On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals

For a class of functionals having the $(p,q)$-growth, we establish an improved range of exponents $p$, $q$ for which the Lavrentiev phenomenon does not occur. The proof is based on a standard mollification argument and Young convolution inequality. Our contribution is two-fold. First, we observe that it is sufficient to regularise only bounded functions. Second, we exploit the $L^{\infty}$ bound on the function rather than the $L^p$ estimate on the gradient. Our proof does not rely on the properties of minimizers to variational problems but it is rather a consequence of the underlying Musielak-Orlicz function spaces. Moreover, our method works for unbounded boundary data, the variable exponent functionals and vectorial problems. In addition, the result seems to be optimal for $p\le d$.


Introduction
We consider a class of functionals with the so-called (p, q)-growth. The prominent example we have in mind is Here, Ω ⊂ R d is a bounded Lipschitz domain, u : Ω → R is an argument of the functional G, a : Ω → [0, ∞) is a given nonnegative function and 1 ≤ p < q < ∞ are given numbers. Functional G is the interesting toy model for studying minimisation of functionals with the so-called non-standard growth. Indeed, depending on whether a = 0 or a > 0, G exhibits either the p-or the q-growth.
A well-known feature of functional G is the so-called Lavrentiev phenomenon. For instance, there exists a function a ∈ C α (Ω) with α ∈ (0, 1), exponents p, q fulfilling p < d < d+ α < q and boundary data u 0 ∈ W 1,q (Ω) such that On the other hand, it is known that if q ≤ p + α p d , the Lavrentiev phenomenon does not occur for the toy model (1.1), see [17]. Under the additional assumption u 0 ∈ L ∞ (Ω), the range of exponents has been improved to q ≤ p + α [13, Proposition 3.6, Remark 5]. The latter work heavily depends on the properties of minimizers and the L ∞ bound for the minimizer of the functional (1.1) form a nontrivial part of the result in [13].
In this paper we prove that neither the assumption u 0 ∈ L ∞ (Ω) nor any further bound on minimizer is irrelevant for the absence of Lavrentiev phenomenon. More precisely, we prove that one does not observe Lavrentiev phenomenon if (1.3) q ≤ p + α max 1, p d and boundary data u 0 ∈ W 1,q (Ω). In this case, we have This significantly improves the available results for the case p < d. Moreover, our proof is elementary as it is based on a simple regularisation argument together with Young's convolution inequality. In particular, we do not use estimates on minimizers of functional (1.1). consequently, our method easily extends to the vector-valued maps and cover variable-exponent functionals as well, see Section 3.2.
The question of whether (1.2) or (1.4) holds true is related to the density of C ∞ c (Ω) in the Musielak-Orlicz-Sobolev space W 1,ψ 0 (Ω) related to the functional (1.1), see (4.1)-(4.3) for definitions. In this context, we prove that the density result hold true for p, q satisfying (1.3) which is again better then so-far known regime of exponents announced in [1].
Let us discuss the result of the paper within the context of previous works related to this topic.
Going back to the functional (1.1), the available results for boundary data u 0 ∈ W 1,q (Ω) provide both positive and negative answers to the question whether (1.4) holds true. On the one hand, if q ≤ p + p α d then (1.4) is indeed valid [16]. On the other hand, if q > p + α max 1, p−1 d−1 then counterexample in [2,Theorem 34] shows that (1.4) is violated (see also [17,Lemma 7] for a weaker result concerning the case p < d < d + α < q obtained with more elementary methods). In this paper we establish (1.4) for q ≤ p + α max 1, p d which partially fills the gap between currently known positive and negative results concerning the Lavrentiev phenomenon. Moreover, in view of [2,Theorem 34], our result is sharp for p ≤ d.
Next, we wish to address two issues that appeared in previous papers on this topic. First, in [14,Lemma 4.1] there is the following claim: for every ε > 0 and ball B r (x) ⊂ Ω, there exists a coefficient a ε ∈ C α (Ω) and a boundary data Although it is a very nice result, it does not prove that range of exponents q ≤ p + α p d is optimal for absence of the Lavrentiev phenomenon and it does not contradict our result about the range stated in (1.3). In fact, authors refer to the counterexample from [17] constructed for exponents satisfying p < d < d + α < q i.e. exponents that do not meet our range because the distance between p and q is greater than α. In fact, it is shown that there exists p ε and q ε but it follows also from the proof that they are constructed in the following way: for δ > 0 to be specified later, we define p ε := d − δ, q ε := d + α + δ and find a proper counterexample constructed in [17]. Then, when p ε ≥ 1, we have Second, we also want compare our result with [13], where authors proved that the Lavrentiev phenomenon is not observed for q ≤ p + α in the particular cases when minimizers of (1.1) are bounded, but this requires an extra assumption on the boundary data, namely that the boundary data u 0 is bounded and apply the maximum principle [21]. In addition, reasoning in [13] is based on the so-called Morrey type estimate on the gradient of minimizer which is not an obvious result itself.
Comparing to our work, we prove that the Lavrentiev phenomenon does not occur independently of the properties of minimizers or boundedness of boundary data. Our methods are elementary and are based on simple estimates on convolutions. We point out that one could naively think that our result is a consequence of [13] and a simple approximation argument (boundary data u 0 ∈ W 1,q (Ω) is approximated with a sequence {u 0,n } n∈N ⊂ W 1,q (Ω) ∩ L ∞ (Ω)) but it is not necessarily true that sequence of minimizers has then a subsequence converning again to a minimizer of the limit problem.
Finally, we want to point out and emphasize the main novelties of the paper. Standard methods [14,16] for proving (1.4) are based on regularization of arbitrary function u ∈ W 1,p 0 (Ω) satisfying G(u, Ω) < ∞ with a sequence of smooth functions u ε = u * η ε and passing to the limit G(u ε , Ω) → G(u, Ω) as ε → 0. The latter is not trivial because the integrand in (1.1) is x-dependent. Therefore, one approximates locally the integrand with function that does not depend on x, see Lemma 5.4. This approximation requires good estimate on ∇u ε ∞ which results in constraint on exponents p and q. The estimate on gradient is obtained by writing ∇u ε = ∇u * η ε and using the fact that ∇u ∈ L p (Ω). Our main contribution is an observation that it is sufficient to approximate only bounded functions u (i.e. u ∈ L ∞ (Ω)). It turns out that for p < d, it is more optimal to write ∇u ε = u * ∇η ε and exploit the estimate u ∈ L ∞ (Ω) rather that ∇u ∈ L p (Ω). We remark that these observations have been already used in our recent paper on parabolic equations [8] but at that point we did not observe that similar ideas may bring new information to analysis of the Lavrentiev phenomenon.
The structure of the paper is as follows. In Section 2 we present the main result, Theorem 2.3.
The theorem holds true under rather complicated assumption so in Section 3 we discuss two representative examples. In Section 4 we review the most important properties of the Musielak-Orlicz-Sobolev spaces. We explain here why it is sufficient to approximate only bounded functions, see in the general case. Finally, in Section 7 we briefly discuss how to extend our work to the case of vectorial problems.

