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Extending Sobolev functions with partially vanishing traces from locally (ε, δ)-domains and applications to mixed boundary problems
Brewster, K Mitrea, D Mitrea, I Mitrea, M
Brewster, K
Mitrea, D
Mitrea, I
Mitrea, M
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Journal Article
Date
2014-04-01
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Higher-order Sobolev space
Linear extension operator
Locally (epsilon, delta)-domain
Higher-order boundary trace operator
Real and complex interpolation
Besov and Triebel-Lizorkin spaces
Bessel potential space and capacity
Ahlfors regular set
Synthesis
Mixed boundary value problem
Higher-order elliptic system
Linear extension operator
Locally (epsilon, delta)-domain
Higher-order boundary trace operator
Real and complex interpolation
Besov and Triebel-Lizorkin spaces
Bessel potential space and capacity
Ahlfors regular set
Synthesis
Mixed boundary value problem
Higher-order elliptic system
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10.1016/j.jfa.2014.02.001
Abstract
We prove that given any k∈N, for each open set Ω⊆Rn and any closed subset D of Ω- such that Ω is locally an (ε, δ)-domain near ∂Ω. D, there exists a linear and bounded extension operator Ek,D mapping, for each p∈. [1, ∞], the space WDk,p(Ω) into WDk,p(Rn). Here, with O denoting either Ω or Rn, the space WDk,p(O) is defined as the completion in the classical Sobolev space Wk,p(O) of (restrictions to O of) functions from Cc∞(Rn) whose supports are disjoint from D. In turn, this result is used to develop a functional analytic theory for the class WDk,p(Ω) (including intrinsic characterizations, boundary traces and extensions results, interpolation theorems, among other things) which is then employed in the treatment of mixed boundary value problems formulated in locally (ε, δ)-domains. Finally, we also prove extension results on the scales of Besov and Bessel potential spaces on (ε, δ)-domains with partially vanishing traces on Ahlfors regular sets and explore some of the implications of such extension results. © 2014 Elsevier Inc.
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Elsevier BV
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Journal of Functional Analysis
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