A Generalization of Aubry-Mather Theory to Partial Differential Equations and Pseudo-Differential Equations

De La Llave, Rafael; Valdinoci, Enrico

doi:10.1016/j.anihpc.2008.11.002

De La Llave, Rafael ; Valdinoci, Enrico

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 4, pp. 1309-1344.

MR Zbl | 2 citations dans Numdam

DOI : 10.1016/j.anihpc.2008.11.002

@article{AIHPC_2009__26_4_1309_0,
     author = {De La Llave, Rafael and Valdinoci, Enrico},
     title = {A {Generalization} of {Aubry-Mather} {Theory} to {Partial} {Differential} {Equations} and {Pseudo-Differential} {Equations}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1309--1344},
     publisher = {Elsevier},
     volume = {26},
     number = {4},
     year = {2009},
     doi = {10.1016/j.anihpc.2008.11.002},
     mrnumber = {2542727},
     zbl = {1171.35372},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2008.11.002/}
}

TY  - JOUR
AU  - De La Llave, Rafael
AU  - Valdinoci, Enrico
TI  - A Generalization of Aubry-Mather Theory to Partial Differential Equations and Pseudo-Differential Equations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2009
SP  - 1309
EP  - 1344
VL  - 26
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2008.11.002/
DO  - 10.1016/j.anihpc.2008.11.002
LA  - en
ID  - AIHPC_2009__26_4_1309_0
ER  -

%0 Journal Article
%A De La Llave, Rafael
%A Valdinoci, Enrico
%T A Generalization of Aubry-Mather Theory to Partial Differential Equations and Pseudo-Differential Equations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2009
%P 1309-1344
%V 26
%N 4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2008.11.002/
%R 10.1016/j.anihpc.2008.11.002
%G en
%F AIHPC_2009__26_4_1309_0

De La Llave, Rafael; Valdinoci, Enrico. A Generalization of Aubry-Mather Theory to Partial Differential Equations and Pseudo-Differential Equations. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 4, pp. 1309-1344. doi : 10.1016/j.anihpc.2008.11.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2008.11.002/

Bibliographie
Cité par

[1] Alberti G., Bellettini G., A Nonlocal Anisotropic Model for Phase Transitions. I. the Optimal Profile Problem, Math. Ann. 310 (3) (1998) 527-560. | MR | Zbl

[2] Angenent S. B., Monotone Recurrence Relations, Their Birkhoff Orbits and Topological Entropy, Ergodic Theory Dynam. Systems 10 (1) (1990) 15-41. | MR | Zbl

[3] Appell J., Zabrejko P. P., Nonlinear Superposition Operators, Cambridge Tracts in Math., vol. 95, Cambridge University Press, Cambridge, 1990. | MR | Zbl

[4] Bangert V., The Existence of Gaps in Minimal Foliations, Aequationes Math. 34 (2-3) (1987) 153-166. | MR | Zbl

[5] Bangert V., On Minimal Laminations of the Torus, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (2) (1989) 95-138. | Numdam | MR | Zbl

[6] Bates P. W., On Some Nonlocal Evolution Equations Arising in Materials Science, in: Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., vol. 48, Amer. Math. Soc., Providence, RI, 2006, pp. 13-52. | MR | Zbl

[7] Bellettini G., Buttà P., Presutti E., Sharp Interface Limits for Non-Local Anisotropic Interactions, Arch. Ration. Mech. Anal. 159 (2) (2001) 109-135. | MR | Zbl

[8] Berti M., Matzeu M., Valdinoci E., On Periodic Elliptic Equations With Gradient Dependence, Comm. Pure Appl. Anal. 7 (3) (2008) 601-615. | MR

[9] Bessi U., Many Solutions of Elliptic Problems on $R^{n}$ of Irrational Slope, Comm. Partial Differential Equations 30 (10-12) (2005) 1773-1804. | MR | Zbl

[10] Birindelli I., Valdinoci E., The Ginzburg-Landau Equation in the Heisenberg Group, Commun. Contemp. Math. 10 (5) (2008) 1-50. | MR | Zbl

[11] Cabré X., Solà-Morales J., Layer Solutions in a Half-Space for Boundary Reactions, Comm. Pure Appl. Math. 58 (12) (2005) 1678-1732. | MR | Zbl

