Approximation by mappings with singular Hessian minors

TL;DR

This paper constructs smooth approximations of functions with controlled Hessian rank, showing that such approximations can be made while preserving certain regularity and convergence properties, contrasting with known regularity limitations.

Contribution

It introduces a method to approximate functions in $W^{2,p}$ with Hessian minors of rank less than $k$, maintaining regularity and convergence, which was previously not well-understood.

Findings

Constructed sequences with Hessian rank less than k almost everywhere.

Achieved convergence in $C^{1,eta}$ and weakly in $W^{2,p}.

Contrasts with known regularity restrictions for functions with the same Hessian rank constraints.

Abstract

Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$ and any $u\in W^{2,p}(\Omega)$ belonging to the little H\"older class $c^{1,\alpha}$, we construct a sequence $u_j$ in the same space with $\operatorname{rank}D^2u_j<k$ 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 $W^{2,p}$, $p\geq k$, satisfying the same rank inequality.