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A Radial Uniqueness Theorem for Sobolev Functions
Author(s) -
Koskela P.
Publication year - 1995
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/27.5.460
Subject(s) - mathematics , sobolev space , zero (linguistics) , unit sphere , uniqueness , exponent , combinatorics , sobolev inequality , mathematical analysis , open set , zero set , constant (computer programming) , philosophy , linguistics , computer science , programming language
We show that continuous functions u in the Sobolev spaceW p 1 ( B ) , 1 < p ⩽ n , which have the limit zero in a certain weak sense in a set of positive p ‐capacity on ∂ B with∫ B e| ∇ u | p d x ⩽ C ɛ p ( log [ 1 ɛ ] )p ‐ 1, where B is the open unit ball of R n andB ɛ = { x ∈ B : | u ( x ) | < ɛ }for 0 > ɛ > ½, are identically zero. Conversely, we produce for each 1 > p ⩾ n and each positive ∂ a non‐constant function u inW p 1 ( B ) , continuous in B ¯ , and a compact set E ⊂∂ B of positive p ‐capacity such that u = 0 in E and the above inequality holds with exponent p − l + δ.