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. Namely, we formulate proper weakly regular anisotropic classes for boundary conditions, so that the boundary value problems appear to be well-posed. Finally, the analogous results are formulated for the p-parabolic problems.

Introduction This article is devoted to a study of a class of anisotropic Sobolev -Slobodetskii spaces and their dual spaces .Here domain is a polycylinder in , i.e., it meets the following requirements.

The Trace Theorem in Anisotropic Spaces
Condition C.
• B i are nonempty bounded open sets with Lipschitz boundaries ∂B i .
Multi-indices s = (s 1 , s 2 , . . ., s k ) and p = (p 1 , p 2 , . . ., p k ) are such that s i ∈ (0, 1] and 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][2][3][4][5][6].Anisotropy p i = p j means that diffusion rates differ in different directions x i and x j .
The general theory of isotropic Sobolev Slobodetskii spaces was built in sixtieth of 20th century [7][8][9].Within its framework, regularity properties of traces of functions φ: R N x → R on (N − 1)-dimensional manifolds were investigated in detail (see, for example, [7, Theorem 5.1], [10, Theorems 2.21 and 2.22]).Some applications of trace theorems to the Dirichlet, Neumann, and Robin problems for the isotropic p-Laplacian equation can be found in [11,Chapter 2].Embedding in anisotropic Sobolev spaces of the first order were studied in [12], and the following result was established [12,Inequality (12)] (see also [1,Lemma 3.6 and Remark 3.6]): x be a bounded domain with the smooth boundary Then where Also, worth to notice the results on traces in anisotropic spaces that were achieved in [13,14].
The present article is organized as follows.
In Section 1, we formulate the trace theorem for anisotropic Sobolev Slobodetskii spaces (Theorem 1).Then we outline a brief scheme of its proof.This theorem somewhat extends the result of Proposition 1, in particular.
In Section 2, we give some examples of application of this theorem to the basic nonhomogeneous boundary value problems.
where x i ∈ B i (i = 1, . . ., k), and denote For the sake of conciseness, for φ: Ō → R we write when it is suitable.
Quite analogously, we also introduce further the notation s i and p i .

Remark 1. In line with the above introduced notation, remark that ∂O
Of course, in the right-hand side here we mean that the order of Cartesian product is proper, that is, by , where, as usually, for sufficiently regular Ψ.
Further, for s i ∈ (0, 1) and sufficiently smooth φ: For s i = 1 we canonically define Now we are in a position to define anisotropic Sobolev Slobodetskii spaces W s,p (O) in a rigorous way.
Definition 1.The space of functions φ: O → R equipped with the Sobolev Slobodetskii norm In order to study properties of traces of φ ∈ W s,p (O) on (N −1)-dimensional manifolds M ⊂ ∂O, we also introduce the notion of W γ,r (M).To this end, we naturally induce the norm of W s,p (O) onto any (N −1)-dimensional Lipschitz manifold M ⊂ ∂O via its atlas, and set (1) where Obviously, restriction of φ ∈ W γ,r (∂O) to M ⊂ ∂O (M Lipschitz) belongs to W γ,r (M).If M ∩ ∂B i = ∅ then γ i and r i are dumb indices and therefore can be taken arbitrarily.
is called the interior boundary trace operator.
Here D( Ō) is the space of C ∞ functions with compact support contained in Ō.
The following theorem is the first main result of the article.

The Trace Theorem in Anisotropic Spaces
Theorem 1. (The Trace Theorem in Anisotropic Spaces.)Let O ⊂ R N x be a polycylinder that meets the requirements of Condition C. Then the following assertions hold true.(i) For any p = (p 1 , p 2 , . . ., p k ), p i ∈ (1, +∞), the interior boundary trace operator γ int 0 , defined by (2) for φ ∈ D(O), admits a continuous extension hereby there is a constant c T > 0 such that (Constant c T is independent of φ.) (ii) For any p = (p 1 , p 2 , . . ., p k ) p i ∈ (1, +∞), the interior boundary trace operator γ int 0 has a continuous right inverse operator (called the lift operator) satisfying γ int 0 Eψ = ψ for all ψ ∈ W γ,p (∂O) as well as (Constant c IT is independent of ψ.Notation γ is the same as in (4).) Remark 4. Let M ⊂ ∂O be a (N − 1)-dimensional Lipschitz manifold.On the strength of Remark 2, the both assertions of Theorem 1 hold true with M on the place of ∂O.
1.4.Brief scheme of proof of Theorem 1.The idea of the proof is very simple.Firstly, we directly apply the Lions Magenes Trace Theorem [7, Theorem 5.1] (in the isotropic case) to the space of functions (x i → φ( x i ; x i )) ∈ W 1,pi (B i ) for a.e.x i ∈ Ō xi and construct the trace operators Secondly, we notice that, since O is polycylinder, then variables x 1 , x 2 , . . ., x k are separated, so that the operator γ int 0 defined by the rule 3) and is the trace operator by construction.Thirdly, thus constructed γ int 0 is surjective since all γ int,Bi 0 are surjective and the sets ∂B i × Ō xi do not overlap each other.Hence the right inverse operator E is also defined and satisfies (5).

On Boundary Value Problems for the
Anisotropic p-Laplacian Equations.In this section we revisit the theory of weak generalized solutions to the Dirichlet, Neumann, and Robin problems for the isotropic p-Laplacian equation.This theory was mainly built in [11,Chapter 2].With the help of Theorem 1, we somewhat extend it onto the anisotropic case.
2.1.Formulations of the basic problems for the anisotropic p-Laplacian equation.Let O ⊂ R N x be a polycylinder that meets Condition C. Let p = (p 1 , p 2 , . . ., p k ), We consider the anisotropic p-Laplacian equation where g = g(x) is given, or the Neumann boundary condition where h = h(x) is given, or the Robin boundary condition , H = H(x) and κ = const > 0 are given.In ( 9), (10), and further, ν i is the unit outward normal to ∂B i .(If x ∈ ∂O but x ∈ ∂B i then in (9) and (10) we simply take zero vector for ν i (x).) In the isotropic case, the following existence and uniqueness results were established in [11 Trace Theorem, and of the fact that W 1,α (O) → W 1,β (O) for α > β, in the general anisotropic case leads to the assertions of Proposition 2 with g ∈ W 1/p * ,p * (∂O), h ∈ W −1/p * ,p * (∂O), H ∈ W −1/p * ,p * (∂O), and u ∈ W 1,p * (O) on the places of g ∈ W 1/r ,r (∂O), h ∈ W −1/r ,r (∂O), H ∈ W −1/r ,r (∂O), and u ∈ W  (6). The following assertions hold true (i) Whenever g ∈ W γ,p (∂O), the Dirichlet problem (7), ( 8) has the unique weak generalized solution u ∈ W 1,p (O).

1 .
The Trace Theorem in Anisotropic Sobolev Slobodetskii Spaces 1.1.Some useful notation.In Introduction we have entered a polycylindrical domain O ⊂ R N x satisfying Condition C. Now let us introduce some relevant convenient notation in order to study traces of functions from W
Remark 2. The important case is M = ∂O.