The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile

Autor: Jean Louet, Luigi De Pascale, Filippo Santambrogio
Přispěvatelé: Dipartimento di Matematica [Pisa], University of Pisa - Università di Pisa, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Financement PGMO (EDF & FMJH) : projet MACRO (FRANCE), project 2010A2TFX2 {\it'Calcolo delle Variazioni'}, financed by the Italian Ministry of Research (ITALY)
Jazyk: angličtina
Rok vydání: 2014
Předmět:
General Mathematics
02 engineering and technology
01 natural sciences
Measure (mathematics)
density of smooth maps
Monge problem
Mathematics - Analysis of PDEs
0202 electrical engineering
electronic engineering
information engineering
Optimal transport
FOS: Mathematics
Optimal transport
Monge problem
monotone transport
Γ-convergence
density of smooth maps
Order (group theory)
49J45
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Limit (mathematics)
0101 mathematics
Mathematics - Optimization and Control
Mathematics
Applied Mathematics
010102 general mathematics
Mathematical analysis
smooth maps
Dirichlet's energy
Lipschitz continuity
Sobolev space
Monotone polygon
Γ-convergence
Optimization and Control (math.OC)
monotone transport
density of
020201 artificial intelligence & image processing
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
AMS subject classification 49J30
Analysis of PDEs (math.AP)
Zdroj: Journal de Mathématiques Pures et Appliquées
Journal de Mathématiques Pures et Appliquées, Elsevier, 2016, 106, pp.237-279. ⟨10.1016/j.matpur.2016.02.009⟩
ISSN: 0021-7824
DOI: 10.1016/j.matpur.2016.02.009⟩
Popis: We investigate the approximation of the Monge problem (minimizing ∫ Ω | T ( x ) − x | d μ ( x ) among the vector-valued maps T with prescribed image measure T # μ ) by adding a vanishing Dirichlet energy, namely e ∫ Ω | DT | 2 . We study the Γ-convergence as e → 0 , proving a density result for Sobolev (or Lipschitz) transport maps in the class of transport plans. In a certain two-dimensional framework that we analyze in details, when no optimal plan is induced by an H 1 map, we study the selected limit map, which is a new “special” Monge transport, possibly different from the monotone one, and we find the precise asymptotics of the optimal cost depending on e, where the leading term is of order e | log ⁡ e | .
Databáze: OpenAIRE
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