American Institute of Mathematical Sciences

Advanced
July  2011, 29(3): 823-838. doi: 10.3934/dcds.2011.29.823

On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional

Isabeau Birindelli 1, and  Enrico Valdinoci 2,

1. 

Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro, 2, I-00185 Roma, Italy
2. 

Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano

Received  November 2009 Revised  July 2010 Published  November 2010

We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate.
   However, they are not necessarily one-dimensional, as a counter-example shows.
Keywords: Allen-Cahn equation, Grushin plane., Symmetry problmes.
Mathematics Subject Classification: Primary: 35J15, 35J62; Secondary: 35H20, 53A3.
Citation: Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823
References:
[1]

G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., 65 (2001), 9-33. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday.  Google Scholar

[2]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat., 22 (1991), 1-37. doi: 10.1007/BF01244896.  Google Scholar

[3]

T. Bieske, Fundamental solutions to $P$-Laplace equations in Grushin vector fields, preprint (2008), http://shell.cas.usf.edu/~tbieske/. Google Scholar

[4]

I. Birindelli and E. Lanconelli, A negative answer to a one-dimensional symmetry problem in the Heisenberg group, Calc. Var. Partial Differ. Equ., 18 (2003), 357-372.  Google Scholar

[5]

I. Birindelli and J. Prajapat, Monotonicity results for nilpotent stratified groups, Pacific J. Math., 204 (2002), 1-17. doi: 10.2140/pjm.2002.204.1.  Google Scholar

[6]

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

[7]

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

[8]

F. Ferrari and E. Valdinoci, Geometric PDEs in the Grushin plane: Weighted inequalities and flatness of level sets, Int. Math. Res. Not. IMRN, 22 (2009), 4232-4270.  Google Scholar

[9]

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

[10]

B. Franchi and R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: A geometrical approach, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1988), 527-568.  Google Scholar

[11]

V. V. Grushin, On a class of hypoelliptic operators, Math. USSR-Sb., 12 (1970), 458-476. doi: 10.1070/SM1970v012n03ABEH000931.  Google Scholar

[12]

Q. Han and F. Lin, "Elliptic Partial Differential Equations," Courant Lecture Notes in Mathematics, 1, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[13]

N. V. Krylov, Hölder continuity and $L^p$ estimates for elliptic equations under general Hörmander's condition, Topol. Methods Nonlinear Anal., 9 (1997), 249-258.  Google Scholar

[14]

N. Th. Varopoulos, L. Coulhon and T. Saloff-Coste, "Analysis and Geometry on Groups," Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992.  Google Scholar

show all references

References:
[1]

G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., 65 (2001), 9-33. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday.  Google Scholar

[2]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat., 22 (1991), 1-37. doi: 10.1007/BF01244896.  Google Scholar

[3]

T. Bieske, Fundamental solutions to $P$-Laplace equations in Grushin vector fields, preprint (2008), http://shell.cas.usf.edu/~tbieske/. Google Scholar

[4]

I. Birindelli and E. Lanconelli, A negative answer to a one-dimensional symmetry problem in the Heisenberg group, Calc. Var. Partial Differ. Equ., 18 (2003), 357-372.  Google Scholar

[5]

I. Birindelli and J. Prajapat, Monotonicity results for nilpotent stratified groups, Pacific J. Math., 204 (2002), 1-17. doi: 10.2140/pjm.2002.204.1.  Google Scholar

[6]

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

[7]

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

[8]

F. Ferrari and E. Valdinoci, Geometric PDEs in the Grushin plane: Weighted inequalities and flatness of level sets, Int. Math. Res. Not. IMRN, 22 (2009), 4232-4270.  Google Scholar

[9]

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

[10]

B. Franchi and R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: A geometrical approach, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1988), 527-568.  Google Scholar

[11]

V. V. Grushin, On a class of hypoelliptic operators, Math. USSR-Sb., 12 (1970), 458-476. doi: 10.1070/SM1970v012n03ABEH000931.  Google Scholar

[12]

Q. Han and F. Lin, "Elliptic Partial Differential Equations," Courant Lecture Notes in Mathematics, 1, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[13]

N. V. Krylov, Hölder continuity and $L^p$ estimates for elliptic equations under general Hörmander's condition, Topol. Methods Nonlinear Anal., 9 (1997), 249-258.  Google Scholar

[14]

N. Th. Varopoulos, L. Coulhon and T. Saloff-Coste, "Analysis and Geometry on Groups," Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992.  Google Scholar

[1]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[2]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[3]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[4]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[5]

Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205
[6]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
[7]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577
[8]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319
[9]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
[10]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024
[11]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
[12]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077
[13]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[14]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[15]

Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113
[16]

Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391
[17]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111
[18]

Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159
[19]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154
[20]

Xiaofeng Yang. Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences