A boundary uniqueness property for weighted Sobolev functions (Q1348169)

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 \(u\equiv 0\).