Currents carried by the subgradient graphs of semi-convex functions and applications to Hessian measure

TL;DR

This paper investigates currents associated with subgradient graphs of semi-convex functions, establishing a weak continuity theorem and providing a novel approach to calculating Hessian measures using currents.

Contribution

It introduces a new framework for analyzing subgradient graphs via currents and offers an alternative method for computing Hessian measures.

Findings

Established weak continuity of currents under pointwise convergence

Derived a new method to compute Hessian measures using currents

Extended the understanding of semi-convex functions and their subgradients

Abstract

In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on a $n$-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the $k$-Hessian measures are calculated by a different method in terms of currents.