Regularity for minimizers for functionals of double phase with variable exponents

The functionals of double phase type \[ \mathcal{H} (u):= \int \left(|Du|^{p} + a(x)|Du|^{q} \right) dx, ( q>p>1, a(x)\geq 0) \] are introduced in the epoch-making paper by Colombo-Mingione for constants $p$ and $q$, and investigated by them and Baroni. They obtained sharp regularity results for minimizers of such functionals. In this paper we treat the case that the exponents are functions of $x$ and partly generalize their regularity results.


Introduction and main theorem
The main goal of this paper is to provide a regularity theorem for minimizers of a class of integral functionals of the calculus of variations called of double phase type with variable exponents defined for u ∈ W 1,1 (Ω; R N ) (Ω ∈ R n , n, N ≥ 2) as F(u, Ω) := Ω |Du| p(x) + a(x)|Du| q(x) dx, q(x) ≥ p(x) > 1, a(x) ≥ 0, where p(x), q(x) and a(x) are assumed to be Hölder continuous. They do not only have strongly non-uniform ellipticity but also discontinuity of growth order at points where a(x) = 0. The above functional is provided by the following type of functionals with variable exponent growth u → g(x, Du)dx, λ|z| p(x) ≤ g(x, z) ≤ Λ(1 + |z|) p(x) , Λ ≥ λ > 0, which are called of p(x)-growth. These p(x)-growth functionals have been introduced by Zhikov [2] (in this article α(x) is used as variable exponents) in the setting of Homogenization theory. He showed higher integrability for minimizers and, on the other hand, he gave an example of discontinuous exponent p(x) for which the Lavrentiev phenomenon occurs ( [3,4]).
Such functionals provide a useful prototype for describing the behaviour of strongly inhomogeneous materials whose strengthening properties, connected to the exponent dominating the growth of the gradient variable, significantly change with the point. In [3], Zhikov pointed out the relationship between p(x)-growth functionals and some physical problems including thermistor. As another application, the theory of electrorheological materials and fluids is known. About these objects see, for example, [5,6,7,8].
For the continuous variable exponent case, nowadays many results on the regularity for minimizer are known, see [21,22,23,24]. Further results in this direction can be, for instance, found in [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36,37,38,39,40,41] for partial regularity results for p(x)-energy type functionals: In 2015 a new class of functional so-called functionals of double phase are introduced by Colombo-Mingione [1]. In the primary model they have in mind are u → H(u; Ω) := H(x, Du)dx, H(x, z) := |z| p + a(x)|z| q , where p and q are constants with q ≥ p > 1 and a(·) is a Hölder continuous non-negative function. By Colombo-Mingione [1,42,43] and 45,46] many sharp results are given about the regularity of local minimizers of the functional defined as where G(x, u, z) : Ω × R × R n → R is a Carathéodory function satisfying the following growth condition for some constants Λ ≥ λ > 0 besides several natural assumptions: For the scalar valued case, in [46] regularity results are given comprehensively. Under the conditions a(·) ∈ C 0,α (Ω), α ∈ (0, 1] and q p ≤ 1 + α n , they showed that a local minimizer of G defined as (1.1) is in the class C 1,β for some β ∈ (0, 1). For the scaler valued case, see also [47]. They proved Harnack's inequality and the Hölde continuity for quasiminimizer of the functional fo type where ϕ is the so-called Φ-function. We mention that Harnack's inequality is not valid in the vector valued cases which we are considering in the present paper.
In this paper we deal with a typical type of functionals of double phase with variable exponents and show a regularity result for minimizers.
In our opinion these results present new and interesting features from the point of view of regularity theory.
