The Trace Theorem for Anisotropic Sobolev — Slobodetskii Spaces with Applications to Nonhomogeneous Elliptic BVPs

Е.В Саженкова,С.А Саженков

doi:10.14258/izvasu(2018)4-19

Abstract

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.DOI 10.14258/izvasu(2018)4-19

Highlights

This article is devoted to a study of a class of anisotropic Sobolev — Slobodetskii spaces and their dual spaces
We focus on the question about regularity properties of traces of functions from W s,p(O) on subsets of ∂O
In order to study properties of traces of φ ∈ W s,p(O) on (N −1)-dimensional manifolds M ⊂ ∂O, we introduce the notion of W γ,r(M)

Summary

Introduction

We focus on the question about regularity properties of traces of functions from W s,p(O) on subsets of ∂O Our interest to this question is motivated by applications to p-elliptic equations (see Eq (7) in Section 2.) supplemented by either Dirichlet or Neumann nonhomogeneous conditions. Such equations arise in modelling of heat transfer, gas diffusion, etc [1–6]. Embedding in anisotropic Sobolev spaces of the first order were studied in [12], and the following result was established [12, Inequality (12)] (see [1, Lemma 3.6 and Remark 3.6]): Proposition 1.

The Trace Theorem in Anisotropic Sobolev Slobodetskii Spaces

Notion of anisotropic Sobolev