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Form estimates for the $ p(x)$-Laplacean

Author: W. Allegretto
Journal: Proc. Amer. Math. Soc. 135 (2007), 2177-2185
MSC (2000): Primary 35P15; Secondary 35J60, 35J25
Published electronically: March 1, 2007
MathSciNet review: 2299495
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Abstract: We consider the problem of establishing conditions on $ p(x)$ that ensure that the form associated with the $ p(x)$-Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that - unlike the $ p=$ constant case - this is not possible if $ p$ has a strict extrema in the domain. They also considered the closely related problem of eigenvalue existence and estimates. Our main tool is the adaptation of a technique, employed by Protter for $ p=2,$ involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in $ \Omega \subset R^n$ with $ n\ge 1,$ and while we focus on Dirichlet boundary conditions we also indicate how our approach can be used in cases of mixed boundary conditions, of unbounded domains and of discontinuous $ p(x).$ Our basic criteria involve restrictions on $ p(x)$ and its gradient.

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Additional Information

W. Allegretto
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Keywords: $p(x)$-Laplacean, arbitrary vector field, Picone, positive forms
Received by editor(s): December 14, 2005
Received by editor(s) in revised form: March 21, 2006
Published electronically: March 1, 2007
Additional Notes: Research supported by NSERC Canada.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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