Symplectic Homogenization

Claude Viterbo

doi:10.5802/jep.214

Symplectic Homogenization
[Homogénéisation symplectique]

Claude Viterbo ¹

¹ DMA, UMR 8553 du CNRS, École Normale Supérieure, PSL University 45 Rue d’Ulm, 75230 Paris Cedex 05, France

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 67-140.

Résumé
Abstract

Soit $H (q, p)$ un hamiltonien défini sur $T^{*} T^{n}$ . Sous des hypothèses convenables, on montre que la suite ${(H_{k})}_{k \geq 1}$ définie par $H_{k} (q, p) = H (k q, p)$ converge pour la topologie $γ$ , définie dans [Vit92], vers un hamiltonien intégrable $\bar{H} (p)$ . Ceci s’étend au cas de hamiltoniens non-autonomes, et au cas où seulement certaines variables sont homogénéisées : par exemple la suite $H_{k} (k x, y, p_{x}, p_{y})$ qui dans ce cas aura une limite $\bar{H} (y, p_{x}, p_{y})$ , qui est un « hamiltonien effectif ». Le but de cet article est de démontrer la convergence de ces suites, ainsi que les premières propriétés de l’opérateur d’homogénéisation et d’en donner des applications aux solutions d’équations de Hamilton-Jacobi, aux quasi-états symplectiques, etc. On démontre aussi que lorsque $H$ est convexe en $p$ , la fonction $\bar{H}$ coïncide avec la fonction $α$ de Mather (cf. [Mat91]) associée au dual de Legendre de $H$ . Cela redémontre, dans le cas du tore, que cette fonction est symplectiquement invariante, comme l’avait démontré P. Bernard ([Ber07]) dans le cas général.

Let $H (q, p)$ be a Hamiltonian on $T^{*} T^{n}$ . Under suitable assumptions on $H$ , we show that the sequence ${(H_{k})}_{k \geq 1}$ defined by $H_{k} (q, p) = H (k q, p)$ converges in the $γ$ -topology—defined in [Vit92]—to an integrable continuous Hamiltonian $\bar{H} (p)$ . This is extended to the case of non-autonomous Hamiltonians, and the more general setting in which only some of the variables are homogenized: we consider the sequence $H (k x, y, q, p)$ and prove it has a $γ$ -limit $\bar{H} (y, q, p)$ , thus yielding an “effective Hamiltonian”. The goal of this paper is to prove convergence of the above sequences, state the first properties of the homogenization operator, and give some applications to solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that when $H$ is convex in $p$ , the function $\bar{H}$ coincides with Mather’s $α$ function defined in [Mat91] and associated to the Legendre dual of $H$ . This gives a new proof—in the torus case—of its symplectic invariance first discovered by P. Bernard in [Ber07].

Reçu le : 2022-07-16
Accepté le : 2022-12-02
Publié le : 2022-12-16

DOI : 10.5802/jep.214

Classification : 37J05, 53D35, 35F20, 49L25, 37J40, 37J50
Keywords: Homogenization, symplectic topology, Hamiltonian flow, Hamilton-Jacobi equation, variational solutions
Mot clés : Flot hamiltonien, homogénéisation, Hamilton-Jacobi, symplectique

Affiliations des auteurs :

Claude Viterbo ¹

¹ DMA, UMR 8553 du CNRS, École Normale Supérieure, PSL University 45 Rue d’Ulm, 75230 Paris Cedex 05, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2023__10__67_0,
     author = {Claude Viterbo},
     title = {Symplectic {Homogenization}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {67--140},
     publisher = {\'Ecole polytechnique},
     volume = {10},
     year = {2023},
     doi = {10.5802/jep.214},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.214/}
}

TY  - JOUR
AU  - Claude Viterbo
TI  - Symplectic Homogenization
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2023
SP  - 67
EP  - 140
VL  - 10
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.214/
DO  - 10.5802/jep.214
LA  - en
ID  - JEP_2023__10__67_0
ER  -

%0 Journal Article
%A Claude Viterbo
%T Symplectic Homogenization
%J Journal de l’École polytechnique — Mathématiques
%D 2023
%P 67-140
%V 10
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.214/
%R 10.5802/jep.214
%G en
%F JEP_2023__10__67_0

Claude Viterbo. Symplectic Homogenization. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 67-140. doi : 10.5802/jep.214. https://jep.centre-mersenne.org/articles/10.5802/jep.214/

Bibliographie
Cité par

[Aar91] J. F. Aarnes - “Quasi-states and quasi-measures”, Adv. Math. 86 (1991) no. 1, p. 41-67 | DOI | MR | Zbl

