American Institute of Mathematical Sciences

Advanced
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 - A, 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, (1992). Google Scholar

[5]

A. Farina, "Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires,", Habilitation à Diriger des Recherches, (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. 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, (2009), 74. 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. Google Scholar

[10]

E. H. Lieb and M. Loss, "Analysis,", Graduate Studies in Mathematics, 14 (1997). Google Scholar

[11]

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

[12]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. 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. Google Scholar

[14]

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, (1992). Google Scholar

[5]

A. Farina, "Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires,", Habilitation à Diriger des Recherches, (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. 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, (2009), 74. 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. Google Scholar

[10]

E. H. Lieb and M. Loss, "Analysis,", Graduate Studies in Mathematics, 14 (1997). Google Scholar

[11]

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

[12]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. 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. Google Scholar

[14]

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

[1]

Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505
[2]

Pavel Krejčí, Songmu Zheng. Pointwise asymptotic convergence of solutions for a phase separation model. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 1-18. doi: 10.3934/dcds.2006.16.1
[3]

Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
[4]

Kota Kumazaki, Akio Ito, Masahiro Kubo. Generalized solutions of a non-isothermal phase separation model. Conference Publications, 2009, 2009 (Special) : 476-485. doi: 10.3934/proc.2009.2009.476
[5]

Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541
[6]

Li Ma, Chong Li, Lin Zhao. Monotone solutions to a class of elliptic and diffusion equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 237-246. doi: 10.3934/cpaa.2007.6.237
[7]

Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198
[8]

Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166
[9]

Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293
[10]

Christian Heinemann, Christiane Kraus. Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2565-2590. doi: 10.3934/dcds.2015.35.2565
[11]

Yasuhito Miyamoto. Global bifurcation and stable two-phase separation for a phase field model in a disk. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 791-806. doi: 10.3934/dcds.2011.30.791
[12]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[13]

Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025
[14]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815
[15]

Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure & Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361
[16]

Pavel Krejčí, Elisabetta Rocca, Jürgen Sprekels. Phase separation in a gravity field. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 391-407. doi: 10.3934/dcdss.2011.4.391
[17]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237
[18]

Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715
[19]

Anatoli F. Ivanov, Bernhard Lani-Wayda. Periodic solutions for three-dimensional non-monotone cyclic systems with time delays. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 667-692. doi: 10.3934/dcds.2004.11.667
[20]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences