Planar Sobolev extension domains
This doctoral thesis deals with geometric characterizations of bounded planar simply connected Sobolev extension domains. It consists of three papers. In the ﬁrst and third papers we give full geometric characterizations of W 1, p-extension domains for 1 < p < 2 and p = 1, respectively. The second paper establishes a density result for Sobolev functions on planar domains, necessary for the solution for the case p = 1. Combining with the known results, we obtain a full geometric characterization of W 1, p-extension domains for every 1 ≤ p ≤ ∞.
PublisherUniversity of Jyväskylä
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Rajala, Tapio (Springer, 2021)We show that any bounded domain in a doubling quasiconvex metric space can be approximated from inside and outside by uniform domains.
Koskela, Pekka; Koski, Aleksis; Onninen, Jani (Elsevier Inc., 2020)Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the ...
Nandi, Debanjan; Rajala, Tapio; Schultz, Timo (Springer Netherlands, 2019)We show that in a bounded simply connected planar domain Ω the smooth Sobolev functions Wk,∞(Ω) ∩ C∞(Ω) are dense in the homogeneous Sobolev spaces Lk,p(Ω).
Gigli, Nicola; Pasqualetto, Enrico; Soultanis, Elefterios (Elsevier, 2020)We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove ...
Zhu, Zheng (Springer Science and Business Media LLC, 2021)In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of classical quasidisks. After that, we also find some applications of this property.