Variable Exponent Sobolev Spaces and Regularity of Domains

We study the embeddings of variable exponent Sobolev and Hölder function spaces over Euclidean domains, providing necessary and/or sufficient conditions on the regularity of the exponent and/or the domain in various contexts. Concerning the exponent, the relevant condition is log-Hölder continuity; concerning the domain, the relevant condition is the measure density condition.


Introduction
In Euclidean domains, variable exponent Lebesgue-Sobolev and Hölder spaces have been intensively studied during the last years (see the books [3,5] for a gentle introduction, and [6] for an overview on the history of the subject). These spaces of functions provide a useful tool for the description of non-linear phenomena in elastic mechanics [21], fluid mechanics [18] and image restoration [16], for example. In some descriptions of the aforementioned phenomena, functionals of Dirichlet type with variable exponent are used; since the regularity of the critical points of those functionals is relevant, the embedding or inclusion problem between these type of spaces must be studied in depth. Note that this theory has also been developed in Riemannian manifolds [9,10], and also in metric-measure spaces [8,12,17].
In this work, we consider inclusions between Lebesgue, Sobolev, and Hölder spaces with variable exponent on Euclidean domains, and obtain sufficient and/or necessary conditions on the regularity of the exponent and/or on the domain.
In Sect. 2, we provide a brief description of variable exponent Sobolev and Hölder spaces in Euclidean domains, together with some of their elementary properties. To have a point of reference, we overview the problem concerning the density of smooth functions in variable and classical (or constant exponent) Sobolev spaces; in the variable exponent case, at least when the domain is regular enough, the regularity of the exponent is fundamental: the log-Hölder regularity turns out to be a sufficient condition.
Then we move to the subject of this work, the inclusion or embedding problem between spaces of functions. When the exponents are constant, there are many results concerning necessary and/or sufficient conditions on the regularity of the domain to achieve the embeddings [1,11]; in the variable exponent case, and at least when the domain is regular enough, the log-Hölder regularity of the exponent appears as a sufficient condition again. To clarify our contribution, we describe the (nowadays) canonical notions of regularity for domains and exponents in detail, and quote the known results for embeddings between Sobolev and/or Hölder spaces using these notions. After this has been done, we state our main results, that could be seen as a mixture of sufficient conditions on the regularity of the exponent, and necessary conditions on the regularity of the domain. The proofs of these results are given in Sect. 3.1.
In Sect. 4, we infer some consequences of our main results, providing generalizations and different proofs of some results to be found in [1].

Preliminaries
In this work, L n is the n-dimensional Lebesgue measure, and is a Lebesgue measurable subset of R n . If A is a L n -measurable subset of , we write |A| instead of A dL n (x) = L n (A). A variable exponent, or simply an exponent, is a bounded L n -measurable function p : → [1, ∞[, usually written as p(·). For such a p(·), whenever A is a L n -measurable subset of define If A = , we simply write p + and p − , respectively. In this context, the generalized Lebesgue space L p(·) ( ) is the vector space of measurable functions u : → R for which the functional is finite. The functional ρ p(·) is convex, and L p(·) ( ) is a Banach space with the norm One can compare the functionals L p(·) ( ) and ρ p(·) ( ) using the inequalities and the unit ball property follows: u L p(·) ( ) ≥ 1 if and only if ρ p(·) (u) ≥ 1. Another basic result is Hölder's inequality: if r (·), p(·) and q(·) are exponents satisfying L n -almost everywhere on , then The Sobolev space L p(·) 1 ( ) is the vector space of those functions u in L p(·) ( ) for which their distributional gradient (that we denote by ∇u if no confusion arises) is also in L p(·) ( ). L p(·) 1 ( ) is also a Banach space with the norm Despite the apparent similarity between these spaces and the classical Lebesgue and Sobolev spaces, some properties of the latter are not valid in the variable exponent case: for instance, if p + is finite, then for every , the smooth functions are dense in L p(·) ( ), see [15]; however, sufficient conditions on p(·) to ensure the density of the smooth functions in L p(·) 1 ( ) even when is a bounded and smooth domain in R n are more delicate. The first counterexample of this type was found by Zhikov for variational problems in the 2-dimensional disk: the infimum over the smooth functions of a functional of p(·)-Dirichlet type is strictly larger than the infimum of that functional over all the functions in L p(·) 1 ( ), see [21,22]; in those examples, the exponent p(·) is not continuous.
If the domain has a locally Lipschitz boundary, then a sufficient condition for the aforementioned density was found by Samko and Diening on provided that p(·) satisfies the log-Hölder condition (see [4,5,13,14]): p(·) is of log-Hölder type if there exists a constant C log such that where |x − y| is the Euclidean distance between the elements x and y in .
To relate log-Hölder continuity with other types of uniform continuity, different moduli of continuity are used: given a continuous function φ : In the quest for the necessity of the log-Hölder condition for the density of the smooth functions on L p(·) 1 ( ), Hästö constructs on the unit disk in R 2 , a uniformly continuous exponent whose modulus of continuity is slightly weaker than the one associated to the log-Hölder condition, but for which the smooth functions are not dense in L p(·) 1 ( ), see [13]. The modulus used by Hästö is equivalent to giving a satisfactory but still partial answer to the problem, see [6,13,14].
In the counterexamples of Zhikov and Hästö, the domain is the standard smooth domain, namely the unit disk. Beside smoothness, some properties or conditions for the regularity of domains in R n are as follows: 1. The strong local Lipschitz condition, 2. The segment condition, 3. The uniform cone condition, 4. The cone condition, 5. The weak cone condition, and 6. The measure density condition.
The definition of the first five notions can be found in [1,20]. Concerning the sixth condition, a subset of R n is said to satisfy the s-measure density condition, for some s > 0, if there exists a positive constant c such that for every x in¯ and each R in ]0, 1] one has If s = n, then we shall simply say that satisfies the measure density condition.
If A and B are any of the mentioned properties for domains, write A ⇒ B if whenever is a domain with property A, then also has property B. The relations between the mentioned properties can be read from the next picture:

