The Trace Theorem for Anisotropic Sobolev — Slobodetskii Spaces with Applications to Nonhomogeneous Elliptic BVPs
In this paper, anisotropic Sobolev — Slobodetskii spaces in poly-cylindrical domains of any dimension N are considered. In the first part of the paper we revisit the well-known Lions — Magenes Trace Theorem (1961) and, naturally, extend regularity results for the trace and lift operators onto the anisotropic case. As a byproduct, we build a generalization of the Kruzhkov — Korolev Trace Theorem for the first-order Sobolev Spaces (1985). In the second part of the paper we observe the nonhomogeneous Dirichlet, Neumann, and Robin problems for p-elliptic equations. The well-posedness theory for these problems can be successfully constructed using isotropic theory, and the corresponding results are outlined in the paper. Clearly, in such a unilateral approach, the anisotropic features are ignored and the results are far beyond the critical regularity. In the paper, the refinement of the trace theorem is done by the constructed extension.
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