W2,1 regularity for solutions of the Monge-Ampère equation

Guido de Philippis; Alessio Figalli

doi:10.1007/s00222-012-0405-4

W^2,1 regularity for solutions of the Monge-Ampère equation

Guido de Philippis, Alessio Figalli

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Ampère equation, with right-hand side bounded away from zero and infinity, is W^2,1_loc. This is obtained by showing higher integrability a priori estimates for D²u, namely D²u∈Llog^kL for any k∈ℕ.

Original language	English (US)
Pages (from-to)	55-69
Number of pages	15
Journal	Inventiones Mathematicae
Volume	192
Issue number	1
DOIs	https://doi.org/10.1007/s00222-012-0405-4
State	Published - Apr 2013

ASJC Scopus subject areas

General Mathematics

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abstract = "In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp{\`e}re equation, with right-hand side bounded away from zero and infinity, is W2,1loc. This is obtained by showing higher integrability a priori estimates for D2u, namely D2u∈LlogkL for any k∈ℕ.",

author = "{de Philippis}, Guido and Alessio Figalli",

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AB - In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Ampère equation, with right-hand side bounded away from zero and infinity, is W2,1loc. This is obtained by showing higher integrability a priori estimates for D2u, namely D2u∈LlogkL for any k∈ℕ.

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W^2,1 regularity for solutions of the Monge-Ampère equation

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