# American Institute of Mathematical Sciences

August  2013, 33(8): 3473-3496. doi: 10.3934/dcds.2013.33.3473

## Geometric inequalities and symmetry results for elliptic systems

 1 SISSA - International School for Advanced Studies, Sector of Mathematical Analysis Via Bonomea, 265, 34136 Trieste

Received  July 2012 Revised  September 2012 Published  January 2013

We obtain some Poincaré type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form \begin{eqnarray*} \left\{ \begin{array}{ll} div\left( a\left( |\nabla u|\right) \nabla u\right) = F_1(u, v), \\ div\left( b\left( |\nabla v|\right) \nabla v\right) = F_2(u, v), \end{array} \right. \end{eqnarray*} where $F ∈ C^{1,1}_{loc}(\mathbb{R}^2)$.
Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in $\mathbb{R}^2$.
Citation: Serena Dipierro. Geometric inequalities and symmetry results for elliptic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3473-3496. doi: 10.3934/dcds.2013.33.3473
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