaces
Received by editor(s): March 16, 2012
Published electronically: October 11, 2013
Additional Notes: The research of the first author was partially supported by the National Science Foundation under Grant No. DMS-1156393.
The second author has been partially supported by a 2011-2012 Go8 European Fellowship, Australia, and by CNCS-UEFISCDI Grant No. PN-II-ID-PCE-2012-4-0021, “Variable Exponent Analysis: Partial Differential Equations and Calculus of Variations”.
Communicated by: Walter Craig
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.