Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 80, pp. 1–30. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GRADIENT REGULARITY FOR NON-AUTONOMOUS FUNCTIONALS WITH DINI OR NON-DINI CONTINUOUS COEFFICIENTS PAOLO BARONI, ALESSANDRA COSCIA Communicated by Giovanni Molica Bisci Abstract. We prove C1 regularity for local vectorial minimizers of the non- autonomous functional w ∈W 1,1 loc (Ω;RN ) 7→ ∫ Ω b(x) [ |Dw|p + a(x)|Dw|p log(e+ |Dw|) ] dx , with Ω open subset of Rn, n ≥ 2 , p > 1, 0 ≤ a(·) ≤ ‖a‖L∞(Ω) < ∞, and 0 < ν ≤ b(·) ≤ L. The result is valid provided that the function a(·) is log- Dini continuous and that the coefficient b(·) is Dini continuous or it is weakly differentiable and its gradient locally belongs to the Lorentz space Ln,1(Ω;Rn). 1. Introduction and results A recent and important object of investigation in the calculus of variations is the study of the regularity of minimizers of non-autonomous functionals of the type u ∈W 1,1 loc (Ω) 7→ ∫ Ω f(x, u,Du) dx , (1.1) being Ω a bounded domain in Rn and f : Ω×R×RN → R a Carathéodory function satisfying unbalanced growth conditions of (p, q)-type 1 c |ξ|p ≤ f(x, u, ξ) ≤ c ( 1 + |ξ|q ) 1 < p < q , (1.2) for almost every x ∈ Ω and all (u, ξ) ∈ R × RN , with c ≥ 1. General energies of this type (i.e., when no structural assumptions are made) were initially introduced and studied by Marcellini [41, 42, 43] in the late eighties but the theory has seen deep and substantial contributions in the recent years, see for instance the Schauder estimates in [29, 30], the series of regularity results under sharp bounds on p and q by Bella and Schäffner [8, 9, 35, 45], the interesting approach developed in [7] also covering more general cases and the new boundary regularity results in [11, 31]. Starting from [18, 19, 20], a substantial amount of work has been put in the study of a special structure having (p, q)-growth, the so-called double phase structure, where the peculiar form of the energy has allowed a much more precise, clean 2020 Mathematics Subject Classification. 35J15, 35J60, 35J99. Key words and phrases. Non-autonomous functionals; gradient continuity; Dini continuous coefficients. ©2022. This work is licensed under a CC BY 4.0 license. Submitted November 1, 2022. Published November 23, 2022. 1 2 P. BARONI, A. COSCIA EJDE-2022/80 and careful analysis of several aspects of regularity. We refer with double phase structure the fact that the energy density has the form f(x, u, ξ) ≈ |ξ|p + a(x)|ξ|q (1.3) with p and q as in (1.2) and a(·) ≥ 0 continuous and sufficiently regular to compen- sate the non-uniform ellipticity of the functional [28]; see [6] for a regularity theory for local minimizers in a general framework, [25, 26] for interesting borderline cases, [27] for extension to manifold-valued minimizers, [44] for ω-minimizers, [22, 24] for the fully nonlinear counterpart of the theory, and [33, 34] for far-reaching extensions of such structures. The object of study of this paper is the borderline version of the double phase functional (1.1)-(1.3) that was introduced in [4]: u ∈W 1,1 loc (Ω;RN ) 7→ P(u,Ω) := ∫ Ω b(x)H(x,Du) dx = ∫ Ω b(x) [ |Du|p + a(x)|Du|p log(e+ |Du|) ] dx (1.4) for p > 1, N ≥ 1, and the aspect of regularity we are interested in is the plain continuity of the gradient of local minimizers. The basic assumptions we impose on a(·), b(·) are as follows: a : Ω→ R continuous and b : Ω→ R measurable with 0 ≤ a(x) ≤ ‖a‖L∞ < +∞, 0 < ν ≤ b(x) ≤ L < +∞ for a.e. x ∈ Ω. (1.5) Because of the soft non-uniform ellipticity of the energy in (1.4), the regularity of minimizers is strictly entangled with the behavior of the quantity ωlog(R) = ωa(R) log ( 1 R ) (1.6) where ωa : [0, 1]→ [0, 2‖a‖L∞ ] is a modulus of continuity for a(·): a concave (thus continuous in (0, 1)) and increasing function, continuous in zero and with ωa(0) = 0, such that |a(x)− a(y)| ≤ ωa(|x− y|) for all x, y ∈ Ω with |x− y| ≤ 1 . (1.7) We define in a totally analogous way ωb : [0, 1]→ [0, 2] as the modulus of continuity of b(·), if b is (uniformly) continuous (note that all the results we are mentioning and proving are local; it is therefore not restrictive to assume that if b is continuous, then it has a modulus of continuity): |b(x)− b(y)| ≤ Lωb(|x− y|) for all x, y ∈ Ω with |x− y| ≤ 1 . (1.8) More in detail, schematically summarizing the results in [4, 5, 21], one has that scalar local minimizer to (1.4) (i.e., N = 1) in the case (1.5) holds are such that • if lim supρ↘0 ωlog(R) <∞, then u ∈ C0,α loc (Ω), Du ∈ Lp(1+δ0) loc (Ω;Rn) for some constants α, δ0 ∈ (0, 1) depending on the data and the Harnack’s inequality holds for positive solutions; • if lim supR↘0 ωlog(R) = 0 and b(·) is continuous (i.e., it has a modulus of continuity), then u ∈ C0,α loc (Ω) for every α ∈ (0, 1); EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 3 • if ωlog(R) + ωb(R) ≤ cRγ for some γ ∈ (0, 1) and c ≥ 1, then u ∈ C1,β loc (Ω) for some exponent β ∈ (0, 1) depending on the data. Note that some of the previous results have also a vectorial counterpart, but some do not. Other noticeable results for minimizers of the functionals in (1.4) and measure data problems associated to its Euler equation can be found in [12, 13, 14] by Byun, Youn and collaborators; see also [1] for a significant counterexample. A natural borderline question would be regarding the assumptions to be imposed on b(·), a(·) in order to ensure gradient continuity for local minimizers and a first answer is given in the next theorem. Theorem 1.1. Let u ∈ W 1,p loc (Ω;RN ), N ≥ 1, a local minimizer of (1.4) under the assumption (1.5); suppose moreover that b(·) is Dini continuous and that a(·) is log-Dini, in the sense, respectively, that both ωb(ρ) and ωlog(ρ) are integrable in zero with respect to the measure dρ/ρ:∫ 1 0 ωb(ρ) dρ ρ <∞, ∫ 1 0 ωlog(ρ) dρ ρ = ∫ 1 0 ωa(ρ) log (1 ρ ) dρ ρ <∞ . (1.9) Du is continuous in Ω and the following local boundedness estimate holds: for every K b Ω, there exists a radius R0 depending on n,N, p, L/ν, ωb(·), ωa(·),dist(K, ∂Ω) and ‖H(·, Du)‖L1(K) such that for almost every x0 ∈ K and each ball BR(x0) ⊂ K with radius R ∈ (0, R0] it holds H(x0, Du(x0)) ≤ c(n,N, p, L/ν)− ∫ BR(x0) H(x,Du(x)) dx . (1.10) Note that the scalar case of the previous result is a consequence of [13, Theorem 1.2] and the fact that a SOLA is also an energy, weak solution to the Euler equation of