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 & Continuous Dynamical Systems - A, 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.   Google Scholar

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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.   Google Scholar

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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.   Google Scholar

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

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

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N. Th. Varopoulos, L. Coulhon and T. Saloff-Coste, "Analysis and Geometry on Groups,", Cambridge Tracts in Mathematics, 100 (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.   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.  doi: 10.1007/BF01244896.  Google Scholar

[3]

T. Bieske, Fundamental solutions to $P$-Laplace equations in Grushin vector fields,, preprint (2008), (2008).   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.   Google Scholar

[5]

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

[6]

E. De Giorgi, Convergence problems for functionals and operators,, in, (1979), 131.   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.   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.   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.   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.   Google Scholar

[11]

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

[12]

Q. Han and F. Lin, "Elliptic Partial Differential Equations,", Courant Lecture Notes in Mathematics, 1 (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.   Google Scholar

[14]

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

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