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September  2017, 37(9): 4625-4636. doi: 10.3934/dcds.2017199

Analytic dependence on parameters for Evans' approximated Weak KAM solutions

1. 

Dipartimento di Matematica "Tullio Levi-Civita", Università degli Studi di Padova, Via Trieste, 63 -35121 Padova, Italy

2. 

Department of Mathematics, The University of Tulsa, 800 South Tucker Drive, Tulsa -Oklahoma 74104, USA

Received  June 2016 Revised  April 2017 Published  June 2017

We consider a variational principle for approximated Weak KAM solutions proposed by Evans. For Hamiltonians in quasi-integrable form $h(p)+\varepsilon f(\varphi,p)$, we prove that the map which takes the parameters $(\varepsilon,P,\varrho)$ to Evans' approximated solution $u_{\varepsilon,P,\varrho}$ is real analytic. In the mechanical case, we compute a recursive system of periodic partial differential equations identifying univocally the coefficients for the power series of the perturbative parameter $\varepsilon$.

Citation: Olga Bernardi, Matteo Dalla Riva. Analytic dependence on parameters for Evans' approximated Weak KAM solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4625-4636. doi: 10.3934/dcds.2017199
References:
[1]

H. Ammari and H. Kang, Polarization and Moment Tensors, Applied Mathematical Sciences, Applied Mathematical Sciences, 162 Springer, New York, 2007. Google Scholar

[2]

V. I. Arnol'd, Proof of A. N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian, Russian Math. Surv., 18 (1963), 13-40. Google Scholar

[3]

O. Bernardi, F. Cardin and M. Guzzo, New estimates for Evans' variational approach to weak KAM theory, Commun. Contemp. Math. , 15 (2013), 1250055, 36 pp. doi: 10.1142/S0219199712500551. Google Scholar

[4]

M. Dalla Riva, A family of fundamental solutions of elliptic partial differential operators with real constant coefficients, Integral Equations Oper. Theor., 76 (2013), 1-23. doi: 10.1007/s00020-013-2052-6. Google Scholar

[5]

M. Dalla Riva and P. Musolino, Real analytic families of harmonic functions in a planar domain with a small hole, J. Math. Anal. Appl., 422 (2015), 37-55. doi: 10.1016/j.jmaa.2014.08.037. Google Scholar

[6]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. Google Scholar

[7]

L. C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 17 (2003), 159-177. doi: 10.1007/s00526-002-0164-y. Google Scholar

[8]

L. C. Evans, Further PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 35 (2009), 435-462. doi: 10.1007/s00526-008-0214-1. Google Scholar

[9]

L. C. Evans, New identities for Weak KAM theory, Chin. Ann. Math. Ser. B, 38 (2017), 379-392. doi: 10.1007/s11401-017-1074-9. Google Scholar

[10]

A. Fathi, Thórème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris, 324 (1997), 1043-1046. doi: 10.1016/S0764-4442(97)87883-4. Google Scholar

[11]

D. A. Gomes and J. Saúde, Mean field games models -a brief survey, Dyn. Games Appl., 4 (2014), 110-154. doi: 10.1007/s13235-013-0099-2. Google Scholar

[12]

D. Gomes and H. S. Morgado, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc., 366 (2014), 903-929. doi: 10.1090/S0002-9947-2013-05936-3. Google Scholar

[13]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2. Google Scholar

[14]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511546730. Google Scholar

[15]

F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience Publishers, New York-London, 1955. Google Scholar

[16]

A. N. Kolmogorov, On the preservation of conditionally periodic motions, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530. Google Scholar

[17]

M. Lanza de Cristoforis, Asymptotic behavior of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach, Complex Var. Elliptic Equ, 52 (2007), 945-977. doi: 10.1080/17476930701485630. Google Scholar

[18]

M. Lanza de Cristoforis and P. Musolino, A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients, Far East J. Math. Sci. (FJMS), 52 (2011), 75-120. Google Scholar

[19]

M. Lanza de Cristoforis and P. Musolino, A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, ZAMM Z. Angew. Math. Mech., 96 (2016), 253-272. doi: 10.1002/zamm.201400035. Google Scholar

[20]

J. M. Lasry and P. L. Lions, Jeux a champ moyen. I. Le cas stationnaire, C.R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019. Google Scholar

[21]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil.Mat, 28 (1997), 141-153. doi: 10.1007/BF01233389. Google Scholar

[22]

J. N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4. Google Scholar

[23]

J. Moser, New aspects in the theory of stability of hamiltonian systems, Comm. on Pure and Appl. Math., 11 (1958), 81-114. doi: 10.1002/cpa.3160110105. Google Scholar

[24]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. doi: 10.1007/BF01399536. Google Scholar

[25]

P. Musolino, A singularly perturbed Dirichlet problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, In A. Almeida, L. Castro, F. Speck, editors, Advances in Harmonic Analysis and Operator Theory, The Stefan Samko Anniversary Volume, Operator Theory: Advances and Applications, 229, 269-289, Birkhäuser Verlag, Basel, (2013). doi: 10.1007/978-3-0348-0516-2_15. Google Scholar

[26]

N. N. Nekhoroshev, Exponential estimates of the stability time of near-integrable Hamiltonian systems, Russ. Math. Surveys, 32 (1977), 5-66,287. Google Scholar