P. The extension of the results in [13] to the vectorial case is completely non- trivial and therefore we decided to present and prove Theorem 1.1, which follows without effort from the approach we developed to prove the forthcoming Theorem 1.2. Dini continuity of the coefficient is a natural assumption ensuring gradient continuity and pointwise potential estimates whose best possible consequence is gradient continuity, see [13, 38, 39] and the counterexample in [36]; the log-Dini continuity of a(·) is also quite natural in view of the fact that, at least heuristically, it is simply needed a logarithmic correction on the assumption on b(·) for the regularity of a(·) [13]; see also [3] for a declination of this principle from the point of view of Sobolev-like assumptions. Maybe more interesting and less expected of the previous one is the forthcoming Theorem 1.2, where we replace the assumption of Dini continuity of b(·) with an assumption of integral-Sobolev type. In particular we shall assume that b(·) is weakly differentiable and that its gradient belongs to the Lorentz space Ln,1 locally in Ω, which means that for every K b Ω it holds∫ ∞ 0 ∣∣{x ∈ K : |Db(x)| > λ} ∣∣ 1 n dλ <∞ . We emphasize that a by-now classic Sobolev-type sharp embedding in Lorentz spaces ensures that in this case b(·) is continuous but not Dini continuous; see [17, Remark 3.6]. Therefore, Theorem 1.1 does not follow by embedding from 4 P. BARONI, A. COSCIA EJDE-2022/80 Theorem 1.2. Let u ∈ W 1,p loc (Ω;RN ) a minimizer of (1.4) under the assumption (1.5). Suppose that a(·) is log-Dini continuous and that b ∈ W 1,1 loc (Ω) with Db ∈ Ln,1loc (Ω;Rn); then Du is continuous in Ω and a local boundedness estimate as (1.10) holds with the sole difference that R0 here depends on |Db(·)|/ν instead of ωb(·) (the other dependencies remain unchanged). We stress that it is also possible to impose Sobolev-Lorentz-Zygmund type as- sumptions on the switching coefficient a(·) and this was done in [3] by the first author. In particular in [3] the case without coefficient is treated (i.e., b ≡ 1) but gradient continuity of minimizers of (1.4) is proved not supposing a(·) log-Dini continuous, but weakly differentiable with∫ ∞ 0 ∣∣{x ∈ Ω : |Da(x)| > λ} ∣∣ 1 n log+ λ dλ <∞, (log+ λ = max{log λ, 0}) , that is, by a nice result by Bennett and Rudnick [10], Da ∈ Ln,1 logL(Ω;Rn) (or at least locally in Ω). We believe that it is possible to consider on both a(·), b(·) assumptions of integral type, but this goes beyond the scopes of this work and will be investigated in the future. The concept allowing to prove gradient continuity under the assumptions of Theorem 1.1 finds its roots in the basic perturbation idea going back to Campanato: we fix a ball BR ≡ BR(x0) b Ω and we quantify, in integral terms, how much our minimizer is distant from a regular minimizer of a reference variational problem (see (3.3)-(3.4)). This is measured in terms of the quantities ωb(R) and ωlog(R) (see (5.1)). Both the gradient boundedness and its continuity come from a telescopic summation argument and this ultimately boils down to testing the summability of ωb + ωlog along a sequence of radii Rj = δjR for some starting radius