A note on the a.e. second-order differentiability of rank-one convex functions

TL;DR

This paper extends Alexandrov's theorem to rank-one convex functions in Euclidean space, using viscosity techniques, and reduces the problem to the almost everywhere differentiability of one-dimensional monotone functions.

Contribution

It introduces a novel approach based on viscosity methods to prove second-order differentiability of rank-one convex functions, expanding classical convex analysis.

Findings

Rank-one convex functions are twice differentiable almost everywhere.

Viscosity techniques are effective in analyzing second-order differentiability.

The classical theorem is reducible to one-dimensional monotone function differentiability.

Abstract

In the Euclidean setting, the well-known Alexandrov theorem states that convex functions are twice differentiable almost everywhere. In this note, we extend this theorem to rank-one convex functions. Our approach is novel in that it draws more from viscosity techniques developed in the context of fully nonlinear elliptic equations. As a byproduct, the original Alexandrov theorem can essentially be reduced to the a.e. differentiability of one-dimensional monotone functions, as presented in the appendix.