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 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.

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