Main result
Let us first set notation. We always assume that Ω ⊂ R d is a bounded Lipschitz domain and d is the dimension of the space. We write B for the unit open ball centered at 0. For balls with radius r we use B r and if the center is at some general point x, we write B r (x) so that B 1 (0) = B and B r (0) = B r . Concerning function spaces, we write C ∞ c (Ω) for the space of smooth compactly supported functions, W 1,p (Ω) and W 1,p 0 (Ω) are usual Sobolev spaces, W 1,ψ (Ω) and W 1,ψ 0 (Ω) are the Musielak-Orlicz-Sobolev spaces defined in Section 4 while C α (Ω) is the space of Hölder continuous functions on Ω with exponent α ∈ (0, 1]. Finally, η ε : R d → R is a usual mollification kernel. We already introduced the key motivation of the paper, i.e., the functional (1.1), but the main result concern more general cases. We focus in the paper on functionals being of the form where ψ is the so-called N -function and it satisfies the following assumptions: Assumption 2.1. We assume that an N -function ψ : Ω × R + → R + satisfies: (A1) (vanishing at 0) ψ(x, ξ) = 0 if and only if ξ = 0, (A3) (autonomous lower-bound) there is a strictly increasing and continuous function m ψ : (A4) (p − q growth) there exist exponents 1 < p < q < ∞ and ξ 0 ≥ 1 and constants C 1 and C 2 such that  The main result of this paper reads: 3. Examples of N -functions satisfying Assumption 2. 3.1. Standard double phase functionals. In this section, we prove that the N -function ϕ(x, ξ) = |ξ| p + a(x) |ξ| q .
satisfies Assumption 2.2 provided that a ∈ C α (Ω) and q ≤ p + α max 1, p d . The related functional reads To show it, we use the following lemma, whose assumptions are evidently satisfied by the example given above.

