Extending Sobolev functions with partially vanishing traces from locally [formula omitted]-domains and applications to mixed boundary problems

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.