David Swanson, William P. Ziemer
Abstract
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Let Ω⊂Rn be an arbitrary open set. In this paper it is shown that if a Sobolev function f∈W1, p(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then f is weakly zero on ϖΩ in the sense that f∈W ${}_{0}^{1,p}$ (Ω).
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Let Ω⊂Rn be an arbitrary open set. In this paper it is shown that if a Sobolev function f∈W1, p(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then f is weakly zero on ϖΩ in the sense that f∈W ${}_{0}^{1,p}$ (Ω).
Key concepts: Sobolev space, Zero (linguistics), Zero set, TRACE (psycholinguistics), Mathematics, Boundary (topology), Lebesgue integration, Trace operator