Let Ω ⊂ R n be a bounded domain, p(x), q(x) and a(x) functions on Ω satisfying where p 0 is a fixed constant strictly larger than one and a ∈ C 0,α (Ω), a(x) ≥ 0, (1.6) for α, σ ∈ (0, 1]. Moreover, we assume that p(x) and q(x) satisfy at every x ∈ Ω (compare these conditions with (1.2)). Let F : We consider the functional with double phase and variable exponents defined for u : Ω → R N and D ⋐ Ω as For a bounded open set Ω ⊂ R n and a function p : Ω → [1, +∞), we define L p(x) (Ω; R N ) and W 1,p(x) (Ω; R N ) as follows: In what follows we omit the target space R N . We also define L p(x) loc (Ω) and W 1,p(x) loc (Ω) similarly. As mentioned in [48], if p(x) is uniformly continuous and ∂Ω satisfies uniform cone property, then Let us define local minimizers of F as follows: for any ϕ ∈ W
In order to prove the above theorem, we employ a freezing argument; namely we consider a frozen functional which is given by freezing the exponents, and compare a minimizer of the original functional under consideration with that of frozen one.

Preliminary results
In what follows, we use C as generic constants, which may change from line to line, but does not depend on the crucial quantities. When we need to specify a constant, we use small letter c with index.
For double phase functional with constant exponents, namely for we prepare the following Sobolev-Poincaré inequality which is a slightly generalised version of [1, Theorem 1.6] due to Colombo-Mingione.
From the above theorem, we have the following corollary.
Corollary 2.2. Assume that all conditions of Theorem 2.1 are satisfied, and let D be a subset of B R with positive measure. Then, there exists a constant C depending only on n, p, q, [a] 0,β , R n /|D| and Du L p (B R ) and exponents d 1 > 1 > d 2 depending only on n, p, q, β such that the following inequality holds whenever Proof.
Choosing ω so that and applying Theorem 2.1, we get the assertion.

Remark 2.3.
In [1, Theorem 6.1], and therefore also in the above theorem and corollary, the exponent d 2 ∈ (0, 1) is chosen so that the following conditions hold: In fact, in [1], they choose a constant γ ∈ (1, p) so that (see [1, (3.6), (3.14)]), and put d 2 = 1/γ. Let us mention the that if d 2 satisfies (2.4) and (2.5) for some q = q 0 and p = p 0 , then the same d 2 satisfies these inequalities for any q and p with q/p ≤ q 0 /p 0 .
For any y ∈ Ω and R > 0 with B R (x) ⊂ Ω let us put We prove interior higher integrability of the gradient of a minimizer, similar results are contained in [54].
(Ω) be a local minimizer of F. Then, for any compact subset K ⊂ Ω, F (x, Du) ∈ L 1+δ 0 (K) and there exists a positive constant δ 0 and C depending only on the given data and K such that Proof. Let K ⊂ Ω be a compact subset and R 0 ∈ (0, dist(K, ∂Ω)) a constant such that For any x 0 ∈ • K, put The above estimate (2.11) implies that where c 0 is a constant depending only on max K q(x). On the other hand, since F (x, Du) ∈ L 1 , we have . Thus, mentioning also that w = u outside B s (y), we see that F (x, Dw) ∈ L 1 (K), namely w is an admissible function. In the following part of the proof, let us abbreviate Then, we have We can use hole-filling method. Add c 0 Bs(y)\Bt(y) F (x, Du)dx to the both side and divide them by c 0 + 1, then we get (2.14) Using an iteration lemma [55, Lemma 6.1], we see, for some constant C = C(c 0 , p 2 , q 2 ), that Putting s = R and t = R/2, we have Since R p 1 −p 2 and R q 1 −q 2 are bounded because of the Hölder continuity of exponents p(x) and q(x), puttingã In order to get the boundedness of R p 1 −p 2 and R q 1 −q 2 the so-called "log-Hölder continuity" (see [56, section 4.1]) is sufficient. On the other hand by virtue of the Hölder continuity of q(·), we have thatã ∈ C 0,β (β = min{α, σ}). Let d 2 ∈ (0, 1) be a constant satisfying (2.4) and (2.5) for β = min{α, σ}, q = q 2 (x 0 , R 0 ) and p = p 1 (x 0 , R 0 ). Then, for any B R (y) ⊂ B R 0 (x 0 ), this d 2 satisfy (2.4) and (2.5) with q = q 2 (y, R) and p = p 2 (y, R). By Theorem 2.1, we can estimate II as follows.