[AB84] E. Acerbi & G. Buttazzo - “On the limits of periodic Riemannian metrics”, J. Analyse Math. 43 (1983/84), p. 183-201 | DOI | MR | Zbl

[AB02] O. Alvarez & M. Bardi - “Viscosity solutions methods for singular perturbations in deterministic and stochastic control”, SIAM J. Control Optim. 40 (2001/02) no. 4, p. 1159-1188 | DOI | MR | Zbl

[AB03] O. Alvarez & M. Bardi - “Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result”, Arch. Rational Mech. Anal. 170 (2003) no. 1, p. 17-61 | DOI | MR | Zbl

[Ban80] V. Bangert - “Closed geodesics on complete surfaces”, Math. Ann. 251 (1980) no. 1, p. 83-96 | DOI | MR | Zbl

[Bar94] G. Barles - Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications, vol. 17, Springer-Verlag, Paris, 1994

[BCBA99] A. Boudaoud, Y. Couder & M. Ben Amar - “A self-adaptative oscillator”, European Phys. J. B 9 (1999), p. 159-165 | DOI

[BCD97] M. Bardi & I. Capuzzo-Dolcetta - Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997 | DOI

[Ben88] V. Benci (1988), Talk at the Workshop of Symplectic Geometry, M.S.R.I., Berkeley, CA, USA

[Ber03] P. Bernard - “The action spectrum near positive definite invariant tori”, Bull. Soc. math. France 131 (2003) no. 4, p. 603-616 | DOI | Numdam | MR | Zbl

[Ber07] P. Bernard - “Symplectic aspects of Mather theory”, Duke Math. J. 136 (2007) no. 3, p. 401-420 | DOI | MR | Zbl

[Bir27] G. D. Birkhoff - Dynamical systems, Amer. Math. Soc. Colloquium Publ., vol. IX, American Mathematical Society, Providence, R.I., 1927, (reprinted 1966)

[Bis19] M. R. Bisgaard - “Mather theory and symplectic rigidity”, J. Modern Dyn. 15 (2019), p. 165-207 | DOI | MR | Zbl

[BM58] N. N. Bogoliubov & Y. A. Mitropolski - Asymptotic methods in the theory of nonlinear oscillations, Fizmatlit, Moscow, 1958, English transl.: Gordon and Breach, New York, 1964

[Bou02] A. Boudaoud - “De la corde au film de savon: de l’auto-adaptation dans les systèmes vibrants”, Images de la Physique (2002), p. 78-83, may be retrieved from https://www.imo.universite-paris-saclay.fr/~claude.viterbo/Imphy.pdf

[Bra02] A. Braides - $Γ$ -convergence for beginners, Oxford Lecture Series in Math. and its Applications, vol. 22, Oxford University Press, Oxford, 2002 | DOI

[Bru91] M. Brunella - “On a theorem of Sikorav”, Enseign. Math. (2) 37 (1991) no. 1-2, p. 83-87 | MR | Zbl

[BS13] L. Buhovsky & S. Seyfaddini - “Uniqueness of generating Hamiltonians for topological Hamiltonian flows”, J. Symplectic Geom. 11 (2013) no. 1, p. 37-52 | DOI | MR | Zbl

[Cha84] M. Chaperon - “Une idée du type ‘géodésiques brisées’ pour les systèmes hamiltoniens”, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) no. 13, p. 293-296 | Zbl

[Cha91] M. Chaperon - “Lois de conservation et géométrie symplectique”, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991) no. 4, p. 345-348 | MR | Zbl

[Che96] Y. V. Chekanov - “Critical points of quasifunctions, and generating families of Legendrian manifolds”, Funktsional. Anal. i Prilozhen. 30 (1996) no. 2, p. 56-69, 96 | DOI

[CIPP98] G. Contreras, R. Iturriaga, G. P. Paternain & M. Paternain - “Lagrangian graphs, minimizing measures and Mañé’s critical values”, Geom. Funct. Anal. 8 (1998) no. 5, p. 788-809 | DOI | Zbl

[CL83] M. G. Crandall & P.-L. Lions - “Viscosity solutions of Hamilton-Jacobi equations”, Trans. Amer. Math. Soc. 277 (1983) no. 1, p. 1-42 | DOI | MR | Zbl

[Con96] M. C. Concordel - “Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula”, Indiana Univ. Math. J. 45 (1996) no. 4, p. 1095-1117 | DOI | MR | Zbl

[CV08] F. Cardin & C. Viterbo - “Commuting Hamiltonians and Hamilton-Jacobi multi-time equations”, Duke Math. J. 144 (2008) no. 2, p. 235-284 | DOI | MR | Zbl

