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.

中文翻译：

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局部非紧度量空间中的面积最小化圆盘

我们在每个度量空间中解决了经典的高原问题，该空间在其自身的超完备性中是 $1$-互补的。这包括所有真度量空间以及许多局部非紧度量空间，特别是所有对偶 Banach 空间、一些非对偶 Banach 空间（如 $L^1$）、所有 Hadamard 空间等等。我们的结果将 Lytchak 和第二作者的相应结果从适当的度量空间的设置推广到局部非紧致空间的设置。我们进一步在同一类空间中解决狄利克雷问题。我们证明中的主要新成分是对 Rellich-Kondachov 紧致定理的适当推广，我们从中推导出关于 Sobolev 映射序列的超极限的结果。