R > 0 and a very small constant δ � 1. A simple computation shows that ∞∑ j=0 ωb(Rj) ≈ ∫ 2R 0 ωb(ρ) dρ ρ , ∞∑ j=0 ωlog(Rj) ≈ ∫ 2R 0 ωlog(ρ) dρ ρ (see (4.2) and (5.2)); it is immediate now to realize why the assumptions of Dini and log-Dini continuity come into play. Theorem 1.2 is based on a much more refined, and quite new, perturbation argument. Thanks to the use of higher integrability, the aforementioned distance from the regular minimizer is measured, for what concern the coefficient b(·) (the switching coefficient is indeed treated exactly as in Theorem 1.1), in terms of the Ls-excess, for s� 1 a very large constant:( − ∫ BR ∣∣b− (b)BR ∣∣s dx)1/s , (b)BR = − ∫ BR b(y) dy . (1.11) One has here to realize that in the case Db ∈ Ln,1, the dyadic summation of this quantity is bounded, in view of Sobolev-Poincaré’s inequality and Lemma 4.1 (which is a simple consequence of the characterization of Ln,1 in terms of rearrangements): for q = s∗ = ns/(n+ s) < n as in Lemma 3.6 and Bj ≡ BRj (x0) ∞∑ j=0 ( − ∫ Bj ∣∣b− (b)Bj ∣∣s dx)1/s . ∞∑ j=0 Rj ( − ∫ Bj |Db|q dx )1/q . ‖Db‖Ln,1(B2R) . EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 5 The idea of considering coefficient with controlled excess (in the sense just de- scribed) is not totally new: equations and variational problems with VMO coeffi- cients - that is, for which their excess as defined in (1.11) is vanishing as R↘ 0+, uniformly with respect to the center of the balls - have been largely studied since the seminal works [15, 16]. Much more recent is the observation that one can also consider summability properties of the excess along dyadic sequences of radii be- yond its qualitative smallness. In [38] it is proved that systems of p-Laplacian type have C1 loc solutions provided the right-hand side belongs to Ln,1: div ( |Du|p−2Du ) = f, p > 1 . This is a consequence, for equations, of the celebrated linear potential estimates in [37] and can also be proved as corollary of the more general vectorial potential estimates of [40]. In any case, this is the affirmative response to a long-standing conjecture by Uraltseva, claiming that the condition on f ensuring Lipschitz reg- ularity for solution should be independent of p. We choose to mention [38] since the telescopic summation technique therein introduced was the main inspiration for the papers [2, 3] of the first author and also for the present one. A further step is the extension of this idea to the coefficients of uniformly elliptic problems, and this can be done (see [2] and the current paper) realizing that, at least by formal differentiation, coefficients can be treated as right-hand sides. In [28, Theorem 1.8] it is shown that solutions to uniformly elliptic vectorial problems of the type div ( b(x) ϕ′(|Du|) |Du| Du ) = f are Lipschitz (and therefore C1, after computations of standard flavor) regular if both |f | and |Db| belong to Ln,1; the approach therein is based on Moser’s iteration. See [2] again for the two dimensional case and a perturbative approach to the question. 