A strictly convex Sobolev function with null Hessian minors

作者：Liu Zhuomin; Maly Jan^*

来源：Calculus of Variations and Partial Differential Equations, 2016, 55(3): 58.

DOI：10.1007/s00526-016-0994-7

Given 1 <= p < k <= n, we construct a strictly convex function f is an element of W-2,W- p((0, 1)(n)) with alpha-Holder continuous derivative for any 0 < alpha < 1 such that rank del(2) f < k almost everywhere in (0, 1)(n). In particular, the mapping F = del f is an example of a W-1,W- 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 is an element of W-2,W- p, p >= k, and rank del(2)g < k a.e., then g cannot be strictly convex on any open portion of the domain.

• 出版日期2016-6