Approximation by mappings with singular Hessian minors

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a Lipschitz domain. Given $1\le p<k\le n$ and any $u\in W^{2,p}(\Omega )$ belonging to the little Hölder class $c^{1,\alpha }$, we construct a sequence $u_j$ in the same space with $\mathrm{rank}{D}^2{u}_j<k$ almost everywhere such that $u_j\to u$ in $C^{1,\alpha }$ and weakly in $W^{2,p}$. This result is in strong contrast with known regularity behavior of functions in $W^{2,p}$, $p\ge k$, satisfying the same rank inequality.

18 pages