Trace Theorems for Sobolev Spaces on Lipschitz Domains.

A famous theorem of E. Gagliardo gives the characterization of traces for Sobolev spaces W 1 ,p (Ω) for 1 ≤ p < ∞ when Ω ⊂ R N is a Lipschitz domain. The extension of this result to W m,p (Ω) for m ≥ 2 and 1 < p < ∞ is now well-known when Ω is a smooth domain. The situation is more complicated for polygonal and polyhedral domains since the characterization is given only in terms of local compatibility conditions at the vertices, edges, .... Some recent papers give the characterization for general Lipschitz domains for m=2 in terms of global compatibility conditions. Here we give the necessary compatibility conditions for m ≥ 3 and we prove how the local compatibility conditions can be derived.


Introduction
Let Ω be a Lipschitz bounded and connected subset of R N whose bounded and orientable boundary is denoted by Γ.For 1 ≤ p < ∞ and m integer W m, p (Ω) denotes the Sobolev space of functions of L p (Ω) whose distributional derivatives up to the order m also belong to L p (Ω).For m ≥ 1 the restriction γ 0 (u) = u |Γ to Γ of a function u ∈ W m, p (Ω) is well-defined and belongs to L p (Γ).A famous result of E. Gagliardo [6] gives, for m = 1, the caracterization of the range of γ 0 .More precisely, Gagliardo proves that the operator γ 0 is linear and continuous from W 1, p (Ω) onto W 1−1/p, p (Γ) for 1 ≤ p < ∞ and has a continuous right inverse for p > 1.
For general Lipschitz domains a first characterization of the range of (γ 0 , γ 1 ) has been obtained for N = 2 and p = 2 in [7] as a byproduct of the study of the Airy function; this result has been extended to general p > 1 by [5].During a visit at the Istituto di Analisi Numerica del CNR in Pavia the following equivalent statement came out after some discussions the with F. Brezzi and A. Buffa.
A first consequence of (1.1) has been a general characterization of the range of (γ 0 , γ 1 ) for N = 3 (see [2]) that also works for all N ≥ 3.
Let us remark that the compatibility conditions at a corner follow from the characterization of W 1−1/p, p (Γ) and the exchange of n and t at the crossing of the corner.
The statement of the theorem 1.1 allows an easy interpretation of the necessity of the condition (1.1).Indeed, when u ∈ W 2, p (Ω) then grad u 2 and γ 0 (grad u) ∈ W 1−1/p, p (Γ)

2
. Hence the necessity of (3.1) follows from In this paper we give general necessary conditions for the traces of the elements of W m, p (Ω) for all integer m ≥ 2, all p > 1 and all N ≥ 2. The proof of [7] can be adapted in order to prove that these conditions are also sufficient when N = 2 and p = 2.
The author is indebted to many people.At first to F. Krasucki whose questions about the mechanical meaning of Grisvard results on the traces of Airy functions were the starting point for the extension of the Gagliardo theorem to W m, p (Ω) in the case m=2 and p=2 for 2-dimensional domains [7].During a visit to the IAN of the CNR the discussions with F. Brezzi and particularly with A. Buffa allowed the extension to the case m=2 and 1 < p < ∞ for general N-dimensional domains [2].At last the discussions with F. Murat during the Colloque were stimulating to obtain the actual formulation of the conditions for m=3.

Preliminaries
Let Ω be a Lipschitz bounded and connected subset of R N whose bounded and orientable boundary is denoted by Γ.This means (see [9], [8] and [1]) that for every x ∈ Γ there exists a neighborhood V (x) of x in R N and a new orthonormal coordinate system {y 1 , ..., y Since Γ is compact there exists M open and connected subsets Γ i such that Γ = M i=1 Γ i and there exists M points a i ∈ Γ i and M Lipschitz continuous function ϕ i such that This parametrization induces N − 1 linearly independent tangent vectors defined a.e. on Γ i : defined as follows for k = 1: and for k = 2, . . ., N − 1, (2.3) At a.e. point x = (y , ϕ i (y )) ∈ Γ i the vectors (T 1 , . . ., T N −1 ) span the tangent space T x Γ who is an hyperplane of R N .Any other orthonormal basis of T x Γ is obtained applying a rotation to the previous one.The unit outward normal vector n = (n 1 , n 2 , . . ., n N ) ∈ (L ∞ (Γ)) N is defined a.e. on Γ i by y −→ (∂ϕ i /∂y 1 , . . ., ∂ϕ i /∂y N −1 , −1) For a.e.x ∈ Γ the vectors (T 1 , . . ., T N −1 , n) are a positively oriented basis of T x Ω ≈ R N .As in the case of regular domains the definitions are intrinsic since not depending from the choice of the parametrization Since Γ is a Lipschitz-continuous manifold without boundary the Sobolev spaces W s, p (Γ) are well defined (independently from the parametrization (ϕ i , Γ i )) for −1 ≤ s ≤ 1 and 1 < p < ∞.More precisely, ψ ∈ L p (Γ) means that for i = 1, . . ., M one has ψ (y , ϕ i (y )) ∈ L p (V (x i )) and ψ ∈ W 1, p (Γ) when ψ (y , ϕ i (y )) ∈ W 1, p (V (x i )).This means that for k = 1, . . ., N − 1 one has ∂ (ψ (y , ϕ i (y ))) /∂y k ∈ L p (V (x i )) where ∂ψ (y , ϕ i (y )) Using the tangent fields T k we define the tangential derivatives and for ψ ∈ W 1, p (Γ) the tangential vector field ) N and its definition does not depend on the parametrization (ϕ i , Γ i ).

