A boundary uniqueness property for weighted Sobolev functions
Boundary behavior of mappings in Sobolev spaces has been studied extensively from various points of view. The author mentions the papers of \textit{J. Jenkins} [Indiana Univ. Math. J. 41, 1077-1080 (1992; Zbl 0766.30028)], \textit{P. Koskela} [Bull. Lond. Math. Soc. 27, 460-466 (1995; Zbl 0838.31007)], \textit{V. M. Miklyukov} and \textit{M. Vuorinen} [Tohoku Math. J., II Ser. 50, 503-511 (1998; Zbl 0914.31003)] and \textit{Y. Mizuta} [Proc. Am. Math. Soc. 126, 1043-1047 (1998; Zbl 0890.31005)] as motivation for this work. Indeed, the purpose of this paper is to obtain a weighted version of a Beurling type uniqueness theorem in these papers concerning the following effect: if the weighted Dirichlet \(p\)-integral of functions \(u \in W^{1,p }(B^n)\) has a growth regulated by suitable function \(\varphi\) and the function tends to 0 along almost every curve terminating at the points of a set \(E \subset \partial B^n\) of positive capacity, then \(u\equiv 0\).
scientific article; zbMATH DE number 1741726