Abstract
In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11971026
Award Identifier / Grant number: 11871381
Funding statement: François Hamel is partially supported by the Excellence Initiative of Aix-Marseille University – A*MIDEX, a French “Investissements d’Avenir” programme, the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) ERC Grant Agreement no. 321186 – ReaDi – Reaction-Diffusion Equations, Propagation and Modelling, and the ANR NONLOCAL project (ANR-14-CE25-0013). Yong Liu is partially supported by “The Fundamental Research Funds for the Central Universities WK3470000014”and NSFC grant no. 11971026. Pieralberto Sicbaldi is partially supported by the grant “Ramón y Cajal 2015” RYC-2015-18730 and the grant “Analisis geométrico” MTM 2017-89677-P. Kelei Wang is supported by NSFC grant no. 11871381. Juncheng Wei is partially supported by NSERC of Canada.
Acknowledgements
Part of the paper was finished while Y. Liu was visiting the University of British Columbia in 2019, and he appreciates the institution for its hospitality and financial support. Part of this work was also completed while F. Hamel and J. Wei were visiting the University of Granada in 2019 in occasion of the conference “Geometry and PDE in front of the Alhambra”, and they also appreciate the institution for the hospitality and the financial support. Finally, part of this work has been carried out in the framework of Archimède Labex of Aix-Marseille University. The authors are also grateful to Alberto Farina for pointing out his results [14, 16] and their relation with our paper, and to the referee for valuable suggestions of improvement of the manuscript.
References
[1] O. Agudelo, M. del Pino and J. Wei, Higher-dimensional catenoid, Liouville equation, and Allen–Cahn equation, Int. Math. Res. Not. IMRN (2016), no. 23, 7051–7102. 10.1093/imrn/rnv350Search in Google Scholar
[2] 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. 10.1023/A:1010602715526Search in Google Scholar
[3]
L. Ambrosio and X. Cabré,
Entire solutions of semilinear elliptic equations in
[4] H. Berestycki, L. A. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 5 (1997), 69–94. Search in Google Scholar
[5] H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math. 50 (1997), no. 11, 1089–1111. 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6Search in Google Scholar
[6] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, Contemp. Math. 446, American Mathematical Society, Providence (2007), 101–123. 10.1090/conm/446/08627Search in Google Scholar
[7] X. Cabré, Uniqueness and stability of saddle-shaped solutions to the Allen–Cahn equation, J. Math. Pures Appl. (9) 98 (2012), no. 3, 239–256. 10.1016/j.matpur.2012.02.006Search in Google Scholar
[8] L. Caffarelli, N. Garofalo and F. Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), no. 11, 1457–1473. 10.1002/cpa.3160471103Search in Google Scholar
[9] T. H. Colding and W. P. Minicozzi, II, The Calabi–Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), no. 1, 211–243. 10.4007/annals.2008.167.211Search in Google Scholar
[10] E. De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome 1978), Pitagora, Bologna (1979), 131–188. Search in Google Scholar
[11] D. De Silva and O. Savin, Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2D, Indiana Univ. Math. J. 58 (2009), no. 1, 301–315. 10.1512/iumj.2009.58.3396Search in Google Scholar
[12] M. del Pino, M. Kowalczyk and F. Pacard, Moduli space theory for the Allen–Cahn equation in the plane, Trans. Amer. Math. Soc. 365 (2013), no. 2, 721–766. 10.1090/S0002-9947-2012-05594-2Search in Google Scholar
[13]
M. del Pino, M. Kowalczyk and J. Wei,
On De Giorgi’s conjecture in dimension
[14]
A. Farina,
Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg–Landau systems in
[15]
A. Farina,
Symmetry for solutions of semilinear elliptic equations in
[16]
A. Farina,
Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of
[17] 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. (5) 7 (2008), no. 4, 741–791. 10.2422/2036-2145.2008.4.06Search in Google Scholar
[18] A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, Recent progress on reaction-diffusion systems and viscosity solutions, World Scientific, Hackensack (2009), 74–96. 10.1142/9789812834744_0004Search in Google Scholar
[19] A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. Ration. Mech. Anal. 195 (2010), no. 3, 1025–1058. 10.1007/s00205-009-0227-8Search in Google Scholar
[20] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481–491. 10.1007/s002080050196Search in Google Scholar
[21] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. 10.1007/BF01221125Search in Google Scholar
[22] D. Hoffman and W. H. Meeks, III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373–377. 10.1007/BF01231506Search in Google Scholar
[23]
Y. Liu, K. Wang and J. Wei,
Global minimizers of the Allen–Cahn equation in dimension
[24]
Y. Liu, K. Wang and J. Wei,
On smooth solutions to one phase free boundary problem in
[25] W. H. Meeks, III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362. 10.4310/jdg/1463404121Search in Google Scholar
[26] W. H. Meeks, III and H. Rosenberg, Maximum principles at infinity, J. Differential Geom. 79 (2008), no. 1, 141–165. 10.4310/jdg/1207834660Search in Google Scholar
[27] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 679–684. 10.1002/cpa.3160380515Search in Google Scholar
[28] O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169 (2009), no. 1, 41–78. 10.4007/annals.2009.169.41Search in Google Scholar
[29] R. P. Sperb, Maximum principles and their applications, Math. Sci. Eng. 157, Academic Press, New York 1981. Search in Google Scholar
[30] F. Xavier, Convex hulls of complete minimal surfaces, Math. Ann. 269 (1984), no. 2, 179–182. 10.1007/BF01451417Search in Google Scholar
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