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Some results on minimizers and stable solutions of a variational problem

Published online by Cambridge University Press:  10 June 2011

ALBERTO FARINA  and
ENRICO VALDINOCI
ALBERTO FARINA
Affiliation:
LAMFA – CNRS UMR 6140, Université de Picardie Jules Verne, Faculté des Sciences, 33, rue Saint-Leu, 80039 Amiens CEDEX 1, France (email: [email protected])
ENRICO VALDINOCI
Affiliation:
Università di Roma Tor Vergata, Dipartimento di Matematica, via della ricerca scientifica, 1, I-00133 Rome, Italy (email: [email protected])
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Abstract

We consider the functional in a periodic setting. We discuss whether the minimizers or the stable solutions satisfy some symmetry or monotonicity properties, with special emphasis on the autonomous case when F is x-independent. In particular, we give an answer to a question posed by Victor Bangert when F is autonomous in dimension n≤3 and in any dimension for non-zero rotation vectors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1302 - 1312
Copyright
Copyright © Cambridge University Press 2011

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References

[AAC01]Alberti, G., Ambrosio, L. and Cabré, X.. On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math. 65(1–3) (2001), 9–33, special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday.CrossRefGoogle Scholar
[AC00]Ambrosio, L. and Cabré, X.. Entire solutions of semilinear elliptic equations in ℝ3 and a conjecture of De Giorgi. J. Amer. Math. Soc. 13(4) (2000), 725–739 (electronic).CrossRefGoogle Scholar
[AJM02]Alessio, F., Jeanjean, L. and Montecchiari, P.. Existence of infinitely many stationary layered solutions in ℝ2 for a class of periodic Allen–Cahn equations. Comm. Partial Differential Equations 27(7–8) (2002), 1537–1574.CrossRefGoogle Scholar
[AM05]Alessio, F. and Montecchiari, P.. Entire solutions in ℝ2 for a class of Allen–Cahn equations. ESAIM Control Optim. Calc. Var. 11(4) (2005), 633–672 (electronic).CrossRefGoogle Scholar
[Aub83]Aubry, S.. The twist map, the extended Frenkel–Kontorova model and the devil’s staircase. Phys. D 7(1–3) (1983), 240–258.CrossRefGoogle Scholar
[Aue01]Auer, F.. Uniqueness of least area surfaces in the 3-torus. Math. Z. 238(1) (2001), 145–176.CrossRefGoogle Scholar
[Ban89]Bangert, V.. On minimal laminations of the torus. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(2) (1989), 95–138.CrossRefGoogle Scholar
[Ban90]Bangert, V.. Laminations of 3-tori by least area surfaces. Analysis, et cetera. Academic Press, Boston, MA, 1990, pp. 85–114.CrossRefGoogle Scholar
[BCN97]Berestycki, H., Caffarelli, L. and Nirenberg, L.. Further qualitative properties for elliptic equations in unbounded domains. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 25(1–2) (1998), 69–94, (1997), dedicated to Ennio De Giorgi.Google Scholar
[Bes05]Bessi, U.. Many solutions of elliptic problems on ℝn of irrational slope. Comm. Partial Differential Equations 30(10–12) (2005), 1773–1804.CrossRefGoogle Scholar
[CC95]Caffarelli, L. A. and Córdoba, A.. Uniform convergence of a singular perturbation problem. Comm. Pure Appl. Math. 48(1) (1995), 1–12.CrossRefGoogle Scholar
[CdlL01]Caffarelli, L. A. and de la Llave, R.. Planelike minimizers in periodic media. Comm. Pure Appl. Math. 54(12) (2001), 1403–1441.CrossRefGoogle Scholar
[DG79]De Giorgi, E.. Convergence problems for functionals and operators. Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978). Pitagora, Bologna, 1979, pp. 131–188.Google Scholar
[dlLV07]de la Llave, R. and Valdinoci, E.. Multiplicity results for interfaces of Ginzburg–Landau–Allen–Cahn equations in periodic media. Adv. Math. 215(1) (2007), 379–426.CrossRefGoogle Scholar
[dPKW08]del Pino, M., Kowalczyk, M. and Wei, J.. A counterexample to a conjecture by De Giorgi in large dimensions. C. R. Math. Acad. Sci. Paris 346(23–24) (2008), 1261–1266.CrossRefGoogle Scholar
[dPKW09]del Pino, M., Kowalczyk, M. and Wei, J.. On De Giorgi conjecture in dimension N≥9. Preprint, 2009, http://eprintweb.org/S/article/math/0806.3141.Google Scholar
[Far07]Farina, A.. Liouville-type theorems for elliptic problems. Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. IV. Elsevier/North-Holland, Amsterdam, 2007, pp. 61–116.CrossRefGoogle Scholar
[FCS80]Fischer-Colbrie, D. and Schoen, R.. The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 33(2) (1980), 199–211.CrossRefGoogle Scholar
[FSV08]Farina, A., Sciunzi, B. and Valdinoci, E.. Bernstein and De Giorgi type problems: new results via a geometric approach. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7(4) (2008), 741–791.Google Scholar
[FV09]Farina, A. and Valdinoci, E.. 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 Publishers, Hackensack, NJ, 2009, pp. 74–96.CrossRefGoogle Scholar
[FV11]Farina, A. and Valdinoci, E.. 1D symmetry for solutions of semilinear and quasilinear elliptic equations. Trans. Amer. Math. Soc. 363(2) (2011), 579–609.CrossRefGoogle Scholar
[GG98]Ghoussoub, N. and Gui, C.. On a conjecture of De Giorgi and some related problems. Math. Ann. 311(3) (1998), 481–491.CrossRefGoogle Scholar
[JGV09]Junginger-Gestrich, H. and Valdinoci, E.. Some connections between results and problems of De Giorgi, Moser and Bangert. Z. Angew. Math. Phys. 60(3) (2009), 393–401.CrossRefGoogle Scholar
[Mat82]Mather, J. N.. Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology 21(4) (1982), 457–467.CrossRefGoogle Scholar
[Mos86]Moser, J.. Minimal solutions of variational problems on a torus. Ann. Inst. H. Poincaré Anal. Non Linéaire 3(3) (1986), 229–272.CrossRefGoogle Scholar
[MP78]Moss, W. F. and Piepenbrink, J.. Positive solutions of elliptic equations. Pacific J. Math. 75(1) (1978), 219–226.CrossRefGoogle Scholar
[Rab04]Rabinowitz, P. H.. A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE. Discrete Contin. Dyn. Syst. 10(1–2) (2004), 507–515.CrossRefGoogle Scholar
[RS03]Rabinowitz, P. H. and Stredulinsky, E.. Mixed states for an Allen–Cahn type equation. Comm. Pure Appl. Math. 56(8) (2003), 1078–1134.CrossRefGoogle Scholar
[RS04]Rabinowitz, P. H. and Stredulinsky, E.. On some results of Moser and of Bangert. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(5) (2004), 673–688.CrossRefGoogle Scholar
[Sav09]Savin, O.. Regularity of flat level sets in phase transitions. Ann. of Math. (2) 169(1) (2009), 41–78.CrossRefGoogle Scholar
[Val04]Valdinoci, E.. Plane-like minimizers in periodic media: jet flows and Ginzburg–Landau-type functionals. J. Reine Angew. Math. 574 (2004), 147–185.Google Scholar