[27]

F. Stoppelli, Sull'esistenza di soluzioni delle equazioni dell'elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio, (Italian) Ricerche Mat, 6 (1957), 241-287. Google Scholar

[28]

F. Stoppelli, Sull'esistenza di soluzioni delle equazioni dell'elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio, Ⅱ, Ⅲ, (Italian) Ricerche Mat, 7 (1958), 71-101. Google Scholar

[29]

T. Valent, Boundary Value Problems of Finite Elasticity, Local Theorems on Existence, Uniqueness and Analytic Dependence on Data, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3736-5. Google Scholar

show all references

References:
[1]

H. Ammari and H. Kang, Polarization and Moment Tensors, Applied Mathematical Sciences, Applied Mathematical Sciences, 162 Springer, New York, 2007. Google Scholar

[2]

V. I. Arnol'd, Proof of A. N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian, Russian Math. Surv., 18 (1963), 13-40. Google Scholar

[3]

O. Bernardi, F. Cardin and M. Guzzo, New estimates for Evans' variational approach to weak KAM theory, Commun. Contemp. Math. , 15 (2013), 1250055, 36 pp. doi: 10.1142/S0219199712500551. Google Scholar

[4]

M. Dalla Riva, A family of fundamental solutions of elliptic partial differential operators with real constant coefficients, Integral Equations Oper. Theor., 76 (2013), 1-23. doi: 10.1007/s00020-013-2052-6. Google Scholar

[5]

M. Dalla Riva and P. Musolino, Real analytic families of harmonic functions in a planar domain with a small hole, J. Math. Anal. Appl., 422 (2015), 37-55. doi: 10.1016/j.jmaa.2014.08.037. Google Scholar

[6]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. Google Scholar

[7]

L. C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 17 (2003), 159-177. doi: 10.1007/s00526-002-0164-y. Google Scholar

[8]

L. C. Evans, Further PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 35 (2009), 435-462. doi: 10.1007/s00526-008-0214-1. Google Scholar

[9]

L. C. Evans, New identities for Weak KAM theory, Chin. Ann. Math. Ser. B, 38 (2017), 379-392. doi: 10.1007/s11401-017-1074-9. Google Scholar

[10]

A. Fathi, Thórème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris, 324 (1997), 1043-1046. doi: 10.1016/S0764-4442(97)87883-4. Google Scholar

[11]

D. A. Gomes and J. Saúde, Mean field games models -a brief survey, Dyn. Games Appl., 4 (2014), 110-154. doi: 10.1007/s13235-013-0099-2. Google Scholar

[12]

D. Gomes and H. S. Morgado, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc., 366 (2014), 903-929. doi: 10.1090/S0002-9947-2013-05936-3. Google Scholar

[13]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2. Google Scholar

[14]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511546730. Google Scholar

[15]

F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience Publishers, New York-London, 1955. Google Scholar

[16]

A. N. Kolmogorov, On the preservation of conditionally periodic motions, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530. Google Scholar

[17]

M. Lanza de Cristoforis, Asymptotic behavior of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach, Complex Var. Elliptic Equ, 52 (2007), 945-977. doi: 10.1080/17476930701485630. Google Scholar

[18]

M. Lanza de Cristoforis and P. Musolino, A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients, Far East J. Math. Sci. (FJMS), 52 (2011), 75-120. Google Scholar

[19]

M. Lanza de Cristoforis and P. Musolino, A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, ZAMM Z. Angew. Math. Mech., 96 (2016), 253-272. doi: 10.1002/zamm.201400035. Google Scholar

[20]

J. M. Lasry and P. L. Lions, Jeux a champ moyen. I. Le cas stationnaire, C.R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019. Google Scholar

[21]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil.Mat, 28 (1997), 141-153. doi: 10.1007/BF01233389. Google Scholar

[22]

J. N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4. Google Scholar

[23]

J. Moser, New aspects in the theory of stability of hamiltonian systems, Comm. on Pure and Appl. Math., 11 (1958), 81-114. doi: 10.1002/cpa.3160110105. Google Scholar

[24]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. doi: 10.1007/BF01399536. Google Scholar

[25]

P. Musolino, A singularly perturbed Dirichlet problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, In A. Almeida, L. Castro, F. Speck, editors, Advances in Harmonic Analysis and Operator Theory, The Stefan Samko Anniversary Volume, Operator Theory: Advances and Applications, 229, 269-289, Birkhäuser Verlag, Basel, (2013). doi: 10.1007/978-3-0348-0516-2_15. Google Scholar

[26]

N. N. Nekhoroshev, Exponential estimates of the stability time of near-integrable Hamiltonian systems, Russ. Math. Surveys, 32 (1977), 5-66,287. Google Scholar

[27]

F. Stoppelli, Sull'esistenza di soluzioni delle equazioni dell'elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio, (Italian) Ricerche Mat, 6 (1957), 241-287. Google Scholar

[28]

F. Stoppelli, Sull'esistenza di soluzioni delle equazioni dell'elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio, Ⅱ, Ⅲ, (Italian) Ricerche Mat, 7 (1958), 71-101. Google Scholar

[29]

T. Valent, Boundary Value Problems of Finite Elasticity, Local Theorems on Existence, Uniqueness and Analytic Dependence on Data, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3736-5. Google Scholar

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