# 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 & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823
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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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