Communications in Analysis and Geometry
Volume 28 (2020)
Area minimizing discs in locally non-compact metric spaces
Pages: 89 – 112
We solve the classical problem of Plateau in every metric space which is $1$‑complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as $L^1$, all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second author from the setting of proper metric spaces to that of locally non-compact ones.We furthermore solve the Dirichlet problem in the same class of spaces. The main new ingredient in our proofs is a suitable generalization of the Rellich–Kondrachov compactness theorem, from which we deduce a result about ultra-limits of sequences of Sobolev maps.
Research partially supported by Swiss National Science Foundation Grants 153599 and 165848.
Received 17 May 2017
Accepted 7 November 2017
Published 12 March 2020