[12] Caffarelli L. A., Further Regularity for the Signorini Problem, Comm. Partial Differential Equations 4 (9) (1979) 1067-1075. | MR | Zbl

[13] Caffarelli L. A., A Note on Nonlinear Homogenization, Comm. Pure Appl. Math. 52 (7) (1999) 829-838. | MR | Zbl

[14] Caffarelli L. A., Cabré X., Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995. | MR | Zbl

[15] Caffarelli L. A., De La Llave R., Planelike Minimizers in Periodic Media, Comm. Pure Appl. Math. 54 (12) (2001) 1403-1441. | MR | Zbl

[16] Caffarelli L. A., De La Llave R., Interfaces of Ground States in Ising Models With Periodic Coefficients, J. Stat. Phys. 118 (3-4) (2005) 687-719. | MR | Zbl

[17] Candel A., De La Llave R., On the Aubry-Mather Theory in Statistical Mechanics, Comm. Math. Phys. 192 (3) (1998) 649-669. | MR | Zbl

[18] Cassandro M., Orlandi E., Picco P., The Optimal Interface Profile for a Non-Local Model of Phase Separation, Nonlinearity 15 (5) (2002) 1621-1651. | MR | Zbl

[19] Chorin A. J., Mccracken M. F., Hughes T. J.R., Marsden J. E., Product Formulas and Numerical Algorithms, Comm. Pure Appl. Math. 31 (2) (1978) 205-256. | MR | Zbl

[20] Collins D. J., Zieschang H., Combinatorial Group Theory and Fundamental Groups, in: Algebra, VII, Springer, Berlin, 1993, pp. 1-166, 233-240. | MR | Zbl

[21] Cordoba D., Nonexistence of Simple Hyperbolic Blow-Up for the Quasi-Geostrophic Equation, Ann. of Math. (2) 148 (3) (1998) 1135-1152. | MR | Zbl

[22] Courant R., Hilbert D., Methods of Mathematical Physics. Vol. II: Partial Differential Equations, Vol. II by R. Courant, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. | MR | Zbl

[23] Craig W., Groves M. D., Hamiltonian Long-Wave Approximations to the Water-Wave Problem, Wave Motion 19 (4) (1994) 367-389. | MR | Zbl

[24] De Cristoforis M. L., Differentiability Properties of the Autonomous Composition Operator in Sobolev Spaces, Z. Anal. Anwend. 16 (3) (1997) 631-651. | MR | Zbl

[25] De La Llave R., Valdinoci E., Critical Points Inside the Gaps of Ground State Laminations for Some Models in Statistical Mechanics, J. Stat. Phys. 129 (1) (2007) 81-119. | MR | Zbl

[26] De La Llave R., Valdinoci E., Ground States and Critical Points for Generalized Frenkel-Kontorova Models in $Z^{d}$ , Nonlinearity 20 (10) (2007) 2409-2424. | MR

[27] De La Llave R., Valdinoci E., Multiplicity Results for Interfaces of Ginzburg-Landau-Allen-Cahn Equations in Periodic Media, Adv. Math. 215 (1) (2007) 379-426. | MR | Zbl

[28] R. de la Llave, E. Valdinoci, Ground states and critical points for Aubry-Mather theory in statistical mechanics, J. Nonlinear Sci., 2009, in press.

[29] Dibenedetto E., $C^{1 + α}$ Local Regularity of Weak Solutions of Degenerate Elliptic Equations, Nonlinear Anal. 7 (8) (1983) 827-850. | MR | Zbl

[30] Dibenedetto E., Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. | MR | Zbl

[31] Dibenedetto E., Partial Differential Equations, Birkhäuser Boston Inc., Boston, MA, 1995. | MR

[32] Duistermaat J. J., Guillemin V. W., The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics, Invent. Math. 29 (1) (1975) 39-79. | MR | Zbl

[33] E W., Aubry-Mather Theory and Periodic Solutions of the Forced Burgers Equation, Comm. Pure Appl. Math. 52 (7) (1999) 811-828. | MR | Zbl

[34] Fathi A., Théorème KAM Faible Et Théorie De Mather Sur Les Systèmes Lagrangiens, C. R. Acad. Sci. Paris Sér. I Math. 324 (9) (1997) 1043-1046. | MR | Zbl

[35] Fefferman C., De La Llave R., Relativistic Stability of Matter. I, Rev. Mat. Iberoamericana 2 (1-2) (1986) 119-213. | MR | Zbl

