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Stable solutions of symmetric systems on Riemannian manifolds


Author: Mostafa Fazly
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
Published electronically: June 9, 2017
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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,

$\displaystyle -\Delta _g u_i = H_i(u_1,\cdots ,u_m) \ $$\displaystyle \ \text {on} \ \ \mathbb{M},$    

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.

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Additional Information

Mostafa Fazly
Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249
Email: mostafa.fazly@utsa.edu

DOI: https://doi.org/10.1090/proc/13656
Keywords: Laplace-Beltrami operator, Riemannian manifolds, nonlinear elliptic systems, qualitative properties of solutions, Liouville theorems
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
Article copyright: © Copyright 2017 American Mathematical Society