Embedding theorems and a variational problem for functions on a metric measure space
In an earlier work [Sib. Math. J. 54, No. 2, 353--367 (2013); translation from Sib. Mat. Zh. 54, No. 2, 450--467 (2013; Zbl 1282.46034)], the author of this article introduced new kinds of Sobolev spaces, describing the idea behind them as follows: we characterize functions by the accuracy, depending on \(\varepsilon\), with which we can approximate them in the \(L_p\)-norm by the piecewise constant functions subordinate to an arbitrary partition of the domain into disjoint measurable sets of diameter at most \(\varepsilon\). The paper under review has its main focus on embeddings of these Sobolev spaces into various other spaces, hereby extending the above mentioned publication. Before addressing questions concerning embeddings, the author first gives a sufficient condition implying that the Sobolev spaces he introduced agree with the standard Sobolev spaces. He also presents a domain where the spaces do not agree. The second section then studies embeddings of the Sobolev spaces into other spaces including Lebesgue spaces with different underlying measures and Sobolev spaces with different exponents. The last section addresses existence and uniqueness of a variational problem in the framework of these new Sobolev spaces. In the study of embeddings of Sobolev spaces, a standard assumption is that the underlying metric spaces are doubling. Remarkably, this paper also considers metric spaces that are not necessarily doubling.
scientific article; zbMATH DE number 6339892
scientific article