A simple proof of the optimal power in Liouville theorems

  1. Villegas Barranco, Salvador 1
  1. 1 Universidad de Granada. Departamento de Análisis Matemático
Revista:
Publicacions matematiques

ISSN: 0214-1493

Año de publicación: 2022

Volumen: 66

Número: 2

Páginas: 883-892

Tipo: Artículo

DIALNET GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Publicacions matematiques

Resumen

Consider the equation div(ϕ2∇σ) = 0 in RN , where ϕ > 0. It is well known [4, 2] that if there exists C > 0 such that R BR (ϕσ) 2 dx ≤ CR2 for every R ≥ 1, then σ is necessarily constant. In this paper we present a simple proof that this result is not true if we replace R2 with Rk for k > 2 in any dimension N. This question is related to a conjecture by De Giorgi.

Referencias bibliográficas

  • G. Alberti, L. Ambrosio, and X. Cabre´, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday, Acta Appl. Math. 65(1-3) (2001), 9–33. DOI: 10.1023/A:1010602715526
  • L. Ambrosio and X. Cabre´, Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13(4) (2000), 725–739. DOI: 10.1090/S0894-0347-00-00345-3
  • M. T. Barlow, On the Liouville property for divergence form operators, Canad. J. Math. 50(3) (1998), 487–496. DOI: 10.4153/CJM-1998-026-9
  • H. Berestycki, L. Caffarelli, and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1-2) (1997), 69–94 (1998).
  • X. Cabre, E. Cinti, and J. Serra ´ . Stable solutions to the fractional Allen–Cahn equation in the nonlocal perimeter regime, Preprint (2021). arXiv:2111.06285
  • L. Caffarelli, N. Garofalo, and F. Segala, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47(11) (1994), 1457–1473. DOI: 10.1002/cpa.3160471103
  • E. De Giorgi, Convergence problems for functionals and operators, in: “Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis” (Rome, 1978), Pitagora, Bologna, 1979, pp. 131–188.
  • M. del Pino, M. Kowalczyk, and J. Wei, On De Giorgi’s conjecture in dimension N ≥ 9, Ann. of Math. (2) 174(3) (2011), 1485–1569. DOI: 10.4007/annals.2011.174.3.3
  • F. Gazzola, The sharp exponent for a Liouville-type theorem for an elliptic inequality, Rend. Istit. Mat. Univ. Trieste 34(1-2) (2002), 99–102 (2003).
  • N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311(3) (1998), 481–491. DOI: 10.1007/s002080050196
  • L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38(5) (1985), 679–684. DOI: 10.1002/cpa.3160380515
  • L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, in: “Partial Differential Equations and the Calculus of Variations”, Vol. II, Progr. Nonlinear Differential Equations Appl. 1, Birkhauser Boston, Boston, MA, 1989, pp. 843–850. DOI: 10.1007/978-1-4615-9831-2_14
  • A. Moradifam, Sharp counterexamples related to the De Giorgi conjecture in dimensions 4 ≤ n ≤ 8, Proc. Amer. Math. Soc. 142(1) (2014), 199–203. DOI: 10.1090/S0002-9939-2013-12040-X
  • O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169(1) (2009), 41–78. DOI: 10.4007/annals.2009.169.41
  • S. Villegas, Sharp Liouville theorems, Adv. Nonlinear Stud. 21(1) (2021), 95–105. DOI: 10.1515/ans-2020-2111
Fuente de los datos: Dialnet