Strong local Lipschitz condition
Uniform cone condition

Cone condition Segment condition
Weak cone condition Measure density condition The relation between the first five properties is given in [1]. To illustrate the relations with the measure density condition, we give some examples: the unit square minus the origin in R 2 . Then 1 satisfies the cone condition, but not the segment condition.

Example 2.2
If K is a compact subset of the square ] − 10, 10[ 2 whose Hausdorff dimension is smaller than 2, then 2 (K ) =] − 10, 10[ 2 ∼ K satisfies the measure density condition. In particular, if K is the von Koch Curve (snowflake) whose Hausdorff dimension is log 4/ log 3, then 2 (K ) satisfies the measure density condition, but not the weak cone condition, nor the segment condition.

Example 2.3 Let
Then 3 satisfies the segment condition but not the weak cone condition nor the measure density condition.
Another situation where there are similarities and differences between standard and variable exponent Lebesgue and Sobolev spaces is the inclusion problem: Find necessary and sufficient conditions on p(·), q(·) and such that there exists a constant C so that In this context, Diening obtained the following result: 1. The boundary of is locally Lipschitz. 1

The exponent p(·) is log-Hölder continuous, with
To prove Theorem 2.1, Diening proceeds, roughly speaking, in two steps. In the first step, he obtains the embedding L p(·) 1 (R n ) → L q(·) (R n ) extending some tools of Potential theory [19] to the case when the exponent p(·) is of log-Hölder type. In the second step, he extends the technique of Extension operators [1] to this setup: as in the case when p is constant, a sufficient condition on for these purposes is the strong local Lipschitz condition.
The inclusion problem between Sobolev and Hölder spaces has also been extended to variable exponent spaces. As in the classical case, define C 0,α(·) ( ), the Hölder space of variable exponent α(·) over , where now α : →]0, 1] is a measurable function: given a bounded and continuous function u on consider its seminorm so that C 0,α(·) ( ) is the vector space made up of those u that are bounded, continuous, and for which the seminorm [u] α(·) is finite. C 0,α(·) ( ) is a Banach space for the norm Using similar ideas and techniques as those used by Diening to obtain Theorem 2.1, Almeida and Samko obtained an inclusion between variable exponent Sobolev and Hölder spaces: On the other hand, concerning necessary conditions on the domain to achieve classical Sobolev and Hölder embeddings, an important progress was achieved by Hajłasz, Koskela and Tuominen, see [11]. Part of their results can be summarized as follows:

Main Results
As mentioned in Sect. 1, our results could be seen as a mixture of the theorems stated in Sect. 2, in the sense that they provide sufficient conditions on { p(·), α(·)} and/or necessary conditions on the domain for the variable exponent Sobolev and Hölder inclusions to hold. We have: Let be an open subset of R n , and suppose that for some s > 1: We refer to the diagram in Sect. 2 to see the gap between the necessary and sufficient conditions on the domain.

