Characterisation of upper gradients on the weighted Euclidean space and applications

TL;DR

This paper investigates Sobolev spaces on Euclidean spaces with arbitrary Radon measures, establishing equivalences among various definitions and characterizing minimal weak upper gradients for Lipschitz functions.

Contribution

It provides a unifying framework for Sobolev spaces in weighted Euclidean spaces and characterizes minimal weak upper gradients, advancing the understanding of analysis on such spaces.

Findings

Proves equivalence of different Sobolev space notions

Characterizes minimal weak upper gradients for Lipschitz functions

Enhances understanding of analysis on weighted Euclidean spaces

Abstract

In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.