Stable solutions of symmetric systems on Riemannian manifolds
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Abstract:
We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\mathbb {M}$ without boundary, \begin{equation*} -\Delta _g u_i = H_i(u_1,\cdots ,u_m) \ \ \text {on} \ \ \mathbb {M}, \end{equation*} when $\Delta _g$ stands for the Laplace-Beltrami operator, $u_i:\mathbb {M}\to \mathbb R$ and $H_i\in C^1(\mathbb R^m)$ for $1\le i\le m$. This system is called symmetric if the matrix of partial derivatives of all components of $H$, that is, $\mathbb H(u)=(\partial _j H_i(u))_{i,j=1}^m$, is symmetric. We prove a stability inequality and a Poincaré type inequality for stable solutions using the Bochner-Weitzenböck formula. Then, we apply these inequalities to establish Liouville theorems and flatness of level sets for stable solutions of the above symmetric system under certain assumptions on the manifold and on solutions.References
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Additional Information
- Mostafa Fazly
- Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249
- MR Author ID: 822619
- Email: mostafa.fazly@utsa.edu
- Received by editor(s): November 1, 2016
- Received by editor(s) in revised form: January 9, 2017
- Published electronically: June 9, 2017
- Additional Notes: The author gratefully acknowledges a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant and a University of Texas at San Antonio Start-up Grant.
- Communicated by: Catherine Sulem
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 5435-5449
- MSC (2010): Primary 58J05, 53B21, 53C21, 35R01, 35J45
- DOI: https://doi.org/10.1090/proc/13656
- MathSciNet review: 3717969