Abstract
Given \(1\le p<k\le n\), we construct a strictly convex function \(f\in W^{2,p}((0,1)^n)\) with \(\alpha \)-Hölder continuous derivative for any \(0<\alpha <1\) such that \({\text {rank}}\nabla ^2 f < k\) almost everywhere in \((0,1)^n\). In particular, the mapping \(F=\nabla f\) is an example of a \(W^{1,p}\) homeomorphism whose differential has rank strictly less than k almost everywhere in the unit cube. This Sobolev regularity is sharp in the sense that if \(g\in W^{2,p}\), \(p\ge k\), and \({\text {rank}}\nabla ^2 g < k\) a.e., then g cannot be strictly convex on any open portion of the domain.
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Acknowledgments
The authors would like to thank Robert Jerrard and Reza Pakzad for many useful discussions, and Camillo De Lellis for an advice how to refine citations in the bibliography about the isometric embedding. Also, we thank the anonymous referee for careful readings and comments how to improve the statements of the results. These comments are reflected in Remarks 3 and 4.
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Communicated by C. De Lellis.
Z. Liu was supported by the ERC CZ Grant LL1203 of the Czech Ministry of Education. J. Malý is supported by the Grant GA ČR P201/15-08218S of the Czech Science Foundation.
