A note on the a.e. second-order differentiability of rank-one convex functions
Jonas Hirsch
TL;DR
The paper investigates the a.e. second-order differentiability of bounded rank-one convex functions $f: B_1 \subset \mathbb{R}^{m\\times n} \to \mathbb{R}$ and extends Alexandrov's theorem to rank-one convexity. It adopts viscosity techniques from fully nonlinear elliptic equations and replaces the Hessian with a two-homogeneous elliptic operator along rank-one directions, together with Lin's interior $W^{2,\\epsilon}$ estimates to obtain quantitative a.e. control. Key steps include paraboloid tangency a.e., a lower-bound-from-upper-bound lemma via a radial comparison, and a Maly-type sup/inf-convolution argument that yields a.e. differentiability of the gradient and hence a.e. second-order differentiability of $f$. The result shows that a.e. differentiability of the Hessian $D^2 f$ can be obtained via a reduction to one-dimensional monotone differentiability, with an appendix clarifying this reduction.
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.