Publication at Faculty of Mathematics and Physics |
2016
Abstract
Given 1 LESS-THAN OR EQUAL TO p < k LESS-THAN OR EQUAL TO n, we construct a strictly convex function f ELEMENT OF W^{2,p}((0,1)^n) with α-Hölder continuous derivative for any 0 < α < 1 such that 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 ELEMENT OF W^{2,p}, p GREATER-THAN OR EQUAL TO k, and rank NABLA^2 g < k a.e., then g cannot be strictly convex on any open portion of the domain.