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August  2013, 33(8): 3473-3496. doi: 10.3934/dcds.2013.33.3473

Geometric inequalities and symmetry results for elliptic systems

Serena Dipierro 1,

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 $\mathbb{R}^2$.
Keywords: phase separation, Poincaré-type inequality., Elliptic systems, stable solutions, monotone solutions.
Mathematics Subject Classification: 35J92, 35J93, 35J5.
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]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

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A. Farina, "Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires," Habilitation à Diriger des Recherches, Paris VI, 2002. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calc. Var. and PDE., ().   Google Scholar

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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.  Google Scholar

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E. H. Lieb and M. Loss, "Analysis," Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997.  Google Scholar

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E. Sernesi, "Geometria 2," Bollati Boringhieri, Torino, 1994. Google Scholar

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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.  Google Scholar

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P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.  Google Scholar

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K. Wang, On the De Giorgi type conjecture for an elliptic system modeling phase separation,, preprint., ().   Google Scholar

show all references

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]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

[5]

A. Farina, "Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires," Habilitation à Diriger des Recherches, Paris VI, 2002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, Calc. Var. and PDE., ().   Google Scholar

[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.  Google Scholar

[10]

E. H. Lieb and M. Loss, "Analysis," Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[11]

E. Sernesi, "Geometria 2," Bollati Boringhieri, Torino, 1994. Google Scholar

[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.  Google Scholar

[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.  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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