On the Dirichlet problem for the one-dimensional ROF functional

TL;DR

This paper investigates conditions under which minimizers of the 1D ROF functional adhere to Dirichlet boundary data, using total variation flow results and providing counterexamples.

Contribution

It establishes sufficient conditions for Dirichlet data adherence and presents counterexamples, advancing understanding of boundary behavior in 1D ROF minimization.

Findings

Minimizers satisfy Dirichlet data under certain conditions

Counterexamples demonstrate boundary behavior exceptions

Results leverage total variation flow properties

Abstract

We provide a number of sufficient conditions for that minimizers of the one-dimensional Rudin-Osher-Fatemi functional satisfy the Dirichlet data in the trace sense. For this purpose we use results specific for the total variation flow. We also show a number of counterexamples.