Geometric inequalities and symmetry results for elliptic systems

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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

Abstract
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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 and 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., ().
[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., ().

show all references

References:

[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., ().

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