On the relaxation of variational integrals in metric Sobolev spaces

TL;DR

This paper extends the theory of relaxation of variational integrals from classical Sobolev spaces to metric Sobolev spaces, providing a general framework and integral representation theorems applicable in metric measure spaces.

Contribution

It introduces a comprehensive framework for relaxed variational integrals in metric Sobolev spaces, extending previous Euclidean results to more general metric measure spaces.

Findings

Established integral representation theorems in metric Sobolev spaces

Extended convex and non-convex variational integral results to metric spaces

Applied the theorems within Cheeger-Keith's differentiable structure

Abstract

We give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for relaxed variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger-Keith's differentiable structure.