[36] Friedman A., Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, NJ, 1964. | MR | Zbl

[37] Friedman A., Partial Differential Equations, original ed., Robert E. Krieger Publishing Co., Huntington, NY, 1976. | MR

[38] Garroni A., Palatucci G., A Singular Perturbation Result With a Fractional Norm, in: Variational Problems in Materials Science, Progr. Nonlinear Differential Equations Appl., vol. 68, Birkhäuser, Basel, 2006, pp. 111-126. | MR | Zbl

[39] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., vol. 224, second ed., Springer-Verlag, Berlin, 1983. | MR | Zbl

[40] Golé C., A New Proof of the Aubry-Mather's Theorem, Math. Z. 210 (3) (1992) 441-448. | MR | Zbl

[41] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin, 1981. | MR | Zbl

[42] Heinonen J., Kilpeläinen T., Martio O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications Inc., Mineola, NY, 2006, Unabridged republication of the 1993 original. | MR | Zbl

[43] Hille E., Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ., vol. 31, American Mathematical Society, New York, 1948. | MR | Zbl

[44] Koch H., De La Llave R., Radin C., Aubry-Mather Theory for Functions on Lattices, Discrete Contin. Dynam. Systems 3 (1) (1997) 135-151. | MR | Zbl

[45] Komatsu H., Fractional Powers of Operators. II. Interpolation Spaces, Pacific J. Math. 21 (1967) 89-111. | MR | Zbl

[46] Kozlov S. M., Ground States of Quasiperiodic Operators, Dokl. Akad. Nauk SSSR 271 (3) (1983) 532-536. | MR | Zbl

[47] Kozlov S. M., Reducibility of Quasiperiodic Differential Operators and Averaging, Tr. Mosk. Mat. Obs. 46 (1983) 99-123. | MR | Zbl

[48] Krylov N. V., Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Mathematical Society, Providence, RI, 1996. | MR | Zbl

[49] Landkof N. S., Foundations of Modern Potential Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 180, Springer-Verlag, New York, 1972, Translated from the Russian by A.P. Doohovskoy. | MR | Zbl

[50] Ladyženskaja O. A., Solonnikov V. A., Ural'Tseva N. N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, American Mathematical Society, Providence, RI, 1967, Translated from the Russian by S. Smith. | MR | Zbl

[51] Lieb E. H., Loss M., Analysis, Grad. Stud. Math., vol. 14, American Mathematical Society, Providence, RI, 1997. | MR | Zbl

[52] Lions J.-L., Quelques Méthodes De Résolution Des Problèmes Aux Limites Non-Linéaires, Dunod, 1969. | MR | Zbl

[53] Lunardi A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progr. Nonlinear Differential Equations Appl., vol. 16, Birkhäuser Verlag, Basel, 1995. | MR | Zbl

[54] Majda A. J., Tabak E. G., A Two-Dimensional Model for Quasigeostrophic Flow: Comparison With the Two-Dimensional Euler Flow, Phys. D 98 (2-4) (1996) 515-522. | MR | Zbl

[55] Mather J. N., Existence of Quasiperiodic Orbits for Twist Homeomorphisms of the Annulus, Topology 21 (4) (1982) 457-467. | MR | Zbl

[56] Mather J. N., Variational Construction of Connecting Orbits, Ann. Inst. Fourier (Grenoble) 43 (5) (1993) 1349-1386. | Numdam | MR | Zbl

[57] J.N. Mather, Graduate course at Princeton, 95-96, and Lectures at Penn State, Spring 96, Paris, Summer 96, Austin, Fall 96.

[58] Moser J., Minimal Solutions of Variational Problems on a Torus, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (3) (1986) 229-272. | Numdam | MR | Zbl

[59] Moser J., Monotone Twist Mappings and the Calculus of Variations, Ergodic Theory Dynam. Systems 6 (3) (1986) 401-413. | MR | Zbl

[60] Moser J., A Stability Theorem for Minimal Foliations on a Torus, Ergodic Theory Dynamical Systems 8* (Charles Conley Memorial Issue) (1988) 251-281. | MR | Zbl

[61] Moser J., Minimal Foliations on a Torus, in: Topics in Calculus of Variations, Montecatini Terme, 1987, Springer, Berlin, 1989, pp. 62-99. | MR | Zbl