Then, using Hölder inequality, we can estimate the first term of the right hand side of (2.18) as follows. (2.20) Since, and u locally minimizes F, B R (y) |Du| p(x) dx is bounded. On the other hand, as mentioned where ω n denotes the volume of a n-dimensional unit ball. Thus, from (2.20) we obtain for some positive constant Similarly, we can estimate the second term of the left hand side of (2.18) as follows.
As above, using local minimality of u and the fact that R −(q 2 −q 1 ) is bounded, we have for a positive constant Thus, we obtain for some positive constant c 4 = c 4 (c 3 , θ) for any B R (y) ⊂ B R 0 ⊂ K ⋐ Ω. Now, by virtue of the reverse Hölder inequality with increasing domain due to Giaquinta-Modica [57], we get the assertion.
For δ 0 determined in Proposition 2.4, in what follows, we always take R > 0 sufficiently small so that We need also higher integrability results on the neighborhood of the boundary. Let us use the following notation: for T > 0 we put We say "f = g on Γ T " when for any η ∈ C ∞ 0 (B T ) we have (f − g)η ∈ W 1,1 0 (B + T ). For y ∈ B T , we write Ω r := B r (y) ∩ B + T . Then, we have the following proposition on the higher integrability near the boundary, independently proved in [58,Lemma 5] , see also [59,Lemma 5] for the manifold constrained case. u ∈ W 1,p (B + T ) be a given function with for some δ 0 >. Assume that v ∈ W 1,p (B + (T )) be a local minimizer of H in the class Then, for any S ∈ (0, T ), there exists a constants δ ∈ (0, δ 0 ) and C > 0 such that for any y ∈ B + S and R ∈ (0, T − S) we have Proof. For convenience, we extend u, v, Du, Dv to be zero in B T \ B + T . Of course, because extended u, v may have discontinuity on Γ T , they are not always in W 1,p loc (B T ), and therefore Du, Dv do not necessarily coincide with distributional derivatives of u, v on B(T ). On the other hand, since u = v on Γ(T ), u − v is in the class W 1,p (B(S)) and Du − Dv can be regarded as the weak derivatives of u − v on B(S) for any S < T .
we see that ϕ ∈ W 1,1 0 (B + T ) with supp ϕ ⊂ B s , and that Then, by virtue of the minimality of v, for a positive constant c 4 depending only on q, we have Using the iteration lemma [55, Lemma 6.1], we get for some constant C = C(c 4 , p, q) Putting t = R/2 and s = R, we have Let us now consider the mean integral in all the terms, we obtain Since we are assuming that x n 0 ≤ 3 4 R we can apply Corollary 2.2 with a constant independent on R for the last term in the right hand side and get Taking into consideration that d 2 < 1 we share in the last term Dv and Du, apply Hölder inequality for the integral of H(x, Du) d 2 , and obtain (2.27) Case 2. Let us deal with the case that x n 0 > 3 4 R. In this case, since B 3R/4 (x 0 ) ⋐ B + T , we can proceed as in [1, 9. Proof of Theorem 1.1:(1.8)], slightly modifying the radii, to get (2.28) Thus, we see that (2.27) holds for every 0 < R < (S − T )/2. Now, the reverse Hölder inequality allows us to obtain . By virtue of [1, Theorem 1.1] and Proposition 2.5, we have the following global higher integrability for functions which minimize H with Dirichlet boundary condition.
Corollary 2.6. Let a(x), q and p satisfy the same conditions in Theorem 2.1 and δ 2 ∈ (0, 1) be a some constant. Assume that u ∈ W 1,(1+δ 1 )p (B R (y)) be a given function with Then, for some δ 2 ∈ (0, δ 1 ) and for any δ 3 ∈ (0, δ 2 ), we have H(x, Dv) ∈ L 1+δ (B R (y)) and Proof. From [1, Theorem 1.1], Proposition 2.5 and covering argument, we have and then, by the minimality of v, Once again we use the Hölder inequality for the first term of the right-hand side that gives us the assertion.

Proof of the main theorem
In this section we prove Theorem 1.2. We employ the so-called direct approach, namely we consider a frozen functional for which the regularity theory has been established in [1] and compare a local minimizer of the frozen functional with u under consideration.
