Published Paper

Inserted: 27 feb 2023
Last Updated: 12 sep 2024

Journal: Arch. Ration. Math. Anal.
Year: 2023
Doi: https://doi.org/10.1007/s00205-023-01938-w

Abstract:

In this paper we analyze in detail a few questions related to the theory of functions with bounded $p$-Hessian-Schatten total variation, which are relevant in connection with the theory of inverse problems and machine learning. We prove an optimal density result, relative to the $p$-Hessian-Schatten total variation, of continuous piecewise linear (CPWL) functions in any space dimension $d$, using a construction based on a mesh whose local orientation is adapted to the function to be approximated. We show that not all extremal functions with respect to the $p$-Hessian-Schatten total variation are CPWL. Finally, we prove existence of minimizers of certain relevant functionals involving the $p$-Hessian-Schatten total variation in the critical dimension $d=2$.