3.2.
Variable exponent double phase functionals. In this section we prove that N -function satisfies our Assumption 2.2. The related functional reads: Assumption 3.2. We assume that: (B3) (α-Hölder continuity) a ∈ C α (Ω) with constant |a| α . q and p such that q ≤ p + α max 1, p d .
Proof. As in the proof of Lemma 3.1, we only need to consider ξ ≥ 1. Let us estimate φ(x,ξ) φ(y,ξ) for x, y ∈ Ω such that |x − y| ≤ min diam Ω, 1 2 . We have where we used max 1, p d min 1, d p = 1 in the last line. We deduce

Musielak-Orlicz-Sobolev spaces
Our results are based on smooth approximation in the Musielak-Orlicz spaces, so we first recall their definitions and basic properties. For more details, we refer to monographs [10,19]. We consider an N -function ψ : Ω × R + → R satisfying (A1)-(A5) in Assumption 2.1. We define the related Luxembourg norm with Finally, the Musielak-Orlicz-Sobolev spaces are defined as the latter one corresponds to the space of functions vanishing at the boundary. These are normed spaces with norm One can think of W 1,ψ (Ω) as the space of functions having gradient integrable with p or q power depending on whether a = 0 or not.
We summarize some properties of the Musielak-Orlicz spaces in the following lemma. They are mainly consequences of (A5) in Assumption 2.1. The proof can be found in many texts on Orlicz spaces [10,19], yet for the sake of completeness we present the proof in Appendix A.2.
on Ω, f ψ < ∞ and the sequence {ψ(x, |f n (x)|} n∈N is uniformly integrable Next two lemmas show that to prove the absence of the Lavrentiev phenomenon, it is sufficient to demonstrate that every u ∈ W 1,ψ 0 (Ω) ∩ L ∞ (Ω) can be approximated in the topology of W 1,ψ by smooth function from C ∞ c (Ω).
First lemma shows, that it is enough to consider only bounded functions. Notice that we do not impose any specific assumption on the N -function ψ here.

Proof of Theorem 2.3 in the special case
In this section we prove Theorem 2.3 in the case when Ω = B (unit ball centered at 0) and the N -function is defined via the formula The corresponding functional then takes the following form Note that, if a ∈ C α (B) and q ≤ p + α max 1, p d , it follows from Lemma 3.1 that ϕ satisfies Assumption 2.2.
The main purpose of this section is that we avoid all technical difficulties and focus only on the main parts of the proof. More precisely, we do not need to take care of difficulties coming from: • geometric properties of general Lipschitz domain Ω, • situation when for general N -function ψ there is no local minimizer of the map x → ψ(x, ξ) valid for all values of ξ.
We start with introducing mollification that will be used to define the approximation.
Before formulating the main theorem of this section, we state and prove two results: a technical lemma concerning approximating sequence and a simple observation concerning N -function ϕ.
Lemma 5.3. Let u ∈ W 1,1 0 (B) be such that G(u) < ∞ and consider its extension to R d . Then, Proof. To see (D1), we note that the convergence holds in the pointwise sense. Moreover, the considered sequence is supported only for x ∈ B 1−2ε . Therefore, to establish convergence in L 1 (R d ), it is sufficient to prove equiintegrability of the sequence ϕ We fix η and arbitrary A ⊂ B. Using change of variables we have where for c ∈ R + , cA denotes a usual scaled set. By assumptions we have G(u) < ∞, so that if we set ω(τ ) := sup then ω(τ ) is a non-decreasing function, continuous at 0. Therefore, we may find τ such that ω(τ ) ≤ 2 −q η. Then, we choose δ = 2 −d τ to conclude the proof of (D1). Finally, the convergence result (D2) follows from Young's convolutional inequality and (D1).
Lemma 5.4. Let ϕ be given by (5.1). Then for all balls Proof. Using continuity of a and compactness of

a(y)
and we choose y * such that inf y∈Bγ (x)∩B a(y) = a(y * ).

Theorem 5.5 (Theorem 2.3 in the special case). Let
Consider sequence u ε as in Definition 5.1 with ε ∈ 0, 1 4 . Then, Proof. The first property follows from construction. To prove convergence, we note that We would like to take mollification out of the function ϕ using its convexity and Jensen's inequality. However, this is not possible as function ϕ depends also on x explicitly. To overcome this problem, we apply Assumption 2.2, which allows us to approximate the function ϕ(x, ξ) locally by a function depending only on ξ. Notice that ψ saisfies Assumption 2.2 thanks to Lemma 3.1 and the structural assumption (5.1).
Case 2: p > d. In this case we have q ≤ p + α p d . Note that Therefore, instead of (5.2), we can compute where p ′ is the usual Hölder conjugate exponent. Using change of variables we obtain: where D := 5 d p ∇u p η p ′ . Using Assumption 2.2 we obtain estimate (5.3). The rest of the proof is exactly the same.

