Sobolev Embedding Theorem for Irregular Domains and Discontinuity of $p \to p^*(p,n)$

Abstract

There are a lot of results on the field of characterization of qΩ(p) for classes of domains. For a Lipschitz domain Ω the function p∗(p) = qΩ(p) is continuous and even smooth, (see (1.1)), this was proven by Sobolev in 1938 [12]. Later, the embedding was examined on some more problematic classes of domains by V. G. Maz’ya [9, 10], O. V. Besov and V. P. Il’in [3], T. Kilpelainen and J. Malý [5], D. A. Labutin [6, 7], B. V. Trushin [13, 14] and others. For further results and motivation we recommend the introduction by O. V. Besov [2]. Even considering somehow irregular domains, examined classes of domains have always qΩ(p) somehow nice and continuous. We construct a domain Ω such that the function of the optimal embedding qΩ(p) is continuous up to some point, has a leap at this point and then it is continuous again. The point of discontinuity p0 ∈ [n,∞) and the size of the leap can be chosen as desired. Our work is inspired by the construction of a domain in [4], but our proof is completely different. The original article shows the construction of such a domain