# American Institute of Mathematical Sciences

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

 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.
Citation: Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete and 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. [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. [3] T. Bieske, Fundamental solutions to $P$-Laplace equations in Grushin vector fields, preprint (2008), http://shell.cas.usf.edu/~tbieske/. [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. [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. [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. [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. [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. [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. [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. [11] V. V. Grushin, On a class of hypoelliptic operators, Math. USSR-Sb., 12 (1970), 458-476. doi: 10.1070/SM1970v012n03ABEH000931. [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. [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. [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.

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. [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. [3] T. Bieske, Fundamental solutions to $P$-Laplace equations in Grushin vector fields, preprint (2008), http://shell.cas.usf.edu/~tbieske/. [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. [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. [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. [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. [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. [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. [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. [11] V. V. Grushin, On a class of hypoelliptic operators, Math. USSR-Sb., 12 (1970), 458-476. doi: 10.1070/SM1970v012n03ABEH000931. [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. [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. [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.
 [1] Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057

2020 Impact Factor: 1.392