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A perturbation argument for a Monge-Ampere type equation arising in optimal transportation
González Nogueras, María del Mar; Caffarelli, Luis; Nguyen, Truyen
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
We prove some asymptotic interior regularity results for potential functions of optimal transportation problems with power costs. We show that our problems are equivalent to optimal transportation problems whose cost functions are sufficiently small perturbations of the quadratic cost but they do not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge- Amp`ere equation in order to obtain interior H¨older estimates for second derivatives of potentials, and a careful understanding of why we might fail to have an Alexandroff weak solution when restricted to subdomains. In particular, we provide some quantitative estimates along the way on how the equation degenerates near the boundary.
Àrees temàtiques de la UPC::Matemàtiques i estadística
Monge-Ampère equation
Attribution-NonCommercial-NoDerivs 3.0 Spain
Artículo - Borrador

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