Article Contents
Article Contents

# Geometric inequalities and symmetry results for elliptic systems

• 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.

 Citation:

•  [1] H. Berestycki, T.-C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties, preprint. [2] H. Berestycki, S. Terracini, K. Wang and J. Wei, Existence and stability of entire solutions of an elliptic system modeling phase separation, preprint. [3] S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, preprint. [4] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [5] A. Farina, "Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires," Habilitation à Diriger des Recherches, Paris VI, 2002. [6] A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741-791. [7] A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems. Recent progress on reaction-diffusion systems and viscosity solutions, in "Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions," World Sci. Publ., Hackensack, NJ, (2009), 74-96.doi: 10.1142/9789812834744_0004. [8] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems, Calc. Var. and PDE. [9] B. Noris, H. Tavares, S. Terracini and G. Verzin, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. [10] E. H. Lieb and M. Loss, "Analysis," Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997. [11] E. Sernesi, "Geometria 2," Bollati Boringhieri, Torino, 1994. [12] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.doi: 10.1007/s002050050081. [13] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85. [14] K. Wang, On the De Giorgi type conjecture for an elliptic system modeling phase separation, preprint.