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June  2014, 34(6): 2639-2656. doi: 10.3934/dcds.2014.34.2639

Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$

1. 

Departamento de Ingeniería Matemática, Universidad de Chile, Blanco Encalada 2120, Santiago, Chile

2. 

Laboratoire d'Analyse, Topologie, Probabilités, Aix-Marseille Université, 9, rue F. Joliot Curie, 13453 Marseille Cedex 13, France

Received  April 2013 Revised  May 2013 Published  December 2013

In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-\Delta_\mathbb{H})^sv=f(v)$ in $\mathbb{H}$, $s\in(0,1)$. We obtain a Poincaré type inequality in connection with a degenerate elliptic equation in $\mathbb{R}^4_+$; through an extension (or ``lifting") procedure, this inequality will be then used for giving a criterion under which the level sets of the above solutions are minimal surfaces in $\mathbb{H}$, i.e. they have vanishing mean curvature.
Citation: Luis F. López, Yannick Sire. Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2639-2656. doi: 10.3934/dcds.2014.34.2639
References:
[1]

N. Arcozzi and F. Ferrari, Metric normal and distance function in the Heisenberg group, Math. Z., 256 (2007), 661-684. doi: 10.1007/s00209-006-0098-8.

[2]

N. Arcozzi and F. Ferrari, The Hessian of the distance from a surface in the Heisenberg group, Ann. Acad. Sci. Fenn. Math., 33 (2008), 35-63. Available from: http://mathstat.helsinki.fi/Annales/Vol33/vol33.html.

[3]

J. Bertoin, Lévy Processes, Cambridge Tracts in Math., 121, Cambridge Univ. Press, Cambridge, 1996.

[4]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007.

[5]

X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733. doi: 10.1016/j.jfa.2005.12.018.

[6]

X. Cabré and Y. Sire, Semilinear equations with fractional Laplacians, in preparation, 2007.

[7]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.

[8]

L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

[9]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[11]

L. Capogna, D. Danielli, S. Pauls and J. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, 259, Birkhäuser Verlag, Basel, 2007.

[12]

J.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4 (2005), 129-177.

[13]

V. Chiadò and F. Serra Cassano, Relaxation of degenerate variational integrals, Nonlinear Anal., 22 (1994), 409-424. doi: 10.1016/0362-546X(94)90165-1.

[14]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2004.

[15]

E. De Giorgi, Convergence problems for functionals and operators, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, 1979, 131-188.

[16]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, translated from the French by C. W. John, Grundlehren Math. Wiss., 219, Springer-Verlag, Berlin-New York, 1976.

[17]

A. Farina, Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires, Habilitation à diriger des recherches, Paris VI, 2002.

[18]

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.

[19]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, preprint, arXiv:1206.0885v4.

[20]

F. Ferrari and E. Valdinoci, A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems, Math. Ann., 343 (2009), 351-370. doi: 10.1007/s00208-008-0274-8.

[21]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204.

[22]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.

[23]

B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 10 (1983), 523-541.

[24]

B. Franchi and R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: A geometrical approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 14 (1987), 527-568.

[25]

N. Garofalo, D. Danielli and D.-M. Nhieu, Notion of convexity in Carnot groups, Comm. Anal. Geom., 11 (2003), 263-341. doi: 10.4310/CAG.2003.v11.n2.a5.

[26]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math., 80, Birkhäuser Verlag, Basel, 1984.

[27]

G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana, 8 (1992), 367-439. doi: 10.4171/RMI/129.

[28]

S. Pauls, Minimal surfaces in the Heisenberg group, Geom. Dedicata, 104 (2004), 201-231. doi: 10.1023/B:GEOM.0000022861.52942.98.

[29]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.

[30]

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

[31]

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.

show all references

References:
[1]

N. Arcozzi and F. Ferrari, Metric normal and distance function in the Heisenberg group, Math. Z., 256 (2007), 661-684. doi: 10.1007/s00209-006-0098-8.

[2]

N. Arcozzi and F. Ferrari, The Hessian of the distance from a surface in the Heisenberg group, Ann. Acad. Sci. Fenn. Math., 33 (2008), 35-63. Available from: http://mathstat.helsinki.fi/Annales/Vol33/vol33.html.

[3]

J. Bertoin, Lévy Processes, Cambridge Tracts in Math., 121, Cambridge Univ. Press, Cambridge, 1996.

[4]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007.

[5]

X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733. doi: 10.1016/j.jfa.2005.12.018.

[6]

X. Cabré and Y. Sire, Semilinear equations with fractional Laplacians, in preparation, 2007.

[7]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.

[8]

L. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

[9]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[11]

L. Capogna, D. Danielli, S. Pauls and J. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, 259, Birkhäuser Verlag, Basel, 2007.

[12]

J.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4 (2005), 129-177.

[13]

V. Chiadò and F. Serra Cassano, Relaxation of degenerate variational integrals, Nonlinear Anal., 22 (1994), 409-424. doi: 10.1016/0362-546X(94)90165-1.

[14]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2004.

[15]

E. De Giorgi, Convergence problems for functionals and operators, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, 1979, 131-188.

[16]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, translated from the French by C. W. John, Grundlehren Math. Wiss., 219, Springer-Verlag, Berlin-New York, 1976.

[17]

A. Farina, Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires, Habilitation à diriger des recherches, Paris VI, 2002.

[18]

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.

[19]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, preprint, arXiv:1206.0885v4.

[20]

F. Ferrari and E. Valdinoci, A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems, Math. Ann., 343 (2009), 351-370. doi: 10.1007/s00208-008-0274-8.

[21]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204.

[22]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.

[23]

B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 10 (1983), 523-541.

[24]

B. Franchi and R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: A geometrical approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 14 (1987), 527-568.

[25]

N. Garofalo, D. Danielli and D.-M. Nhieu, Notion of convexity in Carnot groups, Comm. Anal. Geom., 11 (2003), 263-341. doi: 10.4310/CAG.2003.v11.n2.a5.

[26]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math., 80, Birkhäuser Verlag, Basel, 1984.

[27]

G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana, 8 (1992), 367-439. doi: 10.4171/RMI/129.

[28]

S. Pauls, Minimal surfaces in the Heisenberg group, Geom. Dedicata, 104 (2004), 201-231. doi: 10.1023/B:GEOM.0000022861.52942.98.

[29]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.

[30]

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

[31]

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.

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