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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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/s00209-006-0098-8. Google Scholar

[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. Google Scholar

[3]

J. Bertoin, Lévy Processes,, Cambridge Tracts in Math., (1996). Google Scholar

[4]

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

[5]

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

[6]

X. Cabré and Y. Sire, Semilinear equations with fractional Laplacians,, in preparation, (2007). Google Scholar

[7]

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

[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. doi: 10.4171/JEMS/226. Google Scholar

[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. doi: 10.1007/s00222-007-0086-6. Google Scholar

[10]

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

[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, (2007). Google Scholar

[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. Google Scholar

[13]

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

[14]

R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financ. Math. Ser., (2004). Google Scholar

[15]

E. De Giorgi, Convergence problems for functionals and operators,, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1978), 131. Google Scholar

[16]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, translated from the French by C. W. John, (1976). Google Scholar

[17]

A. Farina, Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires,, Habilitation à diriger des recherches, (2002). Google Scholar

[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. Google Scholar

[19]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups,, preprint, (). Google Scholar

[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. doi: 10.1007/s00208-008-0274-8. Google Scholar

[21]

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

[22]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups,, Mathematical Notes, (1982). Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[26]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Monogr. Math., (1984). Google Scholar

[27]

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

[28]

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

[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. doi: 10.1016/j.jfa.2009.01.020. Google Scholar

[30]

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

[31]

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

show all references

References:
[1]

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

[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. Google Scholar

[3]

J. Bertoin, Lévy Processes,, Cambridge Tracts in Math., (1996). Google Scholar

[4]

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

[5]

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

[6]

X. Cabré and Y. Sire, Semilinear equations with fractional Laplacians,, in preparation, (2007). Google Scholar

[7]

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

[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. doi: 10.4171/JEMS/226. Google Scholar

[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. doi: 10.1007/s00222-007-0086-6. Google Scholar

[10]

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

[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, (2007). Google Scholar

[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. Google Scholar

[13]

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

[14]

R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financ. Math. Ser., (2004). Google Scholar

[15]

E. De Giorgi, Convergence problems for functionals and operators,, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1978), 131. Google Scholar

[16]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, translated from the French by C. W. John, (1976). Google Scholar

[17]

A. Farina, Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires,, Habilitation à diriger des recherches, (2002). Google Scholar

[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. Google Scholar

[19]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups,, preprint, (). Google Scholar

[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. doi: 10.1007/s00208-008-0274-8. Google Scholar

[21]

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

[22]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups,, Mathematical Notes, (1982). Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[26]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Monogr. Math., (1984). Google Scholar

[27]

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

[28]

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

[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. doi: 10.1016/j.jfa.2009.01.020. Google Scholar

[30]

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

[31]

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

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