A Radial Uniqueness Theorem for Sobolev Functions

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.