Rigidity results for stable solutions of symmetric systems
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Abstract:
We study stable solutions of the nonlinear system \[ -\Delta u = H(u) \quad \text {in} \ \ \Omega \] where $u:\mathbb {R}^n\to \mathbb {R}^m$, $H:\mathbb {R}^m\to \mathbb {R}^m$ and $\Omega$ is a domain in $\mathbb R^n$. We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of $H$ is symmetric. It seems that this concept is crucial to prove Liouville theorems, when $\Omega =\mathbb R^n$, and regularity results, when $\Omega =B_1$, for stable solutions of the above system for a general nonlinearity $H \in C^1(\mathbb R ^m)$. Moreover, we provide an improvement for a linear Liouville theorem given by Fazly and Ghoussoub in 2013 that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.References
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Additional Information
- Mostafa Fazly
- Affiliation: Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
- MR Author ID: 822619
- Email: fazly@ualberta.ca
- Received by editor(s): March 29, 2014
- Received by editor(s) in revised form: October 5, 2014
- Published electronically: June 3, 2015
- Additional Notes: The author is pleased to acknowledge the support of the University of Alberta Start-up Grant RES0019810 and the National Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant RES0020463.
- Communicated by: Catherine Sulem
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5307-5321
- MSC (2010): Primary 35J60, 35B35, 35B32, 35D10, 35J20
- DOI: https://doi.org/10.1090/proc/12647
- MathSciNet review: 3411148