An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces

TL;DR

This paper introduces an axiomatic framework for gradient relations in Banach spaces, enabling the analysis of variational problems like Dirichlet and obstacle problems beyond traditional function spaces, with broad applicability.

Contribution

It presents a novel, space-independent approach to generalized gradients, extending variational problem analysis to metric spaces and operator algebras, and explores lattice structures in Banach spaces.

Findings

Established conditions for existence and uniqueness of solutions.

Extended Dirichlet and obstacle problems to non-function space settings.

Provided numerous examples demonstrating the framework's versatility.

Abstract

We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on (ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach.