Metric entropy for functions of bounded total generalized variation

TL;DR

This paper derives a precise estimate of the minimal binary representation size for functions of bounded total generalized variation in metric spaces, linking it to geometric dimensions, and applies it to entropy solutions of scalar conservation laws.

Contribution

It provides a sharp, explicit metric entropy estimate for bounded variation functions in general metric spaces, connecting it to doubling and packing dimensions.

Findings

Explicit entropy bounds in terms of geometric dimensions

Application to entropy solutions of conservation laws

Improved understanding of function complexity in metric spaces

Abstract

We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space $(E,\rho)$ up to an accuracy of $\varepsilon>0$ with respect to the ${\bf L}^1$-distance. Such an estimate is explicitly computed in terms of doubling and packing dimensions of $(E,\rho)$. The obtained result is applied to provide an upper bound on the metric entropy for a set of entropy admissible weak solutions to scalar conservation laws in one-dimensional space with weakly genuinely nonlinear fluxes.