Geometric inequalities and symmetry results for elliptic systems

Serena Dipierro

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arXiv · 1207.4435

Geometric inequalities and symmetry results for elliptic systems

Serena Dipierro

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Abstract

We obtain some Poincar\'{e} 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 {eqnarray*} {{array}{ll} div(a(|\nabla u|) \nabla u) = F_1(u, v), div(b(|\nabla v|) \nabla v) = F_2(u, v), {array}. {eqnarray*} where $F\in C^{1,1}_{loc}(\R^2)$. Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in $\R^2$.

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Serena Dipierro. 2012-07-18. Geometric inequalities and symmetry results for elliptic systems. https://arxiv.org/abs/1207.4435

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