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Geometric inequalities and symmetry results for elliptic systems

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  • 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 $\mathbb{R}^2$.
    Mathematics Subject Classification: 35J92, 35J93, 35J50.


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