2016
DOI: 10.1016/j.na.2015.12.010
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An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces

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 fo… Show more

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Cited by 2 publications
(1 citation statement)
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“…For integral functionals based on the presently introduced Sobolev spaces certain minimisation problems possess a solution. A simple example is given by the Rayleigh quotient where one may check the conditions of the abstract framework of Arnlind, Björn and Björn, see [ABB16,5.3], in the situation of 7.21 if 1 < q < ∞ using 5.26 and 7.22. A more comprehensive study of the problem would include investigation of lower semicontinuity of integral functionals defined on the presently introduced Sobolev spaces in the spirit of quasiconvexity.…”
Section: Minimisation Of Integral Functionalsmentioning
confidence: 99%
“…For integral functionals based on the presently introduced Sobolev spaces certain minimisation problems possess a solution. A simple example is given by the Rayleigh quotient where one may check the conditions of the abstract framework of Arnlind, Björn and Björn, see [ABB16,5.3], in the situation of 7.21 if 1 < q < ∞ using 5.26 and 7.22. A more comprehensive study of the problem would include investigation of lower semicontinuity of integral functionals defined on the presently introduced Sobolev spaces in the spirit of quasiconvexity.…”
Section: Minimisation Of Integral Functionalsmentioning
confidence: 99%