Proof of Theorem 2.3 in the general case
In this section we generalize construction from Section 5 to prove Theorem 2.3 in the general case.
6.1. Second convex conjugate function. For general N -function ψ satisfying Assumption 2.1, Lemma 5.4 is not necessarily true. Therefore, to control mollifications, we need a different method to approximate ψ(x, ξ) with a function depending only on ξ. The construction below is somehow standard and has appeared in many works before, see [11,12].
We start more generally. Let f : R → R. We define convex conjugate f * : Moreover, the second convex conjugate of f * * is defined as .
We now list some basic properties of the convex conjugates cf. [ (F1) f * and f * * are convex functions, (F5) f * * is the gratest convex minorant of f . Now, we apply these notions to N -functions. Given N -function ψ(x, ξ) satisfying Assumption 2.1, we extend it by 0 for ξ < 0 (hence this extension is surely convex), we consider a ball B γ (x) such that B γ (x) ∩ Ω is nonempty and we define One could try to prove Lemma 6.2 by applying property (F3) from Lemma 6.1 to the estimate appearing in Assumption 2.2. However, this estimate is valid only on some bounded interval rather than the whole real line. The correct argument is presented in [10] but since it contains some imperfections, we present it below.
Proof. First, from Step 2 we may assume that a ≥ 0 and from (6.3) we deduce b < ∞ (as function m ψ is superlinear). Second, the reasoning from Step 3 shows that Moreover, by the assumption, there exists t ∈ [0, 1] such that By definition of ψ x, γ , there exist sequences {x a n } n∈N , {x b n } n∈N ⊂ B γ (x) such that (6.4) ψ * * x, γ (η) ≥ t ψ(x a n , a) With these at hand, we proceed to the final proof. By definition and convexity, To apply (6.4), we have to replace ψ(x b n , a) with ψ(x a n , a). This can be done with Assumption 2.2: we note that |x a n − x b n | ≤ 2 γ so if we let D : and Assumption 2.2 implies existence of constants M (D), N (D) (we skip dependence of these constants on p and q as these exponents are fixed) such that It follows from (6.5) that Letting M (D) := max(M (D), 1) and exploiting (6.4) we have Sending n → ∞ we deduce Step 5. Cases considered in Steps 2-4 are the only possible ones.
Proof. Clearly, the tangent line h η touches the epigraph of ψ * * x,γ at least in one point. The case where it is touched exactly at one point was studied in Step 3 while the situation when it is touched along some interval [a, b] was analyzed in Step 4. Now, suppose that there are η < η 1 < η 2 such that Then, ψ * * x,γ is not convex raising contradiction. Figure 1. We assume that there is η > 0 such that functions ψ * * x, γ (black line) and ψ x, γ (grey line) satisfy ψ * * x, γ (η) < ψ x, γ (η) and tangent line h η touches ψ * * x, γ only at η. As ψ x, γ is Lipschitz continuous, we can estimate it from below (dotted lines).
Then, the function obtained by combining ψ * * x, γ and the dashed line is convex. It lies below ψ x, γ and above ψ * * x, γ raising contradiction with Lemma 6.1 (F5).

