An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces
2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 134, 70-104 p.Article in journal (Refereed) PublishedText
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. (C) 2015 Elsevier Ltd. All rights reserved.
Place, publisher, year, edition, pages
PERGAMON-ELSEVIER SCIENCE LTD , 2016. Vol. 134, 70-104 p.
Dirichlet problem; First eigenvalue; Generalized Sobolev space; Gradient relation; Lattice; Metric space; Noncommutative function; Obstacle problem; Operator-valued function; Partial order; Poincare set; Rayleigh quotient; Rellich-Kondrachov cone; Trace class ideal; Variational problem
IdentifiersURN: urn:nbn:se:liu:diva-126128DOI: 10.1016/j.na.2015.12.010ISI: 000370489300004OAI: oai:DiVA.org:liu-126128DiVA: diva2:912076
Funding Agencies|Swedish Research Council2016-03-152016-03-152016-05-04