On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain

Under some reasonable conditions, some trace embedding properties of 
Musielak-Sobolev spaces in a bounded domain are given, including the 
trace on the inner lower dimensional hyperplane and the trace on the 
boundary. Furthermore, a compact trace embedding on the boundary is 
given.


1.
Introduction. In the study of nonlinear differential equations, it is well known that more general functional space can handle differential equations with more complex nonlinearities. If we want to study general forms of differential equations, it is very important to find a proper functional space in which the solutions may exist. Musielak-Orlicz-Sobolev space, or for short, Musielak-Sobolev space is such a general form of Sobolev space that the classical Sobolev spaces, variable exponent Sobolev spaces and Orlicz-Sobolev spaces can be interpreted as its special cases.
The properties and applications of Orlicz-Sobolev spaces and variable exponent Sobolev spaces have been studied extensively in recent years, see for example [3,4,10,11]. In [1,5], analogies for the Sobolev spaces, the Orlicz-Sobolev spaces are studied, including analogies for Sobolev embedding theorems and the trace properties on the lower dimensional hyperplane. In [6] Fan considers the trace embedding of the variable exponent Sobolev space. In a recent paper [3], the authors consider the W 1,p(·) -regularity for elliptic equations with measurable coefficients in non-smooth domains. To our best knowledge, however, the properties of Musielak-Sobolev space have been studied little. In the research of the paper [2], Benkirane and Sidi prove an existence result for some class of variational boundary value problems for quasilinear elliptic equations in the Musielak-Orlicz space. And in that paper, an embedding theorem has also been provided without assuming the ∆ 2 condition. In two recent papers [8,9], Fan gives some properties about this kind of functional space, including an embedding theorem and a compact embedding theorem in a bounded domain. And in [7] Fan and Guan study the uniform convexity of the Musielak-Sobolev spaces and present some applications. As an application of the embedding results in [9], the authors in [12] give the existence of solutions to a kind of quasilinear elliptic equation. Our aim in this paper is to study the trace regularity of the Musielak-Sobolev space in a bounded domain. It is a very important part of the Musielak-Sobolev space theory for the study of the Neumann boundary problems in differential equations.
The paper is organized as follows. In Section 2, for the convenience of the readers we recall some definitions and properties of the Musielak-Sobolev space. In Section 3, based on some estimates for the N (Ω) function (Lemma 3.5 -Lemma 3.8) and the classical trace theory on the hyperplane (Lemma 3.4), we develop the embedding theory for the trace on the inner lower dimensional hyperplane (Theorem 3.3). In Section 4, the boundary trace embedding theorem is given, including a compact embedding. To prove the trace estimate on the boundary, we will not use the method in the classical theories. In the classical theory for the trace on the boundary, one should firstly extend functions defined on the bounded domain Ω to a much larger domain Ω := {x ∈ R n : dist(x, Ω) < }. And ∂Ω is considered as the inner hyperplane of Ω . Then the trace estimate on ∂Ω can be obtained by the classical inner hyperplane trace theory. But our boundary trace estimate for the Musielak-Sobolev space is much more complex now brought by the much more nonlinearity of the N (Ω)-function. By some basic computations, one can figure out that the nonlinearity of the N (Ω)-function does not allow us to develop such an extension theory in the Musielak-Sobolev space as similar as in the classical Sobolev space. Then the theories developed in Section 3 are not valid for the boundary trace theory in Section 4. To our best knowledge, the boundary trace embedding theory even in Orlicz-Sobolev spaces (a special case for Musielak-Sobolev spaces) is not found in the existing mathematical literature. In Section 5, two examples satisfying the conditions in our theories are provided. All of our theories are considered in a bounded domain.
2. The Musielak-Orlicz-Sobolev spaces. In this section we recall some basic definitions and properties about Musielak-Orlicz-Sobolev spaces for the readers to refer. The proofs of these properties can be found in [7,8,9] and the references therein. We emphasis that some of these properties and theorems are a little different in the form from the references therein in order to make an application for our case. For example, assumption (P 3 ), Theorem 2.4 and so on.