2. Preliminaries 2.1. Local and global minimizers. A local minimizer to P is a function u ∈ W 1,1 loc (Ω;RN ), N ≥ 1 such that H(·, Du) ∈ L1 loc(Ω) and the minimality condition P(u, supp(u− v)) ≤ P(v, supp(u− v)) is satisfied whenever v ∈W 1,1 loc (Ω;RN ) is such that supp (u− v) b Ω. Because of the local nature of our results, we may assume that Ω is a bounded domain and that local minimizers u belong to W 1,p(Ω;RN ) with H(·, Du) ∈ L1(Ω) too. As a consequence, we will write ‖Du‖Lp for ‖Du‖Lp(Ω;Rn) and ‖H(·, Du)‖L1 for ‖H(·, Du)‖L1(Ω). The minor changes that lead from this situation to the case described in Theorems 1.1-1.2 are left to the reader. 2.2. Notation and elementary properties. In what follows we denote by c a general positive constant possibly varying from line to line; special occurrences will be denoted by c0, c̄, etc. All such constants will always be larger or equal than one; relevant dependencies on parameters will be highlighted using parentheses, i.e., c ≡ c(n, p, δ) will mean that c depends on n, p and δ. We denote by Br(x0) := {x ∈ RN : |x− x0| < r} 6 P. BARONI, A. COSCIA EJDE-2022/80 the open ball with center x0 and radius r > 0; when not important, or clear from the context, we shall omit denoting the center writing Br ≡ Br(x0). Unless otherwise stated, different balls in the same context will have the same center. With B ⊂ Rn being a measurable set with positive, finite measure |B| > 0, and with g : B → R`, ` ≥ 1, being a locally integrable map, we shall denote by (g)B := − ∫ B g(x) dx = 1 |B| ∫ B g(x) dx its integral average. A well-known property is the following: for any g ∈ Lp(B;R`), p ≥ 1, ` ≥ 1, the estimate − ∫ B |g(x)− (g)B|p dx ≤ 2p − ∫ B |g(x)− ζ|p dx (2.1) holds for each ζ ∈ R`. We use the agreement that N is the set {1, 2, 3, . . . } and N0 := N ∪ {0}. The Sobolev conjugate exponent p∗ is np/(n− p) when p < n. We recall some useful properties of the logarithm function of later frequent use: log(e+Ax) ≤ A log(e+ x) for every x ≥ 0, A ≥ 1 , (2.2) log(e+ xy) ≤ log(e+ x) + log(e+ y) for every x, y ≥ 0 , (2.3) log(e+ xσ) ≤ 1 + max{1, σ} log(e+ x) for every x ≥ 0, σ > 0 . (2.4) The proofs are very simple, we only highlight for (2.4) that distinguishing the cases σ < 1, where log(e + xσ) ≤ log(2(e + x)) ≤ 1 + log(e + x), and σ ≥ 1, where log(e+ xσ) ≤ σ log(e+ x), leads to the result. The following lemma will be very useful in order to treat the logarithmic part of the energy. Lemma 2.1. Let s > 1, σ, β, θ ≥ 0 and let f be a positive function in Ls(Br) for some ball Br(x0) with radius r ≤ e−1. Then there exists a constant c depending on n, β, σ, θ, s such that − ∫ Br f logβ ( e+ fσ ) dx ≤ c ( 1 + rθ‖f‖L1(Br) )β logβ (1 r )( − ∫ Br fs dx )1/s . Proof. Recalling that r ≤ e−1, by (2.4), (2.3), the forthcoming (2.38), basic prop- erties of the logarithm function (for instance (2.2)) we estimate − ∫ Br f logβ(e+ fσ) dx ≤ c(σ, β)− ∫ Br f [ 1 + log (e+ f) ]β dx ≤ c(σ, β)− ∫ Br f [ 1 + log ( e+ f (f)Br ) + log ( e+ (f)Br )]β dx ≤ c(σ, β) [ − ∫ Br f dx+− ∫ Br f logβ ( e+ f (f)Br ) dx+ logβ ( e+− ∫ Br f dx ) − ∫ Br f dx ] ≤ c(n, σ, β, s) ( − ∫ Br fs dx )1/s[ 1 + logβ ( e+− ∫ Br f dx )] ≤ c(n, σ, β, s) ( − ∫ Br fs dx )1/s[ 1 + logβ ( e+ 1 rn ∫ Br f dx )] ≤ c(n, σ, β, s) ( − ∫ Br fs dx )1/s[ 1 + logβ ( e+ (1 + rθ r−θ‖f‖L1(Br)) 1 rn )] EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 