# American Institute of Mathematical Sciences

May  2020, 40(5): 2615-2639. doi: 10.3934/dcds.2020143

## Characterization of minimizable Lagrangian action functionals and a dual Mather theorem

 Université Toulouse 1 Capitole, Unité Mixte de Recherche "Toulouse School of Economics – Research", 1 Esplanade de l'Université, 31080 Toulouse Cedex 08, France

In grateful memory of Professor John N. Mather

Received  March 2019 Revised  December 2019 Published  March 2020

Fund Project: The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement 307062. The author acknowledges the support of ANR-3IA Artificial and Natural Intelligence Toulouse Institute

We show that a necessary and sufficient condition for a smooth function on the tangent bundle of a manifold to be a Lagrangian density whose action can be minimized is, roughly speaking, that it be the sum of a constant, a nonnegative function vanishing on the support of the minimizers, and an exact form.

We show that this exact form corresponds to the differential of a Lipschitz function on the manifold that is differentiable on the projection of the support of the minimizers, and its derivative there is Lipschitz. This function generalizes the notion of subsolution of the Hamilton-Jacobi equation that appears in weak KAM theory, and the Lipschitzity result allows for the recovery of Mather's celebrated 1991 result as a special case. We also show that our result is sharp with several examples.

Finally, we apply the same type of reasoning to an example of a finite horizon Legendre problem in optimal control, and together with the Lipschitzity result we obtain the Hamilton–Jacobi–Bellman equation and the Maximum Principle.

Citation: Rodolfo Ríos-Zertuche. Characterization of minimizable Lagrangian action functionals and a dual Mather theorem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2615-2639. doi: 10.3934/dcds.2020143
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar [2] M.-C. Arnaud, The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents, Annals of Mathematics, 174 (2011), 1571-1601.  doi: 10.4007/annals.2011.174.3.4.  Google Scholar [3] V. Bangert, Minimal measures and minimizing closed normal one-currents, Geometric And Functional Analysis, 9 (1999), 413-427.  doi: 10.1007/s000390050093.  Google Scholar [4] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkh"auser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar [5] P. Bernard, Existence of $C^{1, 1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Annales Scientifiques de l'Ecole Normale Supérieure, 40 (2007), 445–452. doi: 10.1016/j.ansens.2007.01.004.  Google Scholar [6] P. Bernard, Young measures, superposition and transport, Indiana Univ. Math. J., 57 (2008), 247-275.  doi: 10.1512/iumj.2008.57.3163.  Google Scholar [7] P. Bernard and B. Buffoni, The Monge problem for supercritical Mañé potentials on compact manifolds, Advances in Mathematics, 207 (2006), 691-706.  doi: 10.1016/j.aim.2006.01.003.  Google Scholar [8] P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory, Journal of the European Mathematical Society, 9 (2007), 85-121.  doi: 10.4171/JEMS/74.  Google Scholar [9] G. Contreras, A. Figalli and L. Rifford, Generic hyperbolicity of Aubry sets on surfaces, Inventiones Mathematicae, 200 (2015), 201-261.  doi: 10.1007/s00222-014-0533-0.  Google Scholar [10] G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22° Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999.  Google Scholar [11] B. Dacorogna, Introduction to the Calculus of Variations, 2nd edition, Imperial College Press, London, 2009.   Google Scholar [12] M. J. Dias Carneiro and R. O. Ruggiero, On the graph theorem for Lagrangian minimizing tori, Discrete Contin. Dyn. Syst., 38 (2018), 6029–6045, URL http://aimsciences.org//article/id/d34ac573-acb3-4c9c-9472-22c46cd536f1. doi: 10.3934/dcds.2018260.  Google Scholar [13] M. J. Dias Carneiro and R. O. Ruggiero, On Birkhoff theorems for Lagrangian invariant tori with closed orbits, Manuscripta Mathematica, 119 (2006), 411-432.  doi: 10.1007/s00229-005-0619-5.  Google Scholar [14] M. J. Dias Carneiro and R. O. Ruggiero, Birkhoff first theorem for Lagrangian, invariant tori in dimension 3, Preprint. Google Scholar [15] A. Fathi, Weak KAM theorem in Lagrangian dynamics, Preliminary Version Number 10, (2008). Google Scholar [16] A. Fathi, Weak KAM theory: The connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, Proceedings of the International Congress of Mathematicians—Seoul 2014, Kyung Moon Sa, Seoul, 3 (2014), 597–621, URL http://www.icm2014.org/download/Proceedings_Volume_Ⅲ.pdf.  Google Scholar [17] A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.  doi: 10.1007/s00222-003-0323-6.  Google Scholar [18] A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.  doi: 10.1007/s00526-004-0271-z.  Google Scholar [19] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.  Google Scholar [20] A. Griewank and P. J. Rabier, On the smoothness of convex envelopes, Transactions of the American Mathematical Society, 322 (1990), 691-709.  doi: 10.1090/S0002-9947-1990-0986024-2.  Google Scholar [21] Y. Li and L. Nirenberg, The regularity of the distance function to the boundary, arXiv: math/0510577 [math.AP]. Google Scholar [22] Y. Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Communications on Pure and Applied Mathematics, 58 (2005), 85-146.  doi: 10.1002/cpa.20051.  Google Scholar [23] E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar [24] J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.  doi: 10.1007/BF02571383.  Google Scholar [25] C. B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific Journal of Mathematics, 2 (1952), 25-53.  doi: 10.2140/pjm.1952.2.25.  Google Scholar [26] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.  Google Scholar [27] R. Ríos-Zertuche, Deformations of closed measures and variational characterization of measures invariant under the Euler-Lagrange flow, preprint, arXiv: 1810.07838 [math.OC]. Google Scholar [28] S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz, 5 (1993), 206-238.   Google Scholar [29] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969.  Google Scholar

