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Monge-Kantorovich.survey

Monge-Kantorovich.survey - Partial Dierential Equations and...

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Partial Differential Equations and Monge–Kantorovich Mass Transfer Lawrence C. Evans Department of Mathematics University of California, Berkeley September, 2001 version 1. Introduction 1.1 Optimal mass transfer 1.2 Relaxation, duality Part I: Cost = 1 2 (Distance) 2 2. Heuristics 2.1 Geometry of optimal transport 2.2 Lagrange multipliers 3. Optimal mass transport, polar factorization 3.1 Solution of dual problem 3.2 Existence of optimal mass transfer plan 3.3 Polar factorization of vector fields 4. Regularity 4.1 Solving the Monge–Ampere equation 4.2 Examples 4.3 Interior regularity for convex targets 4.4 Boundary regularity for convex domain and target 5. Application: Nonlinear interpolation 6. Application: Time-step minimization and nonlinear diffusion 6.1 Discrete time approximation 6.2 Euler–Lagrange equation Supported in part by NSF Grant DMS-94-24342. This paper appeared in Current Developments in Mathematics 1997, ed. by S. T. Yau 1
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6.3 Convergence 7. Application: Semigeostrophic models in meteorology 7.1 The PDE in physical variables 7.2 The PDE in dual variables 7.3 Frontogenesis Part II: Cost = Distance 8. Heuristics 8.1 Geometry of optimal transport 8.2 Lagrange multipliers 9. Optimal mass transport 9.1 Solution of dual problem 9.2 Existence of optimal mass transfer plan 9.3 Detailed mass balance, transport density 10. Application: Shape optimization 11. Application: Sandpile models 11.1 Growing sandpiles 11.2 Collapsing sandpiles 11.3 A stochastic model 12. Application: Compression molding Part III: Appendix 13. Finite-dimensional linear programming References In Memory of Frederick J. Almgren, Jr. and Eugene Fabes 2
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1 Introduction These notes are a survey documenting an interesting recent trend within the calculus of variations, the rise of differential equations techniques for Monge–Kantorovich type optimal mass transfer problems. I will discuss in some detail a number of recent papers on various aspects of this general subject, describing newly found applications in the calculus of vari- ations itself and in physics. An important theme will be the rather different analytic and geometric tools for, and physical interpretations of, Monge–Kantorovich problems with a uniformly convex cost density (here exemplified by c ( x,y ) = 1 2 | x y | 2 ) versus those prob- lems with a nonuniformly convex cost (exemplified by c ( x,y ) = | x y | ). We will as well study as applications several physical processes evolving in time, for which we can identify optimal Monge–Kantorovich mass transferences on “fast” time scales. The current text corrects some minor errors in earlier versions, improves the exposition a bit, and adds a few more references. The interested reader may wish to consult as well the lecture notes of Urbas [U1] and of Ambrosio [Am] for more. 1.1 Optimal mass transfer The original transport problem, proposed by Monge in the 1780’s, asks how best to move a pile of soil or rubble (“d´ eblais”) to an excavation or fill (“remblais”), with the least amount of work. In modern parlance, we are given two nonnegative Radon measures µ ± on R n ,
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