An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces. (March 2016)

• Record Type: Journal Article
• Authors: Arnlind, Joakim; Björn, Anders; Björn, Jana
• Is Part Of: Nonlinear analysis. Volume 134(2016)
• Page Start: 70
• Page End: 104
• Publication Date: 2016-03
• Subjects: primary 49J27 -- secondary 31E05 35J50 35J66 49Q20 46L51 46L52; Dirichlet problem -- First eigenvalue -- Generalized Sobolev space -- Gradient relation -- Lattice -- Metric space -- Noncommutative function -- Obstacle problem -- Operator-valued function -- Partial order -- Poincaré set -- Rayleigh quotient -- Rellich–Kondrachov cone -- Trace class ideal -- Variational problem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.12.010
• Languages: English
• ISSNs: 0362-546X

Abstract

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.