Geometric inequalities and symmetry results for elliptic systems

Abstract

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 {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 $F\in C_{loc}^{1,1}({\mathbb{R}}^2)$. Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in ${\mathbb{R}}^2$.

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References in corpus (1)

• Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition

Cited by in corpus (1)

• Symmetry results for stable and monotone solutions to fibered systems of PDEs