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, 2013, 33 (8) : 3473-3496. doi: 10.3934/dcds.2013.33.3473
References:
 [1] H. Berestycki, T.-C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties,, preprint., ().   Google Scholar [2] H. Berestycki, S. Terracini, K. Wang and J. Wei, Existence and stability of entire solutions of an elliptic system modeling phase separation,, preprint., ().   Google Scholar [3] S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian,, preprint., ().   Google Scholar [4] Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar [5] Habilitation à Diriger des Recherches, Paris VI, 2002. Google Scholar [6] Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741-791.  Google Scholar [7] in "Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions," World Sci. Publ., Hackensack, NJ, (2009), 74-96. doi: 10.1142/9789812834744_0004.  Google Scholar [8] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calc. Var. and PDE., ().   Google Scholar [9] Comm. Pure Appl. Math., 63 (2010), 267-302.  Google Scholar [10] Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997.  Google Scholar [11] Bollati Boringhieri, Torino, 1994. Google Scholar [12] Arch. Rational Mech. Anal., 141 (1998), 375-400. doi: 10.1007/s002050050081.  Google Scholar [13] J. Reine Angew. Math., 503 (1998), 63-85.  Google Scholar [14] K. Wang, On the De Giorgi type conjecture for an elliptic system modeling phase separation,, preprint., ().   Google Scholar

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References:
 [1] H. Berestycki, T.-C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties,, preprint., ().   Google Scholar [2] H. Berestycki, S. Terracini, K. Wang and J. Wei, Existence and stability of entire solutions of an elliptic system modeling phase separation,, preprint., ().   Google Scholar [3] S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian,, preprint., ().   Google Scholar [4] Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar [5] Habilitation à Diriger des Recherches, Paris VI, 2002. Google Scholar [6] Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741-791.  Google Scholar [7] in "Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions," World Sci. Publ., Hackensack, NJ, (2009), 74-96. doi: 10.1142/9789812834744_0004.  Google Scholar [8] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calc. Var. and PDE., ().   Google Scholar [9] Comm. Pure Appl. Math., 63 (2010), 267-302.  Google Scholar [10] Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997.  Google Scholar [11] Bollati Boringhieri, Torino, 1994. Google Scholar [12] Arch. Rational Mech. Anal., 141 (1998), 375-400. doi: 10.1007/s002050050081.  Google Scholar [13] J. Reine Angew. Math., 503 (1998), 63-85.  Google Scholar [14] K. Wang, On the De Giorgi type conjecture for an elliptic system modeling phase separation,, preprint., ().   Google Scholar
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