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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Entropy solutions for the $ p(x)$-Laplace equation


Authors: Manel Sanchón and José Miguel Urbano
Journal: Trans. Amer. Math. Soc. 361 (2009), 6387-6405
MSC (2000): Primary 35J70; Secondary 35D05, 35D10, 46E35
Published electronically: June 18, 2009
MathSciNet review: 2538597
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a Dirichlet problem in divergence form with variable growth, modeled on the $ p(x)$-Laplace equation. We obtain existence and uniqueness of an entropy solution for $ L^1$ data, as well as integrability results for the solution and its gradient. The proofs rely crucially on a priori estimates in Marcinkiewicz spaces with variable exponent, for which we obtain new inclusion results of independent interest.


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Additional Information

Manel Sanchón
Affiliation: CMUC, Department of Mathematics, University of Coimbra, 3001–454 Coimbra, Portugal
Address at time of publication: Departament de Matemàtica Aplicada i Anàlisi, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, E-08007 Barcelona, Spain
Email: msanchon@mat.uc.pt

José Miguel Urbano
Affiliation: CMUC, Department of Mathematics, University of Coimbra, 3001–454 Coimbra, Portugal
Email: jmurb@mat.uc.pt

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04399-2
PII: S 0002-9947(09)04399-2
Received by editor(s): June 15, 2006
Received by editor(s) in revised form: October 10, 2007
Published electronically: June 18, 2009
Additional Notes: The research of the first author was partially supported by CMUC/FCT and MCYT grants BMF2002-04613-C03, MTM2005-07660-C02.
The research of the second author was partially supported by CMUC/FCT and Project POCI/MAT/57546/2004.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.