Beckmann-type problem for degenerate Hamilton-Jacobi equations
Authors:
Hamza Ennaji, Noureddine Igbida and Van Thanh Nguyen
Journal:
Quart. Appl. Math. 80 (2022), 201-220
MSC (2020):
Primary 35A15, 35D40, 35F21, 65N99
DOI:
https://doi.org/10.1090/qam/1606
Published electronically:
December 21, 2021
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Additional Information
Abstract: The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation $H(x,\nabla u)=0,$ coupled with non-zero Dirichlet condition $u=g$ on $\partial \Omega$. In this article, the Hamiltonian $H$ is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.
References
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- Jean-David Benamou and Guillaume Carlier, Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl. 167 (2015), no. 1, 1–26. MR 3395203, DOI 10.1007/s10957-015-0725-9
- Jean-David Benamou, Guillaume Carlier, and Roméo Hatchi, A numerical solution to Monge’s problem with a Finsler distance as cost, ESAIM Math. Model. Numer. Anal. 52 (2018), no. 6, 2133–2148. MR 3905185, DOI 10.1051/m2an/2016077
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- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
- Andrea Davini, Smooth approximation of weak Finsler metrics, Differential Integral Equations 18 (2005), no. 5, 509–530. MR 2136977
- Luigi De Pascale and Chloé Jimenez, Duality theory and optimal transport for sand piles growing in a silos, Adv. Differential Equations 20 (2015), no. 9-10, 859–886. MR 3360394
- Samer Dweik, Weighted Beckmann problem with boundary costs, Quart. Appl. Math. 76 (2018), no. 4, 601–609. MR 3855823, DOI 10.1090/qam/1512
- Jonathan Eckstein and Dimitri P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming 55 (1992), no. 3, Ser. A, 293–318. MR 1168183, DOI 10.1007/BF01581204
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- Albert Fathi and Antonio Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations 22 (2005), no. 2, 185–228. MR 2106767, DOI 10.1007/s00526-004-0271-z
- Mikhail Feldman and Robert J. McCann, Uniqueness and transport density in Monge’s mass transportation problem, Calc. Var. Partial Differential Equations 15 (2002), no. 1, 81–113. MR 1920716, DOI 10.1007/s005260100119
- Michel Fortin and Roland Glowinski, Augmented Lagrangian methods, Studies in Mathematics and its Applications, vol. 15, North-Holland Publishing Co., Amsterdam, 1983. Applications to the numerical solution of boundary value problems; Translated from the French by B. Hunt and D. C. Spicer. MR 724072
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- Roland Glowinski and Patrick Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM Studies in Applied Mathematics, vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1060954, DOI 10.1137/1.9781611970838
- Noureddine Igbida and Van Thanh Nguyen, Augmented Lagrangian method for optimal partial transportation, IMA J. Numer. Anal. 38 (2018), no. 1, 156–183. MR 3800018, DOI 10.1093/imanum/drw077
- Noureddine Igbida and Van Thanh Nguyen, Optimal partial mass transportation and obstacle Monge-Kantorovich equation, J. Differential Equations 264 (2018), no. 10, 6380–6417. MR 3770053, DOI 10.1016/j.jde.2018.01.034
- Noureddine Igbida, José M. Mazón, Julio D. Rossi, and Julián Toledo, Optimal mass transportation for costs given by Finsler distances via $p$-Laplacian approximations, Adv. Calc. Var. 11 (2018), no. 1, 1–28. MR 3739261, DOI 10.1515/acv-2015-0052
- Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
- José M. Mazón, Julio D. Rossi, and Julián Toledo, An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary, Rev. Mat. Iberoam. 30 (2014), no. 1, 277–308. MR 3186940, DOI 10.4171/RMI/778
- Aldo Pratelli, Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds, Ann. Mat. Pura Appl. (4) 184 (2005), no. 2, 215–238. MR 2149093, DOI 10.1007/s10231-004-0109-5
- Filippo Santambrogio, Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications, vol. 87, Birkhäuser/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling. MR 3409718, DOI 10.1007/978-3-319-20828-2
- Antonio Siconolfi, Hamilton-Jacobi equations and weak KAM theory, Mathematics of complexity and dynamical systems. Vols. 1–3, Springer, New York, 2012, pp. 683–703. MR 3220701, DOI 10.1007/978-1-4614-1806-1_{4}2
- Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
References
- Martin Beckmann, A continuous model of transportation, Econometrica 20 (1952), 643–660. MR 68196, DOI 10.2307/1907646
- Jean-David Benamou and Guillaume Carlier, Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl. 167 (2015), no. 1, 1–26. MR 3395203, DOI 10.1007/s10957-015-0725-9
- Jean-David Benamou, Guillaume Carlier, and Roméo Hatchi, A numerical solution to Monge’s problem with a Finsler distance as cost, ESAIM Math. Model. Numer. Anal. 52 (2018), no. 6, 2133–2148. MR 3905185, DOI 10.1051/m2an/2016077
- Guy Bouchitté and Giuseppe Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 2, 139–168. MR 1831873, DOI 10.1007/s100970000027
- S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends® in Machine Learning 3(2011), no. 1, 1–122.
- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
- Andrea Davini, Smooth approximation of weak Finsler metrics, Differential Integral Equations 18 (2005), no. 5, 509–530. MR 2136977
- Luigi De Pascale and Chloé Jimenez, Duality theory and optimal transport for sand piles growing in a silos, Adv. Differential Equations 20 (2015), no. 9-10, 859–886. MR 3360394
- Samer Dweik, Weighted Beckmann problem with boundary costs, Quart. Appl. Math. 76 (2018), no. 4, 601–609. MR 3855823, DOI 10.1090/qam/1512
- Jonathan Eckstein and Dimitri P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming 55 (1992), no. 3, Ser. A, 293–318. MR 1168183, DOI 10.1007/BF01581204
- Ivar Ekeland and Roger Témam, Convex analysis and variational problems, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. Translated from the French. MR 1727362, DOI 10.1137/1.9781611971088
- Hamza Ennaji, Noureddine Igbida, and Van Thanh Nguyen, Augmented Lagrangian methods for degenerate Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations 60 (2021), no. 6, Paper No. 238. MR 4323131, DOI 10.1007/s00526-021-02092-5
- H. Ennaji, N. Igbida, and V. T. Nguyen, Quasi-convex Hamilton–Jacobi equations via limits of Finsler $p$-Laplace problems as $p\to \infty$, submitted, arXiv:2107.02606, 2021.
- L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653, viii+66. MR 1464149, DOI 10.1090/memo/0653
- Albert Fathi and Antonio Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations 22 (2005), no. 2, 185–228. MR 2106767, DOI 10.1007/s00526-004-0271-z
- Mikhail Feldman and Robert J. McCann, Uniqueness and transport density in Monge’s mass transportation problem, Calc. Var. Partial Differential Equations 15 (2002), no. 1, 81–113. MR 1920716, DOI 10.1007/s005260100119
- Michel Fortin and Roland Glowinski, Augmented Lagrangian methods, Studies in Mathematics and its Applications, vol. 15, North-Holland Publishing Co., Amsterdam, 1983. Applications to the numerical solution of boundary value problems; Translated from the French by B. Hunt and D. C. Spicer. MR 724072
- R. Glowinski and J. Tinsley Oden, Numerical methods for nonlinear variational problems, Springer-Verlag, New York, 1985.
- Roland Glowinski and Patrick Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM Studies in Applied Mathematics, vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1060954, DOI 10.1137/1.9781611970838
- Noureddine Igbida and Van Thanh Nguyen, Augmented Lagrangian method for optimal partial transportation, IMA J. Numer. Anal. 38 (2018), no. 1, 156–183. MR 3800018, DOI 10.1093/imanum/drw077
- Noureddine Igbida and Van Thanh Nguyen, Optimal partial mass transportation and obstacle Monge-Kantorovich equation, J. Differential Equations 264 (2018), no. 10, 6380–6417. MR 3770053, DOI 10.1016/j.jde.2018.01.034
- Noureddine Igbida, José M. Mazón, Julio D. Rossi, and Julián Toledo, Optimal mass transportation for costs given by Finsler distances via $p$-Laplacian approximations, Adv. Calc. Var. 11 (2018), no. 1, 1–28. MR 3739261, DOI 10.1515/acv-2015-0052
- Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
- José M. Mazón, Julio D. Rossi, and Julián Toledo, An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary, Rev. Mat. Iberoam. 30 (2014), no. 1, 277–308. MR 3186940, DOI 10.4171/RMI/778
- Aldo Pratelli, Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds, Ann. Mat. Pura Appl. (4) 184 (2005), no. 2, 215–238. MR 2149093, DOI 10.1007/s10231-004-0109-5
- Filippo Santambrogio, Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications, vol. 87, Birkhäuser/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling. MR 3409718, DOI 10.1007/978-3-319-20828-2
- Antonio Siconolfi, Hamilton-Jacobi equations and weak KAM theory, Mathematics of complexity and dynamical systems. Vols. 1–3, Springer, New York, 2012, pp. 683–703. MR 3220701, DOI 10.1007/978-1-4614-1806-1_42
- Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
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Additional Information
Hamza Ennaji
Affiliation:
Institut de recherche XLIM, UMR-CNRS 7252, Faculté des Sciences et Techniques, Université de Limoges, France
Email:
hamza.ennaji@unilim.fr
Noureddine Igbida
Affiliation:
Institut de recherche XLIM, UMR-CNRS 7252, Faculté des Sciences et Techniques, Université de Limoges, France
MR Author ID:
601440
Email:
noureddine.igbida@unilim.fr
Van Thanh Nguyen
Affiliation:
Department of Mathematics and Statistics, Quy Nhon University, Vietnam
Email:
nguyenvanthanh@qnu.edu.vn
Received by editor(s):
September 1, 2021
Received by editor(s) in revised form:
October 19, 2021
Published electronically:
December 21, 2021
Additional Notes:
Van Thanh Nguyen is the corresponding author
Article copyright:
© Copyright 2021
Brown University
