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On integrable codimension one Anosov actions of $\RR^k$
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 |
However, they are not necessarily one-dimensional, as a counter-example shows.
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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