[62] Moser J., Quasi-Periodic Solutions of Nonlinear Elliptic Partial Differential Equations, Bol. Soc. Brasil. Mat. (N.S.) 20 (1) (1989) 29-45. | MR | Zbl

[63] Naumkin P. I., Shishmarëv I. A., Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, Providence, RI, 1994, Translated from the Russian manuscript by Boris Gommerstadt. | MR | Zbl

[64] Novaga M., Valdinoci E., The Geometry of Mesoscopic Phase Transition Interfaces, Discrete Contin. Dynam. Systems 19 (4) (2007) 777-798. | MR | Zbl

[65] Novaga M., Valdinoci E., Multibump Solutions and Asymptotic Expansions for Mesoscopic Allen-Cahn Type Equations, ESAIM Control Optim. Calc. Var., in press, 2009, DOI: 10.1051/cocv:2008058, http://www.esaim-cocv.org/index.php?option=toc&url=/articles/cocv/abs/first/contents/contents.html. | Numdam | MR

[66] Petrosyan A., Valdinoci E., Density Estimates for a Degenerate/Singular Phase-Transition Model, SIAM J. Math. Anal. 36 (4) (2005) 1057-1079, (electronic). | MR | Zbl

[67] Pollard H., The Representation of $e^{- x^{λ}}$ as a Laplace Integral, Bull. Amer. Math. Soc. 52 (1946) 908-910. | MR | Zbl

[68] Protter M. H., Weinberger H. F., Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984, Corrected reprint of the 1967 original. | MR | Zbl

[69] Rabinowitz P. H., Stredulinsky E., Mixed States for an Allen-Cahn Type Equation, Comm. Pure Appl. Math. 56 (8) (2003) 1078-1134. | MR

[70] Rabinowitz P. H., Stredulinsky E., On Some Results of Moser and of Bangert, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (5) (2004) 673-688. | Numdam | MR | Zbl

[71] Rabinowitz P. H., Stredulinsky E., Mixed States for an Allen-Cahn Type Equation. II, Calc. Var. Partial Differential Equations 21 (2) (2004) 157-207. | MR | Zbl

[72] Rabinowitz P. H., Stredulinsky E., On Some Results of Moser and of Bangert. II, Adv. Nonlinear Stud. 4 (4) (2004) 377-396. | MR

[73] Reed M., Simon B., Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1972. | MR | Zbl

[74] Shen W., Yi Y., Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Mem. Amer. Math. Soc. 136 (647) (1998) x+93. | MR | Zbl

[75] R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, Electronic Monographs in Differential Equations, San Marcos, TX, 1994. Electronic reprint of the 1977 original. | MR | Zbl

[76] Showalter R. E., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, RI, 1997. | MR | Zbl

[77] Y. Sire, E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, preprint, 2008. | MR | Zbl

[78] Slijepčević S., Monotone Gradient Dynamics and Mather's Shadowing, Nonlinearity 12 (4) (1999) 969-986. | MR | Zbl

[79] Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970. | MR | Zbl

[80] Taylor M. E., Partial Differential Equations. I, Basic Theory, Springer-Verlag, New York, 1996. | MR | Zbl

[81] Taylor M. E., Partial Differential Equations. III. Nonlinear Equations, Springer-Verlag, New York, 1997, Corrected reprint of the 1996 original. | MR | Zbl

[82] Tolksdorf P., Regularity for a More General Class of Quasilinear Elliptic Equations, J. Differential Equations 51 (1) (1984) 126-150. | MR | Zbl

[83] Valdinoci E., Plane-Like Minimizers in Periodic Media: Jet Flows and Ginzburg-Landau-Type Functionals, J. Reine Angew. Math. 574 (2004) 147-185. | MR

[84] E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau, PhD thesis, University of Texas at Austin, 2001, MP_ARC. | MR

[85] Weinberger H. F., A First Course in Partial Differential Equations With Complex Variables and Transform Methods, Dover Publications Inc., New York, 1995, Corrected reprint of the 1965 original. | MR | Zbl

[86] Yosida K., On the Differentiability and the Representation of One-Parameter Semi-Group of Linear Operators, J. Math. Soc. Japan 1 (1948) 15-21. | MR | Zbl

[87] Yosida K., Functional Analysis, Grundlehren Math. Wiss., Band 123, fifth ed., Springer-Verlag, Berlin, 1978. | MR | Zbl

Cité par Sources :