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One-dimensional symmetry for semilinear equations with unbounded drift

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  • We consider semilinear equations with unbounded drift in the whole of $R^n$ and we show that monotone solutions with finite energy are one-dimensional.
    Mathematics Subject Classification: Primary: 35J61; Secondary: 35J20, 35B06.


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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 $R^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739.doi: 10.1090/S0894-0347-00-00345-3.


    H. Berestycki, L. A. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1998), 69-94.doi: item?id=ASNSP_1997_4_25_1-2_69_0.


    H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.doi: 10.1215/S0012-7094-00-10331-6.


    A. Bonnet and F. Hamel, Existence of non-planar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118.doi: 10.1137/S0036141097316391.


    A. Cesaroni, M. Novaga and E. ValdinociA simmetry result for the Ornstein-Uhlenbech operator, to appear on Discrete Contin. Dyn. Syst. A, arXiv:1204.0880v2.


    C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.doi: 10.1090/S0002-9939-2011-11351-0.


    G. Da Prato and A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition, J. Differential Equations, 198 (2004), 35-52.doi: 10.1016/j.jde.2003.10.025.


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


    M. del Pino, M. Kowalczyk and J. Wei, On a conjecture by De Giorgi in dimensions 9 and higher, Ann. of Math., 174 (2011), 1485-1569.doi: 10.4007/annals.2011.174.3.3.


    L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities, Nonlinear Anal., 70 (2009), 2882-2888.doi: 10.1016/j.na.2008.12.017.


    L. Dupaigne and A. Farina, Stable solutions of $-\Delta u=f(u)$ in $R^N$, J. Eur. Math. Soc., 12 (2010), 855-882.doi: 10.4171/JEMS/217.


    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., 7 (2008), 741-791.doi: 10.2422/2036-2145.2008.4.06.


    A. Farina, Y. Sire and E. ValdinociStable solutions of elliptic equations on Riemannian manifolds, to appear in J. Geom. Anal. doi: 10.1007/s12220-011-9278-9.


    A. Farina, Y. Sire and E. Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds with Euclidean coverings, Proc. Amer. Math. Soc., 140 (2012), 927-930.doi: 10.1090/S0002-9939-2011-11241-3.


    M. Fazly and N. GhoussoubDe Giorgi type results for elliptic systems, to appear in Calc. Var. Partial Differential Equations. doi: 10.1007/s00526-012-0536-x.


    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.


    F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in $R^N$ with conical-shaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819.doi: 10.1080/03605300008821532.


    A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $R^n$, Studia Mathematica, 128 (1998), 171-198.


    M. Lucia, C. B. Muratov and M. Novaga, Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium, Comm. Pure Appl. Math., 57 (2004), 616-636.doi: 10.1002/cpa.20014.


    M. Lucia, C. B. Muratov and M. Novaga, Existence of traveling wave solutions for Ginzburg-Landau-type problems in infinite cylinders, Arch. Ration. Mech. Anal., 188 (2008), 475-508.doi: 10.1007/s00205-007-0097-x.


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


    A. Pinamonti and E. Valdinoci, A geometric inequality for stable solutions of semilinear elliptic problems in the Engel group, Ann. Acad. Sci. Fenn. Math., 37 (2012), 357-373.doi: 10.5186/aasfm.2012.3733.


    J. M. Roquejoffre, Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. H. Poincarè Anal. Non Linèaire, 14 (1997), 499-552.doi: 10.1016/S0294-1449(97)80137-0.


    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.


    P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.doi: 10.1515/crll.1998.100.


    J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains, Comm. Partial Differential Equations, 18 (1993), 505-531.doi: 10.1080/03605309308820939.

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