Relaxation for a degenerate functional with linear growth in the onedimensional case

TL;DR

This paper investigates the relaxation of a degenerate linear growth functional in one dimension, providing explicit formulas and domain descriptions using weighted Sobolev inequalities without standard doubling or Muckenhoupt conditions.

Contribution

It offers a novel explicit representation of the relaxed functional and its domain for a degenerate linear growth functional with non-standard weights.

Findings

Explicit formula for the relaxed functional

Characterization of the domain of the relaxed functional

Application of weighted Sobolev inequalities without doubling conditions

Abstract

In this work, we study the relaxation of a degenerate functional with linear growth, depending on a weight $w$ that does not exhibit doubling or Muckenhoupt-type conditions. In order to obtain an explicit representation of the relaxed functional and its domain, our main tools for are Sobolev inequalities with double weight.