Form estimates for the -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 | References | Similar Articles | Additional Information

Abstract: We consider the problem of establishing conditions on that ensure that the form associated with the -Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that - unlike the constant case - this is not possible if 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 involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in with 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 Our basic criteria involve restrictions on 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

Email:
wallegre@math.ualberta.ca

DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08718-7

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.