The $P$-Laplace equation on a class of Grushin-type spaces
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- by Thomas Bieske and Jasun Gong PDF
- Proc. Amer. Math. Soc. 134 (2006), 3585-3594 Request permission
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
- André Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1–78. MR 1421822, DOI 10.1007/978-3-0348-9210-0_{1}
- Thomas Bieske, Lipschitz extensions on generalized Grushin spaces, Michigan Math. J. 53 (2005), no. 1, 3–31. MR 2125531, DOI 10.1307/mmj/1114021082
- Bieske, Thomas. Properties of Infinite Harmonic Functions in Riemannian Vector Fields. Preprint.
- Thomas Bieske and Luca Capogna, The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics, Trans. Amer. Math. Soc. 357 (2005), no. 2, 795–823. MR 2095631, DOI 10.1090/S0002-9947-04-03601-3
- Luca Capogna, Donatella Danielli, and Nicola Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), no. 6, 1153–1196. MR 1420920, DOI 10.1353/ajm.1996.0046
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Roberto Monti and Francesco Serra Cassano, Surface measures in Carnot-Carathéodory spaces, Calc. Var. Partial Differential Equations 13 (2001), no. 3, 339–376. MR 1865002, DOI 10.1007/s005260000076
- Nguen Min′Chi, On the Grushin equation, Mat. Zametki 63 (1998), no. 1, 95–105 (Russian, with Russian summary); English transl., Math. Notes 63 (1998), no. 1-2, 84–93. MR 1631852, DOI 10.1007/bf02316146
- Nguyen Minh Tri, Remark on non-uniform fundamental and non smooth solutions of some classes of differential operators with double characteristics, J. Math. Sci. Univ. Tokyo 6 (1999), no. 3, 437–452. MR 1726678
- Wang, C.Y. The Euler Equation of Absolutely Minimizing Lipschitz Extensions for Vector Fields satisfying Hörmander’s Condition. Preprint.
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
- MR Author ID: 792173
- Email: jgong@umich.edu
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 3585-3594
- MSC (2000): Primary 35H20; Secondary 17B70
- DOI: https://doi.org/10.1090/S0002-9939-06-08394-8
- MathSciNet review: 2240671