show all references

##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar [2] M.-C. Arnaud, The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents, Annals of Mathematics, 174 (2011), 1571-1601.  doi: 10.4007/annals.2011.174.3.4.  Google Scholar [3] V. Bangert, Minimal measures and minimizing closed normal one-currents, Geometric And Functional Analysis, 9 (1999), 413-427.  doi: 10.1007/s000390050093.  Google Scholar [4] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkh"auser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar [5] P. Bernard, Existence of $C^{1, 1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Annales Scientifiques de l'Ecole Normale Supérieure, 40 (2007), 445–452. doi: 10.1016/j.ansens.2007.01.004.  Google Scholar [6] P. Bernard, Young measures, superposition and transport, Indiana Univ. Math. J., 57 (2008), 247-275.  doi: 10.1512/iumj.2008.57.3163.  Google Scholar [7] P. Bernard and B. Buffoni, The Monge problem for supercritical Mañé potentials on compact manifolds, Advances in Mathematics, 207 (2006), 691-706.  doi: 10.1016/j.aim.2006.01.003.  Google Scholar [8] P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory, Journal of the European Mathematical Society, 9 (2007), 85-121.  doi: 10.4171/JEMS/74.  Google Scholar [9] G. Contreras, A. Figalli and L. Rifford, Generic hyperbolicity of Aubry sets on surfaces, Inventiones Mathematicae, 200 (2015), 201-261.  doi: 10.1007/s00222-014-0533-0.  Google Scholar [10] G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians, 22° Colóquio Brasileiro de Matemática, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999.  Google Scholar [11] B. Dacorogna, Introduction to the Calculus of Variations, 2nd edition, Imperial College Press, London, 2009.   Google Scholar [12] M. J. Dias Carneiro and R. O. Ruggiero, On the graph theorem for Lagrangian minimizing tori, Discrete Contin. Dyn. Syst., 38 (2018), 6029–6045, URL http://aimsciences.org//article/id/d34ac573-acb3-4c9c-9472-22c46cd536f1. doi: 10.3934/dcds.2018260.  Google Scholar [13] M. J. Dias Carneiro and R. O. Ruggiero, On Birkhoff theorems for Lagrangian invariant tori with closed orbits, Manuscripta Mathematica, 119 (2006), 411-432.  doi: 10.1007/s00229-005-0619-5.  Google Scholar [14] M. J. Dias Carneiro and R. O. Ruggiero, Birkhoff first theorem for Lagrangian, invariant tori in dimension 3, Preprint. Google Scholar [15] A. Fathi, Weak KAM theorem in Lagrangian dynamics, Preliminary Version Number 10, (2008). Google Scholar [16] A. Fathi, Weak KAM theory: The connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, Proceedings of the International Congress of Mathematicians—Seoul 2014, Kyung Moon Sa, Seoul, 3 (2014), 597–621, URL http://www.icm2014.org/download/Proceedings_Volume_Ⅲ.pdf.  Google Scholar [17] A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), 363-388.  doi: 10.1007/s00222-003-0323-6.  Google Scholar [18] A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.  doi: 10.1007/s00526-004-0271-z.  Google Scholar [19] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.  Google Scholar [20] A. Griewank and P. J. Rabier, On the smoothness of convex envelopes, Transactions of the American Mathematical Society, 322 (1990), 691-709.  doi: 10.1090/S0002-9947-1990-0986024-2.  Google Scholar [21] Y. Li and L. Nirenberg, The regularity of the distance function to the boundary, arXiv: math/0510577 [math.AP]. Google Scholar [22] Y. Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Communications on Pure and Applied Mathematics, 58 (2005), 85-146.  doi: 10.1002/cpa.20051.  Google Scholar [23] E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar [24] J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.  doi: 10.1007/BF02571383.  Google Scholar [25] C. B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific Journal of Mathematics, 2 (1952), 25-53.  doi: 10.2140/pjm.1952.2.25.  Google Scholar [26] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.  Google Scholar [27] R. Ríos-Zertuche, Deformations of closed measures and variational characterization of measures invariant under the Euler-Lagrange flow, preprint, arXiv: 1810.07838 [math.OC]. Google Scholar [28] S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz, 5 (1993), 206-238.   Google Scholar [29] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969.  Google Scholar
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