Necessary conditions
From the previous definitions it follows that u −→ (γ 0 (u), . . ., γ m−1 (u)) is a linear and continuous map from A natural question is the characterization of the range of such map.
When the boundary Γ of Ω is more regular (for instance of class C ∞ ), the extension of the Gagliardo's theorem states that, for p > 1, u −→ (γ 0 (u), . . ., γ m−1 (u)) is a linear and continuous map from W m, p (Ω) onto and it has a continuous right inverse.
For polygonal-type domains (when N = 2) and for polyhedral-type domains (when N = 3) the caracterization of the range of the map (γ 0 , . . ., γ m−1 ) has been obtained in terms of compatibility conditions.
Remark 3.2.For Lipschitz domains the two terms of the sum in (3.1) belong separately only to From the previous proposition it then follows the following result.
In order to state the second necessary condition, let be and let define for k = 1, . . ., N − 1 the vector ∂ T k g 0 ∈ (L p (Γ)) N whose components are where the tangential derivatives are defined in (2.7).
Theorem 3.4.Let be 3) and: Proof.Since g 0 = γ 0 (Du) ∈ (W 1, p (Γ)) N it follows from (3.4) and the definition of T k that on V (x i ) one has for h = 1, . . ., N : ∂ 2 u ∂y h ∂y s y , ϕ i y T k s A simple computation then gives: and where A : B = N i,j=1 a ij b ij denotes the scalar product of the symmetric matrices A = (a ij ) and B = (b ij ).Since the vectors (T 1 , . . ., T N −1 , n) are an orthonormal basis of T x Ω, an orthonormal basis of the symmetrized tensor product T x Ω ⊗ S T x Ω is given by: A development of γ 0 (Hu) with respect to this basis and the use of (3.6) and (3.7) gives immediately (3.5).
Example 4.1.We prove that the usual compatibility conditions in a corner ( see e.g.[8]) for u ∈ W 2, p (Ω) follow from (3.1).Indeed this condition becomes: Since the definition of W 1−1/p, p (Γ) is invariant under the Lipschitz transform Γ −→ R, the previous condition means that: Since obviously, df 0 /dx, f 1 ∈ W 1−1/p, p (R ± ), one can verify these conditions with the help of the following proposition, whose proof can be found e.g. in [8].
Proposition 4.2.Let be H ± ∈ W 1−1/p, p (R ± ); and let define: In order to prove that the usual compatibility conditions in a corner for u ∈ W 3, p (Ω) , follow from (3.3)and (3.5) one has at first to express the conditions g 0 h ∈ W 1, p (R) for h = 1, 2 with an analogous of proposition 4.2.Since the orthonormal basis of T x Ω ⊗ S T x Ω is now given by: it is a simple exercice to express the different terms of these symmetric matrices on Γ 1 , Γ 2 and so find the compatibility conditions, always thank to the proposition 4.2.
When N = 3 with an analogous method one can express the compatibility conditions for a vertex or an edge of a polyhedral-type domain.Once more since the definition of W 1−1/p, p (Γ) and W 1, p (Γ) are invariant under a Lipschitz transform one can reduce the study of the vertex behaviour to the following problems.Let be R 2 divided in Λ non overlapping sectors S λ , λ = 1, . . ., Λ, with vertex at the origin.Let be given H λ ∈ W 1−1/p, p (S λ ) (resp.H λ ∈ W 1, p (S λ )) for λ = 1, . . ., Λ; find the conditions such that: for (x, y) ∈ S Λ belongs to W 1−1/p, p R 2 , resp.toW 1, p R 2 .Some partial answers can be found in [3] and in [5].

Remark 3 . 5 .
Let us once more remark that each term of the sum in (3.5)only belongs to L p Γ; M N sym .Remark 3.6.With the same procedure one can write the necessary compatibility conditions for the traces of u ∈ W m, p (Ω), m ≥ 4.