Proof of Theorem 3.1 For a fixed x in¯ define
It is enough to consider the case when |A R | ≤ 1, otherwise |A R | ≥ 1 ≥ R s whenever R ≤ 1 and there is nothing to prove; moreover, it is enough to consider R ≤ 1 4 . For such an R, denote byR < R the smallest real number such that To prove Theorem 3.1, we need the following Lemma: Proof Unless some confusion arises, we omit the domain in the function spaces and norms. So assume that L p(·) 1 → L q(·) ; then there exists a constant C sob > 0 such that 1 , one has the inequality For a fixed x in¯ let u(y) := φ(y − x) be a function of y ∈ , where φ is a cut-off function satisfying: 3. φ| BR (0) = 1, and 4. |∇φ| ≤c/(R −R) for some constantc.
Recall that x is fixed, to deduce the inequalities and Combine those inequalities with the Sobolev inequality (3), and follows. Moreover, use Hölder's inequality (1) (·) and the basic estimates between norm and semimodular in (4), to infer that To conclude the proof of Theorem 3.1, given x in¯ and R in ]0, 1 4 ], construct the sequence {R i } by setting R 0 := R, and then define R i+1 :=R i inductively for i ≥ 0. It follows that With those ingredients in Lemma 3.1, one observes that where the abbreviation η R : has been used.
Note that η R ≥ η := 1 s + 1 q + − 1 q − > 0, to deduce, thanks to the previous observations, that Moreover, since c 2 := 1/ max{1, c 1 1−2 −η } ≤ 1 one has where β R := 1 − sη R . From (6), one sees that if a positive lower bound for R β R /η R is provided, the proof of Theorem 3.1 is finished. To achieve such a lower bound, the log-Hölder continuity of p(·) will be used: by the hypotheses on p(·) and q(·) one has that q(·) is log-Hölder continuous as well, hence taking the supremum over pairs of points in A R one gets But hence using (7) the required bound Proof of Theorem 3.2 We use a notation similar to that in Theorem 3.1. Since q(·) is log-Hölder continuous, q(·) is uniformly continuous, and one can extend q(·) to¯ . Since¯ is compact, there is a finite cover {O α : α in } of¯ by open sets such that Let λ be some Lebesgue number of the covering {O α }, and consider R such that R ≤ min{1/4, λ/2}. Fix some x in , note that B R (x) ⊂ O α for some α ∈ , to define the sequence {R i } in the same way as in the proof of Theorem 3.1: hence by (5) where η R ≥ η α > 0. Proceeding in a similar way as in Theorem 3.1, one obtains where c 2,α := 1/ max{1, c 1 1−2 −ηα }; also for each α in one has Putting things together the claimed estimate |A R | ≥ cR s follows, where From Theorem 3.2, we deduce: and note that 1 ≤ Q(·) ≤ q(·), hence Moreover, since p(·) < q(·) one has that with p + < t. One can use Q(·) instead of q(·) and t instead of s in the hypotheses of Theorem 3.2: the result follows.

Consequences and Questions
In all the results in this work, the measure density condition on the domain is necessary. With the help of Federer's co-area formula, we will characterize some domains that do not satisfy the measure density condition. Such characterization together with the theorems in this work provides a simpler proof of results from [1]; at the same time, an extension of those results to Sobolev and Hölder spaces with variable exponents follows.
According to [1] the domain has a cusp of exponential sharpness at its boundary point x 0 if for every real number s one has Using such notion, the following result is related with Theorem 4.48 in [1]: Let be an open subset of R n , and assume that has a cusp of exponential sharpness at some point of its boundary. Then: 1 ( ) is not embedded in L q(·) for those p(·) and q(·) satisfying the conditions in Theorem 3.1, and 2. L p(·) 1 ( ) is not embedded in C 0,α(·) for those p(·) and α(·) satisfying the conditions in Theorem 3.3.

Question 4.1 Are embeddings between variable exponent Sobolev and/or Hölder spaces possible if the modulus of continuity of the exponents is weaker than log-Hölder, for example, given by the modulus in (2)? If so, for which regularity of the domain might this happen?
The second problem is motivated by Theorem 3.2 and another notion of regularity condition of the exponent: we say that the exponent p(·) is log-Hölder continuous at infinity, if there exist constants C ∞ and p ∞ such that for all x ∈ , | p(x) − p ∞ | ≤ C ∞ / log(e + |x|).

Question 4.2
Can we replace the assumption 1 q − < 1 q + + 1 s in Theorem 3.1 by the assumption that p(·) (or q(·)) is log-Hölder continuous at infinity?