7 ≤ c(n, σ, β, s)(1 + rθ‖f‖L1(Br)) β ( − ∫ Br fs dx )1/s[ 1 + logβ ( e+ 1 rn+θ )] ≤ c(n, σ, β, s)(1 + rθ‖f‖L1(Br)) β ( − ∫ Br fs dx )1/s[ logβ (1 r ) + [ 2(n+ θ) log (1 r )]β] ≤ c(n, σ, β, θ, s)(1 + rθ‖f‖L1(Br)) β logβ (1 r )( − ∫ Br fs dx )1/s and the proof is complete. � 2.3. N-functions setting. In the following we are going to introduce a general class of tools, related to the so-called general class of N -functions. For the results we mention here see for instance [23, 32]. We consider a convex function ϕ : [0,∞)→ [0,∞), such that ϕ ∈ C1([0,∞)) ∩ C2((0,∞)), ϕ(0) = ϕ′(0) = 0, ϕ′(t) is monotone increasing and lim t→∞ ϕ′(t) =∞ . (2.5) In addition we assume that there exists a constant cϕ ≥ 1 such that 1 cϕ ≤ ϕ′′(t)t ϕ′(t) ≤ cϕ, for all t > 0 . (2.6) If the function ϕ verifies (2.5) and (2.6), then we call ϕ an N -function. Notice that every non-decreasing function ϕ : [0,∞)→ [0,∞) satisfies ϕ(t+ s) ≤ ϕ(2t) + ϕ(2s) . (2.7) We define the auxiliary vector field Vϕ : R` → R`, ` ∈ N by Vϕ(z) := (ϕ′(|z|) |z| )1/2 z , where Vϕ is continuously extended to zero when z = 0; Vϕ turns out to be a bijec- tion of R` by (2.5) and under the assumption (2.6) Vϕ describes the monotonicity properties of the map [ϕ′(|z|)/|z|]z: for z1, z2 ∈ R`, z1, z2 6= 0 we have 1 c ∣∣Vϕ(z1)− Vϕ(z2) ∣∣2 ≤ 〈ϕ′(|z1|) |z1| z1 − ϕ′(|z2|) |z2| z2, z1 − z2〉 ≤ c ∣∣Vϕ(z1)− Vϕ(z2) ∣∣2 , (2.8) for a constant c ≥ 1 depending on cϕ. For another constant c ≡ c(cϕ) the following relations (see [32, Lemma 2.4]) hold for every z, z1, z2 ∈ R` with z1 or z2 different from zero (which means |z1|+ |z2| > 0): 1 c ϕ(|z|) ≤ ∣∣Vϕ(z) ∣∣2 ≤ cϕ(|z|) , (2.9) 1 c ϕ′′(|z1|+ |z2|)|z1 − z2|2 ≤ ∣∣Vϕ(z1)− Vϕ(z2) ∣∣2 ≤ cϕ′′(|z1|+ |z2|)|z1 − z2|2 . (2.10) For a constant ā ≥ 0, we are interested in ϕp(t) = tp p , ϕlog(t) = tp p log(e+ t) , and ϕā(t) = ϕp(t) + āϕlog(t) , (2.11) which satisfy all the assumptions (2.5). In addition for t > 0, tϕ′′p(t) ϕ′p(t) = p− 1 , p− 1 ≤ tϕ′′log(t) ϕ′log(t) ≤ 2p , 8 P. BARONI, A. COSCIA EJDE-2022/80 (p− 1)ϕ′ā(t) ≤ tϕ′′ā(t) ≤ 2pϕ′ā(t) , so (2.6) is satisfied with a constant cϕ = cϕ(p) depending only on p and independent of ā. It is also easy to estimate tp−1 log(e+ t) ≤ ϕ′log(t) ≤ 2tp−1 log(e+ t) , (2.12) (p− 1)tp−2 log(e+ t) ≤ ϕ′′log(t) ≤ (p+ 1)tp−2 log(e+ t) . (2.13) Adopting the notation hā(t) = tp−1 + ātp−1 log(e+ t) for every t ≥ 0 , (2.14) from (2.12) and (2.13) we deduce the following estimates: hā(t) ≤ ϕ′ā(t) ≤ 2hā(t) , (2.15) 1 c(p) hā(t) ≤ tϕ′′ā(t) ≤ c(p)hā(t) . (2.16) which will be useful in the proof of Lemma 2.4. In the sequel for p > 1, for every x ∈ Ω and z ∈ R`, we adopt the notation H(x, z) := |z|p + a(x)|z|p log(e+ |z|) = p [ ϕp(|z|) + a(x)ϕlog(|z|) ] , (2.17) Hā(z) := |z|p + ā|z|p log(e+ |z|) = pϕā(|z|) ; (2.18) let us remark that the relations Hā(z) = hā(|z|)|z| , ∂zHā(z) = pϕ′ā(|z|) z |z| (2.19) hold, where the function hā(·) is defined in (2.14). Let us denote by Vp(·) and Vlog(·) the vector fields generated by ϕp and ϕlog, and by Vā(z) := √ ϕ′ā(|z|) |z| z , (2.20) the one generated by ϕā(t), all continuously extended to zero when z = 0. It is easy to verify that |Vp(z)|2 = |z|p = pϕp(|z|) , pϕlog(|z|) = |z|p log(e+ |z|) ≤ |Vlog(z)|2 ≤ 2|z|p log(e+ |z|) = 