Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The $ P$-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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We find the fundamental solution to the $ P$-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.

References [Enhancements On Off] (What's this?)

  • 1. Bellaïche, André. The Tangent Space in Sub-Riemannian Geometry. In Sub-Riemannian Geometry; Bellaïche, André., Risler, Jean-Jacques., Eds.; Progress in Mathematics; Birkhäuser: Basel, Switzerland. 1996; Vol. 144, 1-78. MR 1421822 (98a:53108)
  • 2. Bieske, Thomas. Lipschitz Extensions on Generalized Grushin Spaces. Mich. Math. J., 2005, 53 (1), 3-31. MR 2125531
  • 3. Bieske, Thomas. Properties of Infinite Harmonic Functions in Riemannian Vector Fields. Preprint.
  • 4. Bieske, Thomas.; Capogna, Luca. The Aronsson-Euler Equation for Absolutely Minimizing Lipschitz Extensions with respect to Carnot-Carathéodory Metrics. Trans. Amer. Math. Soc., 2005, 357 (2), 795-823. MR 2095631 (2005g:35036)
  • 5. Capogna, Luca.; Danielli, Donatella.; Garofalo, Nicola. Capacitary Estimates and the Local Behavior of Solutions of Nonlinear Subelliptic Equations. Amer. J. of Math. 1997, 118, 1153-1196. MR 1420920 (97k:35033)
  • 6. Chavel, Isaac. Eigenvalues in Riemannian Geometry; Academic Press: Orlando, 1984. MR 0768584 (86g:58140)
  • 7. Monti, Roberto.; Serra Cassano, Francesco. Surfaces Measures in Carnot-Carathéodory Spaces. Calc. Var. PDE, 2001, 13 (3), 339-376.MR 1865002 (2002j:49052)
  • 8. Tri, Nguyen Minh. On Grushin's Equation. Math. Notes, 1998, 63 (1), 84-93. MR 1631852 (99e:35069)
  • 9. Tri, Nguyen Minh. Remark on Non-Uniform Fundamental and Non-Smooth Solutions of Some Classes of Differential Operators with Double Characteristics. J. Math. Sci. Univ. Tokyo 1999, 6, 437-452.MR 1726678 (2000j:35292)
  • 10. Wang, C.Y. The Euler Equation of Absolutely Minimizing Lipschitz Extensions for Vector Fields satisfying Hörmander's Condition. Preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35H20, 17B70

Retrieve articles in all journals with MSC (2000): 35H20, 17B70

Additional Information

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

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

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.

American Mathematical Society