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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 $ \Theta$ of a homogeneous space $ (\Omega,d)$ have been demonstrated; see the paper by Gutierrez and Lanconelli (2003). These results require the assumption of a Sobolev inequality on $ \Theta$ of the form

$\displaystyle (1)\qquad\qquad\Big\{\displaystyle{\int_\Theta} \vert w(x)\vert^{... ...Theta} \mathcal{Q}(x,\nabla w(x))dx\Big\}^\frac{1}{2}, \qquad\qquad\qquad\qquad$

holding for $ w\in Lip_0(\Theta)$ and some $ \sigma\in (1,2]$. However, it is unclear when such an inequality is valid, as techniques often yield only a local version of (1):

(2)

$\displaystyle \Big\{\displaystyle{\frac{1}{\vert B_r\vert}} \displaystyle{\int_... ...ert B_r\vert}} \displaystyle{\int_{B_r}}\vert v(x)\vert^2dx\Big\}^\frac{1}{2}, $

holding for $ v\in Lip_0(B_r)$, with $ \sigma$ 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 $ \mathcal{Q}(x,\cdot)$. This is achieved via a new technique involving existence of weak solutions, with global estimates, to a 1-parameter family of Dirichlet problems on $ \Theta$ 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.