One-dimensional symmetry for semilinear equations with unbounded drift

Previous Article
Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation
CPAA Home
This Issue
Next Article
Existence of positive steady states for a predator-prey model with diffusion

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

One-dimensional symmetry for semilinear equations with unbounded drift

Annalisa Cesaroni ^1, , Matteo Novaga ^2, and Andrea Pinamonti ^3,

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

Abstract
Related Papers

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.

Keywords: semilinear elliptic PDEs, Symmetry and monotonicity properties of solutions, Ornstein-Uhlenbeck type operators, Liouville theorems, energy estimates, Poincaré-type inequality..

Mathematics Subject Classification: Primary: 35J61; Secondary: 35J20, 35B0.

Citation: Annalisa Cesaroni, Matteo Novaga, Andrea Pinamonti. One-dimensional symmetry for semilinear equations with unbounded drift. Communications on Pure and 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), 9-33. doi: 10.1023/A:1010602715526.
[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), 725-739. doi: 10.1090/S0894-0347-00-00345-3.
[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.
[4]	H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396. doi: 10.1215/S0012-7094-00-10331-6.
[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), 80-118. doi: 10.1137/S0036141097316391.
[6]	A. Cesaroni, M. Novaga and E. Valdinoci, A simmetry result for the Ornstein-Uhlenbech operator,, to appear on Discrete Contin. Dyn. Syst. A, ().
[7]	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.
[8]	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.
[9]	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).
[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), 1485-1569. doi: 10.4007/annals.2011.174.3.3.
[11]	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.
[12]	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.
[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), 741-791. doi: 10.2422/2036-2145.2008.4.06.
[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.
[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), 927-930. doi: 10.1090/S0002-9939-2011-11241-3.
[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.
[17]	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.
[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), 769-819. doi: 10.1080/03605300008821532.
[19]	A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $R^n$, Studia Mathematica, 128 (1998), 171-198.
[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), 616-636. doi: 10.1002/cpa.20014.
[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), 475-508. doi: 10.1007/s00205-007-0097-x.
[22]	O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.
[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), 357-373. doi: 10.5186/aasfm.2012.3733.
[24]	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.
[25]	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.
[26]	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.
[27]	J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains, Comm. Partial Differential Equations, 18 (1993), 505-531. doi: 10.1080/03605309308820939.

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.
[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), 725-739. doi: 10.1090/S0894-0347-00-00345-3.
[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.
[4]	H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396. doi: 10.1215/S0012-7094-00-10331-6.
[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), 80-118. doi: 10.1137/S0036141097316391.
[6]	A. Cesaroni, M. Novaga and E. Valdinoci, A simmetry result for the Ornstein-Uhlenbech operator,, to appear on Discrete Contin. Dyn. Syst. A, ().
[7]	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.
[8]	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.
[9]	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).
[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), 1485-1569. doi: 10.4007/annals.2011.174.3.3.
[11]	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.
[12]	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.
[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), 741-791. doi: 10.2422/2036-2145.2008.4.06.
[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.
[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), 927-930. doi: 10.1090/S0002-9939-2011-11241-3.
[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.
[17]	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.
[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), 769-819. doi: 10.1080/03605300008821532.
[19]	A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $R^n$, Studia Mathematica, 128 (1998), 171-198.
[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), 616-636. doi: 10.1002/cpa.20014.
[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), 475-508. doi: 10.1007/s00205-007-0097-x.
[22]	O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.
[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), 357-373. doi: 10.5186/aasfm.2012.3733.
[24]	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.
[25]	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.
[26]	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.
[27]	J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains, Comm. Partial Differential Equations, 18 (1993), 505-531. doi: 10.1080/03605309308820939.

[1]	Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete and 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 and Continuous Dynamical Systems, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451
[3]	Laura Baldelli, Roberta Filippucci. A priori estimates for elliptic problems via Liouville type theorems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1883-1898. doi: 10.3934/dcdss.2020148
[4]	Alessia E. Kogoj. A Zaremba-type criterion for hypoelliptic degenerate Ornstein–Uhlenbeck operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3491-3494. doi: 10.3934/dcdss.2020112
[5]	Foued Mtiri. Liouville type theorems for stable solutions of elliptic system involving the Grushin operator. Communications on Pure and Applied Analysis, 2022, 21 (2) : 541-553. doi: 10.3934/cpaa.2021187
[6]	Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022
[7]	Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
[8]	Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871
[9]	Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649
[10]	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
[11]	Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure and Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049
[12]	Sami Baraket, Soumaya Sâanouni, Nihed Trabelsi. Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1013-1063. doi: 10.3934/dcds.2020069
[13]	D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure and Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[14]	Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142
[15]	Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3199-3215. doi: 10.3934/dcdsb.2020058
[16]	Mondher Damak, Brice Franke, Nejib Yaakoubi. Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4093-4112. doi: 10.3934/dcds.2020173
[17]	Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651
[18]	Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic and Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013
[19]	Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399
[20]	Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure and Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Annalisa Cesaroni Matteo Novaga Andrea Pinamonti
on Google Scholar
Annalisa Cesaroni Matteo Novaga Andrea Pinamonti

[Back to Top]