# American Institute of Mathematical Sciences

September  2013, 12(5): 2203-2211. doi: 10.3934/cpaa.2013.12.2203

## One-dimensional symmetry for semilinear equations with unbounded drift

 1 Dip. di Matematica Pura e Applicata, Univ. di Padova, via Trieste 63, 35131 Padova 2 Università di Padova, Via Trieste 63, 35121 Padova 3 Dipartimento di Matematica, Università di Padova, Via Trieste 63, Padova, Italy

Received  June 2012 Revised  October 2012 Published  January 2013

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.
Citation: Annalisa Cesaroni, Matteo Novaga, Andrea Pinamonti. One-dimensional symmetry for semilinear equations with unbounded drift. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2203-2211. doi: 10.3934/cpaa.2013.12.2203
##### 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. doi: 10.1023/A:1010602715526. Google Scholar [2] 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. doi: 10.1090/S0894-0347-00-00345-3. Google Scholar [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. doi: item?id=ASNSP_1997_4_25_1-2_69_0. Google Scholar [4] H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations,, Duke Math. J., 103 (2000), 375. doi: 10.1215/S0012-7094-00-10331-6. Google Scholar [5] 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. doi: 10.1137/S0036141097316391. Google Scholar [6] A. Cesaroni, M. Novaga and E. Valdinoci, A simmetry result for the Ornstein-Uhlenbech operator,, to appear on Discrete Contin. Dyn. Syst. A, (). Google Scholar [7] C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations with weights,, Proc. Amer. Math. Soc., 140 (2012), 2003. doi: 10.1090/S0002-9939-2011-11351-0. Google Scholar [8] G. Da Prato and A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition,, J. Differential Equations, 198 (2004), 35. doi: 10.1016/j.jde.2003.10.025. Google Scholar [9] E. De Giorgi, Convergence problems for functionals and operators,, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1979), 131. Google Scholar [10] 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. doi: 10.4007/annals.2011.174.3.3. Google Scholar [11] L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities,, Nonlinear Anal., 70 (2009), 2882. doi: 10.1016/j.na.2008.12.017. Google Scholar [12] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u=f(u)$ in $R^N$,, J. Eur. Math. Soc., 12 (2010), 855. doi: 10.4171/JEMS/217. Google Scholar [13] 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. doi: 10.2422/2036-2145.2008.4.06. Google Scholar [14] A. Farina, Y. Sire and E. Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds,, to appear in J. Geom. Anal., (). doi: 10.1007/s12220-011-9278-9. Google Scholar [15] 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. doi: 10.1090/S0002-9939-2011-11241-3. Google Scholar [16] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, to appear in Calc. Var. Partial Differential Equations., (). doi: 10.1007/s00526-012-0536-x. Google Scholar [17] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems,, Math. Ann., 311 (1998), 481. doi: 10.1007/s002080050196. Google Scholar [18] 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. doi: 10.1080/03605300008821532. Google Scholar [19] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $R^n$,, Studia Mathematica, 128 (1998), 171. Google Scholar [20] 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. doi: 10.1002/cpa.20014. Google Scholar [21] 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. doi: 10.1007/s00205-007-0097-x. Google Scholar [22] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. doi: 10.4007/annals.2009.169.41. Google Scholar [23] 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. doi: 10.5186/aasfm.2012.3733. Google Scholar [24] J. M. Roquejoffre, Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders,, Ann. Inst. H. Poincar\e Anal. Non Lin\eaire, 14 (1997), 499. doi: 10.1016/S0294-1449(97)80137-0. Google Scholar [25] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081. Google Scholar [26] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces,, J. Reine Angew. Math., 503 (1998), 63. doi: 10.1515/crll.1998.100. Google Scholar [27] J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains,, Comm. Partial Differential Equations, 18 (1993), 505. doi: 10.1080/03605309308820939. 