In this work we address the issue of Sobolev regularity of solutions of Monge-Ampere equations in a borderline case and show how this question is related to a rigidity problem for partial di¤erential inclusions.
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Luigi Ambrosio, Guido De Philippis, Bernd Kirchheim, Regularity of optimal transport maps and partial differential inclusions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 3, pp. 311–336DOI 10.4171/RLM/602