We prove that given any k ∈ N, for each open set Ω ⊆ R n 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 operatoris defined as the completion in the classical Sobolev space W k,p (O) of (restrictions to O of) functions from C ∞ c (R n ) whose supports are disjoint from D. In turn, this result is used to develop a functional analytic theory for the class W k,p D (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.

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Abstract. We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R. Brown in (1994). In this context, we obtain results which generalize those by D. Jerison and C. Kenig (1995) as well as E. Fabes, O. Mendez and M. Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

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We give a complete description of the spectra of certain elastostatic and hydrostatic boundary layer potentials in L p , 1 < p < ∞, on bounded curvilinear polygons. In particular, our analysis shows that the spectral radii of these operators on L p , 2 ≤ p < ∞ are less than one. Such results are relevant in the context of constructively solving boundary value problems for the Lamé system of elasticity, the Stokes system of hydrostatics as well as the two dimensional Laplacian on curvilinear polygons. Our approach is based on Mellin transform techniques and Calderón-Zygmund theory.Math Subject Classifications. primary: 45E05, 47A10; secondary: 35J25, 42B20.