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. doi: 10.1023/A:1010602715526. Google Scholar [2] 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. doi: 10.1090/S0894-0347-00-00345-3. Google Scholar [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. doi: item?id=ASNSP_1997_4_25_1-2_69_0. Google Scholar [4] H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations,, Duke Math. J., 103 (2000), 375. doi: 10.1215/S0012-7094-00-10331-6. Google Scholar [5] 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. doi: 10.1137/S0036141097316391. Google Scholar [6] A. Cesaroni, M. Novaga and E. Valdinoci, A simmetry result for the Ornstein-Uhlenbech operator,, to appear on Discrete Contin. Dyn. Syst. A, (). Google Scholar [7] C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations with weights,, Proc. Amer. Math. Soc., 140 (2012), 2003. doi: 10.1090/S0002-9939-2011-11351-0. Google Scholar [8] G. Da Prato and A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition,, J. Differential Equations, 198 (2004), 35. doi: 10.1016/j.jde.2003.10.025. Google Scholar [9] E. De Giorgi, Convergence problems for functionals and operators,, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1979), 131. Google Scholar [10] 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. doi: 10.4007/annals.2011.174.3.3. Google Scholar [11] L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities,, Nonlinear Anal., 70 (2009), 2882. doi: 10.1016/j.na.2008.12.017. Google Scholar [12] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u=f(u)$ in $R^N$,, J. Eur. Math. Soc., 12 (2010), 855. doi: 10.4171/JEMS/217. Google Scholar [13] 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. doi: 10.2422/2036-2145.2008.4.06. Google Scholar [14] A. Farina, Y. Sire and E. Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds,, to appear in J. Geom. Anal., (). doi: 10.1007/s12220-011-9278-9. Google Scholar [15] 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. doi: 10.1090/S0002-9939-2011-11241-3. Google Scholar [16] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems,, to appear in Calc. Var. Partial Differential Equations., (). doi: 10.1007/s00526-012-0536-x. Google Scholar [17] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems,, Math. Ann., 311 (1998), 481. doi: 10.1007/s002080050196. Google Scholar [18] 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. doi: 10.1080/03605300008821532. Google Scholar [19] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $R^n$,, Studia Mathematica, 128 (1998), 171. Google Scholar [20] 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. doi: 10.1002/cpa.20014. Google Scholar [21] 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. doi: 10.1007/s00205-007-0097-x. Google Scholar [22] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. doi: 10.4007/annals.2009.169.41. Google Scholar [23] 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. doi: 10.5186/aasfm.2012.3733. Google Scholar [24] J. M. Roquejoffre, Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders,, Ann. Inst. H. Poincar\e Anal. Non Lin\eaire, 14 (1997), 499. doi: 10.1016/S0294-1449(97)80137-0. Google Scholar [25] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081. Google Scholar [26] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces,, J. Reine Angew. Math., 503 (1998), 63. doi: 10.1515/crll.1998.100. Google Scholar [27] J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains,, Comm. Partial Differential Equations, 18 (1993), 505. doi: 10.1080/03605309308820939. Google Scholar
 [1] Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525 [2] Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451 [3] Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869 [4] Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871 [5] Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649 [6] Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206 [7] Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049 [8] D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499 [9] Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142 [10] Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651 [11] Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013 [12] Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399 [13] Nemat Nyamoradi, Kaimin Teng. Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Communications on Pure & Applied Analysis, 2015, 14 (2) : 361-371. doi: 10.3934/cpaa.2015.14.361 [14] Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Networks & Heterogeneous Media, 2012, 7 (4) : 967-988. doi: 10.3934/nhm.2012.7.967 [15] Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665 [16] Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011 [17] Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017 [18] Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719 [19] Linfen Cao, Wenxiong Chen. Liouville type theorems for poly-harmonic Navier problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3937-3955. doi: 10.3934/dcds.2013.33.3937 [20] Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607

2018 Impact Factor: 0.925