[DG75] E. De Giorgi - “Sulla convergenza di alcune successioni d’integrali del tipo dell’area”, Rend. Mat. (6) 8 (1975), p. 277-294 | MR | Zbl

[DGF75] E. De Giorgi & T. Franzoni - “Su un tipo di convergenza variazionale”, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 58 (1975) no. 6, p. 842-850 | MR | Zbl

[DM93] G. Dal Maso - An introduction to $Γ$ -convergence, Progress in Nonlinear Differential Equations and their Appl., vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993 | DOI

[DMGZ94] G. De Marco, G. Gorni & G. Zampieri - “Global inversion of functions: an introduction”, NoDEA Nonlinear Differential Equations Appl. 1 (1994) no. 3, p. 229-248 | DOI | MR | Zbl

[Eli91] Y. Eliashberg - “New invariants of open symplectic and contact manifolds”, J. Amer. Math. Soc. 4 (1991) no. 3, p. 513-520 | DOI | MR | Zbl

[EP06] M. Entov & L. Polterovich - “Quasi-states and symplectic intersections”, Comment. Math. Helv. 81 (2006) no. 1, p. 75-99 | DOI | MR | Zbl

[Eva89] L. C. Evans - “The perturbed test function method for viscosity solutions of nonlinear PDE”, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) no. 3-4, p. 359-375 | DOI | MR | Zbl

[Fat97] A. Fathi - “Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens”, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) no. 9, p. 1043-1046 | DOI | Zbl

[Fat98] A. Fathi - “Sur la convergence du semi-groupe de Lax-Oleinik”, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) no. 3, p. 267-270 | DOI | MR | Zbl

[Fat08] A. Fathi - “Weak KAM theorem in Lagrangian dynamics”, 2008, Version 10, available from https://www.math.u-bordeaux.fr/~pthieull/Recherche/KamFaible/Publications/Fathi2008_01.pdf

[Ger31] S. Geršgorin - “Über die Abgrenzung der Eigenwerte einer Matrix”, Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et na. 6 (1931), p. 749-754 | Zbl

[Gro99] M. Gromov - Metric structures for Riemannian and non-Riemannian spaces, Progress in Math., vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999, Based on the 1981 French original (3rd ed. 2007)

[GV22] S. Guillermou & N. Vichery - “Viterbo’s spectral bound conjecture for homogeneous spaces”, 2022 | arXiv

[Had06] J. Hadamard - “Sur les transformations ponctuelles”, Bull. Soc. math. France 34 (1906), p. 71-84 | DOI | Numdam | MR | Zbl

[HLS15] V. Humilière, R. Leclercq & S. Seyfaddini - “New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians”, Comment. Math. Helv. 90 (2015) no. 1, p. 1-21 | DOI | MR | Zbl

[Hum08a] V. Humilière - Continuité en topologie symplectique, Ph. D. Thesis, École polytechnique, 2008, https://www.theses.fr/2008EPXX0005

[Hum08b] V. Humilière - “On some completions of the space of Hamiltonian maps”, Bull. Soc. math. France 136 (2008) no. 3, p. 373-404 | DOI | Numdam | MR | Zbl

[Lau92] F. Laudenbach - “On the Thom-Smale complex”, in An extension of a theorem by Cheeger and Müller, Astérisque, vol. 205, Société Mathématique de France, Paris, 1992, p. 219-233 | Numdam

[LPV87] P.-L. Lions, G. C. Papanicolaou & S. R. S. Varadhan - “Homogenization of Hamilton-Jacobi equations”, 1987, Unpublished preprint, available from http://localwww.math.unipd.it/~bardi/didattica/Nonlinear_PDE_%20homogenization_Dott_%202011/LPV87.pdf

[LS85] F. Laudenbach & J.-C. Sikorav - “Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibré cotangent”, Invent. Math. 82 (1985) no. 2, p. 349-357 | DOI | Zbl

[Mat91] J. N. Mather - “Action minimizing invariant measures for positive definite Lagrangian systems”, Math. Z. 207 (1991) no. 2, p. 169-207 | DOI | MR | Zbl

[Mon42] P. Mondrian - “New-York City”, 1942, Centre Pompidou, MNAM-CCI, Paris, https://www.centrepompidou.fr/cpv/resource/c5pRBL/rdyjdr9

[MVZ12] A. Monzner, N. Vichery & F. Zapolsky - “Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization”, J. Modern Dyn. 6 (2012) no. 2, p. 205-249 | DOI | MR | Zbl

