Extendability of functions with partially vanishing trace
Sebastian BECHTEL (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Russell M. BROWN (University of Kentucky)
Robert HALLER (Technische Universität Darmstadt)
Patrick TOLKSDORF (Karlsruhe Institut für Technologie)
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Abstract
Let Ω ⊆ Rd be open and D ⊆ ∂Ω be a closed part of its boundary. Under very mild assumptions on Ω, we construct a bounded Sobolev extension operator for the Sobolev space WD k,p(Ω), 1 ⩽ p < ∞, which consists of all functions in Wk,p(Ω) that vanish in a suitable sense on D. In contrast to earlier work, this construction is global and does not use a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing D and ∂Ω \ D. Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on D.
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