June  2014, 34(6): 2451-2467. doi: 10.3934/dcds.2014.34.2451

A symmetry result for the Ornstein-Uhlenbeck operator

1. 

Dipartimento di Matematica, Università di Padova, Via Trieste 63, I-35121 Padova, Italy

2. 

Dipartimento di Matematica, Universitá di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa, Italy

3. 

Weierstraß Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 11A, D-10117 Berlin, Germany

Received  August 2012 Revised  December 2012 Published  December 2013

In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation $\Delta u=F'(u)$, which are monotone in some direction. In this paper we prove the analogous statement for the equation $\Delta u- \langle x,\nabla u\rangle u=F'(u)$, where the Laplacian is replaced by the Ornstein-Uhlenbeck operator. Our theorem holds without any restriction on the dimension of the ambient space, and this allows us to obtain an similar result in infinite dimensions by a limit procedure.
Citation: Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451
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. doi: 10.1023/A:1010602715526.  Google Scholar

[2]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbbR^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739. doi: 10.1090/S0894-0347-00-00345-3.  Google Scholar

[3]

L. Ambrosio, S. Maniglia, M. Miranda, Jr. and D. Pallara, BV functions in abstract Wiener spaces, J. Funct. Anal., 258 (2010), 785-813. doi: 10.1016/j.jfa.2009.09.008.  Google Scholar

[4]

H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 69-94.  Google Scholar

[5]

V. I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[6]

E. De Giorgi, Convergence problems for functionals and operators, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, 1979, 131-188.  Google Scholar

[7]

M. del Pino, M. Kowalczyk and J. Wei, On a conjecture by De Giorgi in dimensions 9 and higher, in Symmetry for Elliptic PDEs, Contemp. Math., 528, Amer. Math. Soc., Providence, RI, 2010, 115-137. doi: 10.1090/conm/528/10418.  Google Scholar

[8]

K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2), 130 (1989), 453-471. doi: 10.2307/1971452.  Google Scholar

[9]

A. Ehrhard, Symmetrization in Gaussian spaces, Math. Scand., 53 (1983), 281-301.  Google Scholar

[10]

A. Farina, Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires, Habilitation à Diriger des Recherches, Paris VI, 2002. Google Scholar

[11]

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.  Google Scholar

[12]

A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, 2009, 74-96. doi: 10.1142/9789812834744_0004.  Google Scholar

[13]

M. Goldman and M. Novaga, Approximation and relaxation of perimeter in the Wiener space, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 29 (2012), 525-544. doi: 10.1016/j.anihpc.2012.01.008.  Google Scholar

[14]

N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491. doi: 10.1007/s002080050196.  Google Scholar

[15]

M. Ledoux, A short proof of the Gaussian isoperimetric inequality, in High Dimensional Probability (Oberwolfach, 1996), Progr. Probab., 43, Birkhäuser, Basel, 1998, 229-232.  Google Scholar

[16]

A. Lunardi, On the Ornstein-Uhlenbeck operator in $L^2$ spaces with respect to invariant measures, Trans. Amer. Math. Soc., 349 (1997), 155-169. doi: 10.1090/S0002-9947-97-01802-3.  Google Scholar

[17]

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2), 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.  Google Scholar

[18]

P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.  Google Scholar

[19]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400. doi: 10.1007/s002050050081.  Google Scholar

[20]

L. Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata, 151 (2011), 297-303. doi: 10.1007/s10711-010-9535-2.  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), 9-33. doi: 10.1023/A:1010602715526.  Google Scholar

[2]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbbR^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739. doi: 10.1090/S0894-0347-00-00345-3.  Google Scholar

[3]

L. Ambrosio, S. Maniglia, M. Miranda, Jr. and D. Pallara, BV functions in abstract Wiener spaces, J. Funct. Anal., 258 (2010), 785-813. doi: 10.1016/j.jfa.2009.09.008.  Google Scholar

[4]

H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), 69-94.  Google Scholar

[5]

V. I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[6]

E. De Giorgi, Convergence problems for functionals and operators, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, 1979, 131-188.  Google Scholar

[7]

M. del Pino, M. Kowalczyk and J. Wei, On a conjecture by De Giorgi in dimensions 9 and higher, in Symmetry for Elliptic PDEs, Contemp. Math., 528, Amer. Math. Soc., Providence, RI, 2010, 115-137. doi: 10.1090/conm/528/10418.  Google Scholar

[8]

K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2), 130 (1989), 453-471. doi: 10.2307/1971452.  Google Scholar

[9]

A. Ehrhard, Symmetrization in Gaussian spaces, Math. Scand., 53 (1983), 281-301.  Google Scholar

[10]

A. Farina, Propriétés Qualitatives de Solutions d'Équations et Systèmes d'Équations Non-Linéaires, Habilitation à Diriger des Recherches, Paris VI, 2002. Google Scholar

[11]

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.  Google Scholar

[12]

A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, 2009, 74-96. doi: 10.1142/9789812834744_0004.  Google Scholar

[13]

M. Goldman and M. Novaga, Approximation and relaxation of perimeter in the Wiener space, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 29 (2012), 525-544. doi: 10.1016/j.anihpc.2012.01.008.  Google Scholar

[14]

N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491. doi: 10.1007/s002080050196.  Google Scholar

[15]

M. Ledoux, A short proof of the Gaussian isoperimetric inequality, in High Dimensional Probability (Oberwolfach, 1996), Progr. Probab., 43, Birkhäuser, Basel, 1998, 229-232.  Google Scholar

[16]

A. Lunardi, On the Ornstein-Uhlenbeck operator in $L^2$ spaces with respect to invariant measures, Trans. Amer. Math. Soc., 349 (1997), 155-169. doi: 10.1090/S0002-9947-97-01802-3.  Google Scholar

[17]

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2), 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.  Google Scholar

[18]

P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.  Google Scholar

[19]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400. doi: 10.1007/s002050050081.  Google Scholar

[20]

L. Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata, 151 (2011), 297-303. doi: 10.1007/s10711-010-9535-2.  Google Scholar

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