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.

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