[MZ11] A. Monzner & F. Zapolsky - “A comparison of symplectic homogenization and Calabi quasi-states”, J. Topol. Anal. 3 (2011) no. 3, p. 243-263 | DOI | MR | Zbl

[Oh05] Y.-G. Oh - “Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds”, in The breadth of symplectic and Poisson geometry, Progress in Math., vol. 232, Birkhäuser Boston, Boston, MA, 2005, p. 525-570 | DOI | MR | Zbl

[OV95] A. Ottolenghi & C. Viterbo - “Solutions généralisées pour l’équation de Hamilton-Jacobi dans le cas d’évolution”, 1995, Preprint, available from http://www.math.ens.fr/~viterbo/Ottolenghi-Viterbo.pdf

[Roo17] V. Roos - Solutions variationnelles et solutions de viscosité, Ph. D. Thesis, Université de Paris-Dauphine, 2017, http://www.theses.fr/2017PSLED023 and https://basepub.dauphine.fr/handle/123456789/16992

[Sch00] M. Schwarz - “On the action spectrum for closed symplectically aspherical manifolds”, Pacific J. Math. 193 (2000) no. 2, p. 419-461 | DOI | MR | Zbl

[Sey12] S. Seyfaddini - “Descent and $C^{0}$ -rigidity of spectral invariants on monotone symplectic manifolds”, J. Topol. Anal. 4 (2012) no. 4, p. 481-498 | DOI | MR | Zbl

[She22] E. Shelukhin - “Symplectic cohomology and a conjecture of Viterbo”, Geom. Funct. Anal. (2022), published online, doi:10.1007/s00039-022-00619-2 | DOI

[Sik89] J.-C. Sikorav - “Rigidité symplectique dans le cotangent de $T^{n}$ ”, Duke Math. J. 59 (1989) no. 3, p. 759-763 | DOI | MR | Zbl

[Sik90] J.-C. Sikorav (1990), Talk given at Paris 7 seminar

[SV85] J. A. Sanders & F. Verhulst - Averaging methods in nonlinear dynamical systems, Applied Math. Sciences, vol. 59, Springer-Verlag, New York, 1985 | DOI

[SV10] A. Sorrentino & C. Viterbo - “Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms”, Geom. Topol. 14 (2010) no. 4, p. 2383-2403 | DOI | MR | Zbl

[Thé99] D. Théret - “A complete proof of Viterbo’s uniqueness theorem on generating functions”, Topology Appl. 96 (1999) no. 3, p. 249-266 | DOI | MR | Zbl

[Tra94] L. Traynor - “Symplectic homology via generating functions”, Geom. Funct. Anal. 4 (1994) no. 6, p. 718-748 | DOI | MR | Zbl

[Var04] R. S. Varga - Geršgorin and his circles, Springer Series in Computational Math., vol. 36, Springer-Verlag, Berlin, 2004 | DOI

[Vit92] C. Viterbo - “Symplectic topology as the geometry of generating functions”, Math. Ann. 292 (1992) no. 4, p. 685-710 | DOI | MR | Zbl

[Vit95] C. Viterbo - “Solutions d’équations d’Hamilton-Jacobi et géométrie symplectique”, in Séminaire X-EDP, École polytechnique, Palaiseau, 1995, http://www.numdam.org/item/SEDP_1995-1996____A22_0/

[Vit06a] C. Viterbo - “On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows”, Internat. Math. Res. Notices (2006), article ID 34028, 9 pages, Erratum: Ibid., article no. 38784 (4 pages) | DOI | MR | Zbl

[Vit06b] C. Viterbo - “Symplectic topology and Hamilton-Jacobi equations”, in Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Sci. Ser. II Math. Phys. Chem., vol. 217, Springer, Dordrecht, 2006, p. 439-459 | DOI | MR | Zbl

[Vit18] C. Viterbo - “Non-convex Mather theory”, 2018, submitted to Duke Math. J. | arXiv

[Vit21] C. Viterbo - “Stochastic homogenization of variational solutions of Hamilton-Jacobi equations”, 2021 | arXiv

[Vit22] C. Viterbo - “Inverse reduction inequalities for spectral numbers and applications”, 2022 | arXiv

[Wei13] Q. Wei - Solutions de viscosité des équations de Hamilton-Jacobi et minmax itérés, Ph. D. Thesis, Université de Paris 7, 2013, https://tel.archives-ouvertes.fr/tel-00963780

[Zhu96] T. Zhukovskaya - “Metamorphoses of the Chaperon-Sikorav weak solutions of Hamilton-Jacobi equations”, J. Math. Sci. 82 (1996) no. 5, p. 3737-3746 | DOI | MR | Zbl

Cité par Sources :