Wolansky, G. Minimizers of Dirichlet functionals on the \(n\)-torus and the weak KAM theory. (English) Zbl 1173.35047 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 2, 521-545 (2009). This paper deals with the minimization of a Dirichlet-type functional \(F\) on the \(n\)-torus \({\mathbb T}^n\). It is first argued that the supremum of this functional is not attained in \(L^1({\mathbb T}^n)\), but in the set \(\overline{\mathcal M}\) of Borel probability measures on \({\mathbb T}^n\). This motivates the author to extend the domain of \(F\) from the set of nonnegative densities in \(L^1({\mathbb T}^n)\) to \(\overline{\mathcal M}\).First objective of the present paper is to define a generalized minimizer of \(F\). This is mainly done by means of the effective Hamiltonian, in the framework of the weak KAM theory. A second objective of this paper is to relate the generalized minimizer of \(F\) to the minimal Mather measure. Essentially, the minimal Mather measures of a given Lagrangian is connected with the measure which minimizes a certain optimal transportation plan. That is why the third main objective of the present paper is to approximate \(F\) by an optimal transportation function and to establish a combinatorial search algorithm. Reviewer: Vicenţiu D. Rădulescu (Craiova) Cited in 1 Document MSC: 35J20 Variational methods for second-order elliptic equations 37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 49J35 Existence of solutions for minimax problems Keywords:Monge-Kantorovich; optimal mass transport; periodic Lagrangian; effective Hamiltonian; rotation vector; Dirichlet functional; KAM theory PDFBibTeX XMLCite \textit{G. Wolansky}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 2, 521--545 (2009; Zbl 1173.35047) Full Text: DOI arXiv EuDML References: [1] Arriola, E. A.; Soler, J., A variational approach to the Schrödinger-Poisson system: Asymptotic behaviour, Breathers, and Stability, J. Stat. Phys., 103, 5-6, 1069-1106 (2001) · Zbl 0999.82062 [2] Aubry, S., The twist map, the extended Frenkel-Kontrovna model and the devil’s staircase, Physica D, 7, 240-258 (1983) [3] Bernard, P.; Buffoni, B., Optimal mass transportation and mather theory, Preprint · Zbl 1241.49025 [4] Evans, L. C., A survey of partial differential equations methods in weak KAM theory, Comm. Pure Appl. Math., 57, 4, 445-480 (2004) · Zbl 1040.37046 [5] Evans, L. C.; Gomes, D., Effective Hamiltonians and averaging for Hamiltonian dynamics. I, Arch. Ration. Mech. Anal., 157, 1, 1-33 (2001) · Zbl 0986.37056 [6] Evans, L. C.; Gomes, D., Effective Hamiltonians and averaging for Hamiltonian dynamics. II, Arch. Ration. Mech. Anal., 161, 4, 271-305 (2002) · Zbl 1100.37039 [7] Fathi, A., The weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics Series, vol. 88 (2003), Cambridge University Press [8] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001 [9] Gomes, D. A.; Oberman, A. M., Computing the effective Hamiltonian using a variational approach, SIAM J. Control Optim., 43, 3, 792-812 (2004) · Zbl 1081.49024 [10] Govin, M.; Chandre, C.; Jauslin, H. R., Kolmogorov-Arnold-Moser-Renormalization-Group analysis of stability in Hamiltonian flows, Phys. Rev. Lett., 79, 3881-3884 (1997) [11] L. Granieri, On action minimizing measures for the Monge-Kantorovich problem, Preprint, July 2004; L. Granieri, On action minimizing measures for the Monge-Kantorovich problem, Preprint, July 2004 · Zbl 1133.37027 [12] Hedlund, G. A., Geodesics on a 3 dimensional Riemannian manifolds with periodic coefficients, Ann. of Math., 33, 719-739 (1932) · Zbl 0006.32601 [13] Hiriart-Urruty, J. B.; Lemarechal, C., Convex Analysis and Minimization Algorithms II, Grundlehren der Mathematischen Wissenschaften, vol. 306 (1993), Springer-Verlag, (Chapter 10) · Zbl 0795.49002 [14] Illner, R.; Zweifel, P. F.; Lange, H., Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger-Poisson systems, Math. Meth. Appl. Sci., 17, 349-376 (1994) · Zbl 0808.35116 [15] Keller, J. B., Semiclassical mechanics, SIAM Rev., 27, 4, 485-504 (1985) · Zbl 0581.70012 [16] Luigi, D. P.; Stella, G. M.; Granieri, L., Minimal measures, one-dimensional currents and the Monge-Kantorovich problem, Calc. Var. Partial Differential Equations, 27, 1, 1-23 (2006) · Zbl 1096.37033 [17] Mañè, R., On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5, 623-638 (1992) · Zbl 0799.58030 [18] Mather, J. N., Existence of quasi-periodic orbits for twist homeomorphisms on the annulus, Topology, 21, 457-467 (1982) · Zbl 0506.58032 [19] Mather, J. N., Minimal measures, Comment. Math. Helv., 64, 375-394 (1989) · Zbl 0689.58025 [20] Moser, J., Monotone twist mappings and the calculus of variations, Ergodic Theory Dynam. Systems, 6, 401-413 (1986) · Zbl 0619.49020 [21] Rubinstein, J.; Wolansky, G., Eikonal functions: Old and new, (Givoli, D.; Grote, M. J.; Papanicolaou, G., A Celebration of Mathematical Modeling: The Joseph B. Keller Anniversary Volume (2004), Kluwer) [22] Siburg, K. F., The Principle of Least Action in Geometry and Dynamics, Lecture Notes in Mathematics, vol. 1844 (2004), Springer · Zbl 1060.37048 [23] Villani, C., Topics in Optimal Transportation, Graduate Studies in Math., vol. 58 (2003), Amer. Math. Soc. · Zbl 1106.90001 [24] Wolansky, G., Optimal transportation in the presence of a prescribed pressure field (12 Jan 2006), Preprint [25] G. Wolansky, On time reversible description of the process of coagulation and fragmentation, Arch. Rat. Mech., submitted for publication; G. Wolansky, On time reversible description of the process of coagulation and fragmentation, Arch. Rat. Mech., submitted for publication · Zbl 1169.76052 [26] Markowich, P.; Rein, G.; Wolansky, G., Existence and nonlinear stability of stationary states of the Schrödinger-Poisson system, J. Stat. Phys., 106, 1221-1239 (2002) · Zbl 1001.82107 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.