Firstly, we give the definition of "N -function" and "generalized N -function" as follows. ∃λ > 0 such that The Musielak-Sobolev space W 1,A (Ω) can be defined by A is called locally integrable if A(·, t 0 ) ∈ L 1 loc (Ω) for every t 0 > 0. For x ∈ Ω and t ≥ 0, we denote by a(x, t) the right-hand derivative of A(x, ·) at t. Then A(x, t) = A is called the complementary function to A in the sense of Young. It is well known that A ∈ N (Ω) and A is also the complementary function to A.
) for x ∈ Ω and t ∈ [0, +∞); 2. A and A satisfy the Young inequality and the equality holds if s = a(x, t) or t = a −1 + (x, s). Let A, B ∈ N (Ω). We say that A is weaker than B, denoted by A B, if there exist positive constants K 1 , K 2 and a nonnegative function h ∈ L 1 (Ω) such that We say that A and B are equivalent near infinity, if there exist positive constants t 0 , k 1 , k 2 and nonnegative functions h 1 , h 2 ∈ L 1 (Ω) such that Proposition 2 (See [8,13]). Let A, B ∈ N (Ω) and A B.
In the study of the uniform convexity of the space L A (Ω), the following definition is useful.
is called uniformly convex, if there exists a function σ mapping the interval (0, 1) into itself such that for every s > 0, 0 < α < 1 and 0 ≤ β ≤ α, there holds the inequality We customarily call this uniformly convex condition the (UC) 1 condition. If an N (Ω) function satisfies (UC) 1 condition, then this function satisfies the classical (UC) condition, see [7].
is an N -function and h(x) ≡ 0 in Ω in Definition 2.3, then A ∈ ∆ 2 (Ω) if and only if A satisfies the well-known ∆ 2 condition defined in [1,5].
replacing, if necessary, A by another N (Ω)-function equivalent to A near infinity. Similarly as mentioned in [1], since Ω is bounded, (2) places no restrictions on A from the point of view of embedding theory since N (Ω)-functions which are equivalent near infinity determine identical Musielak-Orlicz spaces in that case.
Let X be a metric space and f : X → (−∞, +∞] be an extended real-valued function. For x ∈ X with f (x) ∈ R, the continuity of f at x is well defined. For x ∈ X with f (x) = +∞, we say that f is continuous at x if given any M > 0, there exists a neighborhood U of x such that f (y) > M for all y ∈ U . We say that f : X → (−∞, +∞] is continuous on X if f is continuous at every x ∈ X. Define Dom(f ) = {x ∈ X : f (x) ∈ R} and denote by C 1−0 (X) the set of all locally Lipschitz continuous real-valued functions defined on X.
The following assumptions will also be used. (P 4 ) T : Ω → [0, +∞] is continuous on Ω and T ∈ C 1−0 (Dom(T )); (P 5 ) A * ∈ C 1−0 (Dom(A * )) and there exist positive constants δ 0 < 1 n , C 0 and t 0 < min x∈Ω T (x) such that We give an embedding theorem for Musielak-Sobolev spaces developed by Fan in [9], see also in [12] and remarks therein. 3. Trace on the inner lower dimensional hyperplane. In this section we study the trace regularity on the inner lower dimensional hyperplane in a bounded domain. We always assume the following condition holds. (P p ∂ ) A is an N (Ω)-function such that T (x) defined in equation (3) satisfies And there exists a constant p (1 ≤ p < n) such that the function B defined by B(x, t) = A(x, t 1 p ) is an N (Ω)-function. We recall the trace defined on the lower dimensional hyperplane Ω k (Ω k denotes the intersection of Ω with a k-dimensional hyperplane in R n ) or ∂Ω for the classic W 1,1 (Ω) as follows.
with the constant C depending only on Ω.
The main theorem of this section is the following. Let Ω be a bounded domain in R n satisfying the cone condition, and let Ω k denote the intersection of Ω with a k-dimensional hyperplane in R n .
