The -Laplace equation on a class of Grushin-type spaces

Authors:
Thomas Bieske and Jasun Gong

Journal:
Proc. Amer. Math. Soc. **134** (2006), 3585-3594

MSC (2000):
Primary 35H20; Secondary 17B70

Published electronically:
May 31, 2006

MathSciNet review:
2240671

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Abstract: We find the fundamental solution to the -Laplace equation in Grushin-type spaces. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized Grushin operator in Euclidean space. We then use this solution to find an infinite harmonic function with specific boundary data and to compute the capacity of annuli centered at the singularity. A solution to the 2-Laplace equation in a wider class of spaces is presented.

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Additional Information

**Thomas Bieske**

Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620

Email:
tbieske@math.usf.edu

**Jasun Gong**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Email:
jgong@umich.edu

DOI:
https://doi.org/10.1090/S0002-9939-06-08394-8

Keywords:
Grushin-type spaces,
$P$-Laplacian

Received by editor(s):
December 20, 2004

Received by editor(s) in revised form:
June 28, 2005

Published electronically:
May 31, 2006

Communicated by:
David S. Tartakoff

Article copyright:
© Copyright 2006
American Mathematical Society

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