A degenerate Sobolev inequality for a large open set in a homogeneous space

Author:
Scott Rodney

Journal:
Trans. Amer. Math. Soc. **362** (2010), 673-685

MSC (2000):
Primary 35Hxx

Published electronically:
September 15, 2009

MathSciNet review:
2551502

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Abstract: In current literature, existence results for degenerate elliptic equations with rough coefficients on a large open set of a homogeneous space have been demonstrated; see the paper by Gutierrez and Lanconelli (2003). These results require the assumption of a Sobolev inequality on of the form

holding for and some . However, it is unclear when such an inequality is valid, as techniques often yield only a local version of (1):(2)

holding for , with as above. The main result of this work shows that the global Sobolev inequality (1) can be obtained from the local Sobolev inequality (2) provided standard regularity hypotheses are assumed with minimal restrictions on the quadratic form . This is achieved via a new technique involving existence of weak solutions, with global estimates, to a 1-parameter family of Dirichlet problems on and a maximum principle.

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Additional Information

**Scott Rodney**

Affiliation:
Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

DOI:
http://dx.doi.org/10.1090/S0002-9947-09-04809-0

Received by editor(s):
October 1, 2007

Published electronically:
September 15, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.