1. If either n − p < k ≤ n or p = 1 and n − 1 ≤ k ≤ n, then Remark 2. In our theorem, A satisfies (P p ∂ ) and (P 1 ) − (P 5 ). We claim that, in fact, under the assumption (P p ∂ ), (P 4 ) is automatically satisfied. In the following of this paper we will use the following notations. Denote that where q = np/(n − p) and (y, t) ∈ Ω × [0, +∞), and denote the derivative of the function in a weak sense f yj (y, t) = ∂f ∂y j (y, t) and f t (y, t) = ∂f ∂t (y, t).
We give some lemmas under the assumptions of Theorem 3.3 to prove the main theorem.

Lemma 3.4 (See Theorem 4.12 in [1]).
Let Ω be a domain in R n and, for 1 ≤ k ≤ n, let Ω k be the intersection of Ω with a hyperplane of dimension k in R n . (If k = n, then Ω k = Ω.) Suppose that Ω satisfies the cone condition, p < n and either n − p < k ≤ n or p = 1 and n − 1 ≤ k ≤ n. Then Lemma 3.5 (See [9]).

Let
. Then for every j = 1, 2, . . . , n, the weak derivative g xj of g exists and
Lemma 3.7. For any t > 0, the following estimate holds Proof. By the definition of B, B and A * , one can verify the following equations respectively Lemma 3.8. For any > 0, there exists a constant C( ) > 0 such that On the other hand by the definition of A * , we have By assumption (P p ∂ ) on the function B(x, t) we can see that lim τ →0+ By (16) and (17), for any > 0, there exists a constant C( ) > 0 such that which completes the proof of the lemma.
Based on the above lemmas now we can give the proof of Theorem 3.3. Proof.
Step 1. Suppose that u ∈ W 1,A (Ω) is a bounded function. From the assumption (A p ∂ ), it is easy to check that A k n * is an N (Ω)-function. And by the definition of the norm in L A k n * (Ω), we have since u is bounded. We will show that with a constant C > 0 independent of u. By Theorem 2.4, there exists a constant C > 0 such that u A * ,Ω ≤ C u 1,A,Ω . Then if u A * ,Ω ≥ K, the proof is completed. In the rest of the proof we suppose that and without loss of generality, we assume that u(x) ≥ 0.
It is obvious that K ) ∂xj = 0 for a.e. x ∈ Ω 0 . By Lemma 3.5, we can see for a.e. x ∈ Ω + , Then by Lemma 3.4, there exists a constant C * > 0 independent of u such that By Lemma 3.6 and (19) there exists a constant C > 0, such that By Lemma 3.7 and (19), noticing that Lemma 3.7 holds for t > 0( = 0), we can see that So by the definition of the norm for L B (Ω) (since B ∈ N (Ω)), we conclude Then there exists a constant C > 0 such that Take = 1 4C * in Lemma 3.8. There exists a constant C > 0 such that Then by (19) there exists a constant C > 0 such that Combining (21), (22) and (23) we conclude which implies (18).
Step 2. For a general u ∈ W 1,A (Ω), define Clearly u m ∈ W 1,A (Ω) is bounded. By Step 1, we can see that there exists a constant C > 0 independent of u m , such that From Fatou's lemma Combining (24) and (25) we can see that (18) holds for any u m ∈ W 1,A (Ω). Part (1) of the theorem is proved.
Step 3. Compactness of the embedding. Let {u m } be a bounded sequence in  In this section the following condition will be used.
A main theorem of this section is the following.
Now we can give the proof of Theorem 4.2.
The following compact embedding theorem is one of our main results.
Let > 0 small enough such that p(x) + < n for any x ∈ Ω. Since for t > 0 big enough t p(x) ≤ A(x, t) ≤ t p(x)+ , we can get the estimate for A * (x, t) as follows for any x ∈ Ω. Then A * (x, t) ∈ ∆ 2 (Ω). The conditions in Theorem 4.4 are verified.