NDLI: De Giorgi type results for elliptic systems

Content Provider	SpringerLink
Author	Fazly, Mostafa Ghoussoub, Nassif
Copyright Year	2012
Abstract	We consider the following elliptic system $${\Delta}u = \nabla H (u) \quad {\rm in}\quad \mathbf{R}^N,$$ where $${u : \mathbf{R}^N \to \mathbf{R}^m}$$ and $${H \in C^2(\mathbf{R}^m)}$$ , and prove, under various conditions on the nonlinearity H that, at least in low dimensions, a solution $${u=(u_i)_{i=1}^m}$$ is necessarily one-dimensional whenever each one of its components u i is monotone in one direction. Just like in the proofs of the classical De Giorgi’s conjecture in dimension 2 (Ghoussoub-Gui) and in dimension 3 (Ambrosio-Cabré), the key step is a Liouville theorem for linear systems. We also give an extension of a geometric Poincaré inequality to systems and use it to establish De Giorgi type results for stable solutions as well as additional rigidity properties stating that the gradients of the various components of the solutions must be parallel. We introduce and exploit the concept of an orientable system, which seems to be key for dealing with systems of three or more equations. For such systems, the notion of a stable solution in a variational sense coincide with the pointwise (or spectral) concept of stability.
Ending Page	823
Page Count	15
Starting Page	809
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3-4
Volume Number	47
Language	English
Publisher	Springer-Verlag
Publisher Date	2012-06-22
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Stability Liouville theorems, PhragmÃ©n-LindelÃ¶f theorems Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Second-order elliptic systems Analysis Theoretical, Mathematical and Computational Physics Entire solutions Asymptotic behavior of solutions
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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