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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

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 & 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

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