For a constant p > 1, let us define the auxiliary vector field V p : R n → R n as Let mention that V p satisfies Proof of Theorem 1.2. We divide the proof into two parts. We prove the Hölder continuity of u in Part 1, and of the gradient Du in Part 2.
Part 1. Let K and B R 0 (x 0 ), are as in the Proposition 2.4. For B R (y) ⊂ B 2R (y) ⊂ B R 0 (x 0 ), let us define p i and q i as in the Proposition 2.4. We define a frozen functional F 0 as In what follows, let us abbreviateã(x) = (a(x)) q 2 q(x) as in the proof of Proposition 2.4. Let v ∈ W p 2 (B R (y)) be a minimizer of F 0 in the class Then, by [1,Theorem1.3], for any γ ∈ (0, 1) there exists a constant C > 0 dependent on n, p 2 , q 2 , λ, Λ, [ã] 0,β , ã ∞ , Dv L p 2 (B R (y)) and γ such that (3.5) where we used the minimality of v. Here, we mention that by the coercivity of the functional and the minimality of v we have the following: On the other hand, since we are taking R > 0 sufficiently small so that (2.26) holds, there exists a constant C(p 2 , q 2 ) > 0 such that holds for any (x, ξ) ∈ B R (y) × R nN . Now, by virtue of above 2 estimates and Proposition 2.4, we can see, for a constant C > 0 depending only on the given data on the functional, that Because of the local minimality of u, the last quantity is finite. Consequently, we can regard the constant in (3.5) is a constant depending only on given data and F(u, K). For further convenience, let us mention that from (3.5), is nothing to see that Let us compare Du and Dv. Mentioning the elementary equality for a twice differentiable function as [21, (9)], and using the fact that v satisfies the Euler-Lagrange equation of F 0 , we can see that On the other hand, by the minimality of v, we have Since we are assuming p(x), q(x) ∈ C 0,σ , using the inequality [21, (7)], we can see that, for any ε ∈ (0, 1), there exists a positive constant C such that Similarly we have Now, for δ 0 of Proposition 2.4, choose δ 3 > 0 so that (2.29) of Corollary 2.6 holds, and let us take ε so that ε ∈ (0, min{δ 0 /2, δ 3 }/2). Since we are choosing R so that (2.26) holds, we have (3.14) By Proposition 2.4 and (3.14), we deduce from (3.12) that where we used the fact that for some constant M 0 . The existence of M 0 guaranteed by the local minimality of u. For (3.13) we use Proposition 2.6, Proposition 2.4 and (3.14), to get On the other hand, by the definition of F 0 , we have Du)) .
Part 2. Now, we are going to show the Hölder continuity of the gradient Du. For y ∈ • K let R 1 ∈ (0, R 0 ) be a constant such that B R 1 (y) ⊂ K, and for 0 < R < R 1 /4 let v be as in Part 1. Then, by the estimate given by Colombo-Mingione at [1, p.484, l.-6], we see that there exist constants C > 0, dependent on n, p 2 , q 2 , λ, Λ, ã ∞ , dist(K, ∂Ω), F 0 (v, B R (y)) andα ∈ (0, 1) holds for any ρ ≤ R/2. Here, as in Part 1, let us mention that F 0 (v, B R (y)) can be controlled by F(u, K) as (3.8). So, we can choose the above constant in (3.20) to be dependent only on the given data of the functional, the local minimizer u under consideration and K.
By virtue of (3.20), for ρ and R as above, we get For the case that p 2 ≥ 2, since there exists a constant such that |z 1 − z 2 | p 2 ≤ C |z 1 | p 2 −2 + |z 2 | p 2 −2 |z 1 − z 2 | 2 for any z 1 , z 2 ∈ R n , using (3.17), we can estimate the last term of the right hand side of (3.21) as We use (3.19) replacing ρ by 2R and R by R 0 to see that Since R 0 is determined in the beginning of the proof, we can regard R ζ 0 − B R 0 F 0 (x, Du)dx as a constant. So, we get B 2R (y) F 0 (x, Du)dx ≤ CR n−ζ . For the last inequality we used the following facts: 0 < R ≤ 1, 0 < σ − nε, p 2 > 1.