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

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• 作者：Joakim Arnlind ; ^{joakim.arnlind@liu.se" class="auth_mail" title="E-mail the corresponding author} ; Anders Bjö ; rn ^{anders.bjorn@liu.se" class="auth_mail" title="E-mail the corresponding author} ; Jana Bjö ; rn ^{jana.bjorn@liu.se" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 49J27 ; secondary ; 31E05 ; 35J50 ; 35J66 ; 49Q20 ; 46L51 ; 46L52
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：134
• 期：Complete
• 页码：70-104
• 全文大小：724 K

文摘

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.