Geometric issues.
As Ω is not a ball in general, we cannot define compactly supported approximation by retracting the function to the interior part of Ω as in Definition 5.1. However, one can still do that for star-shaped domains. (2) A bounded domain U ⊂ R d is said to be star-shaped with respect to the ball B γ (x 0 ) if U is star-shaped with respect to all y ∈ B γ (x 0 ).
The following lemma shows that star-shaped domains can be uniformly shrinked which allows for defining compactly supported approximations.
More generally, if U is star-shaped with respect to the ball B R (x 0 ),  Let α be a half of an apex angle of the cone C, see Figure 2. It follows that so it is sufficient to estimate sin(α) from below. Using notation from Figure 2, the length of interval . Therefore, As sin 2 (α) = tan 2 (α) 1+tan 2 (α) we have where we used R 2 ≤ |b| 2 ≤ 3 |b| 2 . We conclude that dist(∂U, c) ≥ 2ε. As this argument can be repeated for all c ∈ ∂(κ ε U ), we obtain dist(∂U, κ ε U ) ≥ 2 ε. The second statement follows from observation that the set U − x 0 is star-shaped with respect to the ball B R .
On star-shaped domain we can define mollification with squeezing as in Definition 5.1: Reader may think about the case x 0 = 0 first. Lemma 6.6. Function S ε U u from Definition 6.5 belongs to C ∞ c (U ).
Proof. The smoothness is clear from standard properties of convolutions. Concerning compact support, we claim that S ε U u is supported in x 0 + κ ε (U − x 0 ) + ε B which is a compact subset of U due to Lemma 6.4. Indeed, let x / ∈ x 0 + κ ε (U − x 0 ) + ε B and suppose that there is y with |y| ≤ ε such that x 0 + x−x0−y κε ∈ U . Then, we can write so that x ∈ x 0 +κ ε (U − x 0 ) + ε B raising contradiction. It follows that for x ∈ x 0 +κ ε (U − x 0 ) + ε B we have either To move from star-shaped domains to Lipschitz ones we will use the following decomposition cf. [ and Ω ∩ U i is star-shaped with respect to some ball B Ri (x i ).

6.3.
Approximating sequence and proof of Theorem 2.3. We are in position to define the approximating sequence. Let Ω be a Lipschitz bounded domain. From Lemma 6.7 we obtain family ..,n are star-shaped domains with respect to balls B R (x i ) (without loss of generality, we may assume that the radii of the balls are the same by taking R := min i=1,...n R i ). In particular, {U i } i=1,...,n is an open covering of Ω so there exists partition of unity related to this covering: family of functions {θ i } i=1,...,n such that Given u ∈ W 1,1 0 (Ω) ∩ L ∞ (Ω) we extend it with 0 as above and we set We note that since u vanishes outside of Ω, function u θ i is supported in Ω ∩ U i which is star-shaped.
Before formulating the main result of this section, we will state and prove two technical lemmas concerning approximating sequence. Proof. Note that for ε ≤ R 8 , we have 1 κε ≤ 2. We compute As |x i − x| ≤ diam(Ω) (the diameter of Ω), we choose C Ω,R := 8 diam(Ω) R + 2.
Lemma 6.9. Let u ∈ W 1,1 0 (Ω) be such that H(u) < ∞ and consider its extension to R d . Then, Proof. Concerning (H1), we note that the convergence holds in the pointwise sense. Moreover, the considered sequence is supported on Ω ∩ U i . Therefore, to establish convergence in L 1 (R d ), it is sufficient to prove equiintegrability of the sequence ψ x i + x−xi κε , (|∇u| θ i ) x i + x−xi κε ε and apply Vitali convergence theorem. To this end, we need to prove We fix η and arbitrary A ⊂ Ω ∩ U i . Using convexity, where A is a set obtained from A after the performed change of variables. Note that measures of these sets satisfy | A| ≤ 1 Having this in mind, we let Function ω(τ ) is a non-decreasing function, continuous at 0 because H(u, Ω) < ∞. Therefore, we may find τ such that ω(τ ) ≤ η. Then, we choose δ = 2 −d τ to concude the proof of (H1 Proof. The first property follows from Lemma 6.6. To prove convergence, we note that To take mollification out of the function ψ we want to use Jensen's inequality and Lemma 6.2. The latter requires estimate on ∇S ε u ∞ Now, we can apply Lemma 6.2 (G1) to obtain estimate (6.8). The rest of the proof is exactly the same.

Extension of Theorem 2.3 to vector-valued maps
Many authors consider variational problems with vector-valued functions. However, in our work functionals depend only on the length of the gradient so there is almost no difficulty in extending our result to the vector case setting. In this section, we write u = (u 1 , ..., u n ) for the map u : Ω → R n . For simplicity, we use the same notation for spaces of vector-valued functions as for spaces of scalar-valued ones.
The main point that needs explanation is a generalisation of Lemma 4.2, where we applied truncation to approximate functions from W 1,ψ (Ω) by functions from W 1,ψ (Ω) ∩ L ∞ (Ω).
Proof. We first prove that C ∞ c (Ω) is dense in the Musielak-Orlicz-Sobolev space W 1,ψ 0 (Ω). This follows from the following facts.
Having density of C ∞ c (Ω) in W 1,ψ 0 (Ω) in hand, the absence of Lavrentiev phenomenon follows as in the proof of Lemma 4.3. In the proof of the main results, we have applied the following corollary. Then, f n → f in L 1 (X, F , µ).