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A symmetry result for the Ornstein-Uhlenbeck operator

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  • 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.
    Mathematics Subject Classification: Primary: 35J15, 35J20; Secondary: 35J61.


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  • [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.


    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.


    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.


    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.


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


    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.


    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.


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


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


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


    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.


    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.


    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.


    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.


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


    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.


    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.


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


    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.


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

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