2pϕlog(|z|) , pϕā(|z|) ≤ |Vā(z)|2 ≤ 2pϕā(|z|) , (2.21) thus in all cases (2.9) holds for a constant c depending only on p. In particular, if a(·) is a function satisfying (1.5), from (2.21) it follows that for every x0 ∈ Ω H(x0, z) ≤ ∣∣Va(x0)(z) ∣∣2 ≤ 2H(x0, z) . (2.22) In addition, since ϕ′′p(t) = (p− 1)tp−2 the estimate in (2.10), which holds whenever z1, z2 ∈ R` , |z1|+ |z2| > 0, becomes 1 c |z1 − z2|2 (|z1|+ |z2|)p−2 ≤ |Vp(z1)− Vp(z2)|2 ≤ c|z1 − z2|2 (|z1|+ |z2|)p−2 (2.23) where c depends only on p. In particular, when p ≥ 2, |z1 − z2|p ≤ c|Vp(z1)− Vp(z2)|2 holds, while for 1 < p ≤ 2 (see [38, Lemma 2]) we will use that |z1 − z2| ≤ c ∣∣Vp(z1)− Vp(z2) ∣∣2/p + c|z1|(2−p)/2 ∣∣Vp(z1)− Vp(z2) ∣∣ (2.24) EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 9 both for a suitable constant c ≡ c(p). Finally, let us remark that combining (2.10) for Vp(·) and Vā(·) it is easy to verify that for every z1, z2 ∈ R`∣∣Vp(z1)− Vp(z2) ∣∣2 ≤ c(p)∣∣Vā(z1)− Vā(z2) ∣∣2 . (2.25) 2.4. Preliminary results. First of all, let us prove that log-Dini continuity (1.9) of a(·) guarantees that a(·) is log-Hölder vanishing (that is, (2.41) holds); later this will allow the application of Theorem 2.7 to the minimizers of P in (1.4). Lemma 2.2. Let a(·) be a function satisfying (1.5) and let ωa(·) be a modulus of continuity of a(·) as in (1.7). If∫ 1 0 ωa(ρ) log (1 ρ )dρ ρ <∞ , (2.26) then limR↘0+ ωa(R) log(1/R) = 0. Proof. It suffices to prove that lim supR↘0+ ωa(R) log ( 1/R ) = 0. Reasoning by contradiction, let us assume that there exists a decreasing sequence {Rk} ⊂ (0, 1/2] such that R↘ 0+ and lim k→+∞ ωa(Rk) log ( 1 Rk ) = l ∈ (0,+∞] ; (2.27) we assume for now that l is finite. Up to a subsequence, we may assume in addition that for every k, ωa(Rk) log ( 1 Rk ) ≥ l 2 and lim k→+∞ Rk+1 Rk = 0 . (2.28) We want to prove that the integral (2.26) is divergent at 0. We can estimate the improper integral (2.26) with a series of integrals on [Rk+1, Rk],∫ 1 0 ωa(ρ) log (1 ρ )dρ ρ ≥ ∞∑ k=0 ∫ Rk Rk+1 ωa(ρ) log (1 ρ )dρ ρ . (2.29) Using the monotonicity of both ωa(R)/R and log(1/R) and (2.28), we estimate ∞∑ k=0 ∫ Rk Rk+1 ωa(ρ) ρ log (1 ρ ) dρ ≥ ∞∑ k=0 ωa(Rk) Rk log ( 1 Rk ) ∫ Rk Rk+1 dρ ≥ l 2 ∞∑ k=0 ( 1− Rk+1 Rk ) . By (2.28) the series with non-negative terms ∑∞ k=0 ( 1− Rk+1 Rk ) is divergent, thus by (2.29) the integral (2.26) is divergent at 0, against the assumption of the Lemma. Finally, if the limit in (2.27) is l = +∞, fixed T > 0 it suffices to replace l/2 in (2.28) with T to gain the same conclusion with the same calculations. � First, we need the following estimates on the function Hā(·), defined in (2.18) and on its gradient. For the proof see [21, (2.44),(2.45) and Lemma 2.2], where all estimates are proved for the function H(·, ·) defined in (2.17), of which the function Hā(·) is a particular case. Lemma 2.3. Let Hā : R` → R be the function defined in (2.18). Then the following estimates hold for every A ≥ 1 and every z, λ ∈ R`: Hā(Az) ≤ Ap+1Hā(z) , (2.30) Hā(z1 ± z2) ≤ 2p+1(Hā(z1) +Hā(z2)) , (2.31)∣∣〈∂zHā(z), λ〉 ∣∣ ≤ (p+ 1) ( Hā(z) +Hā(λ) ) . (2.32) 10 P. BARONI, A. COSCIA EJDE-2022/80 We collect in next lemma some monotonicity and growth results that link the vector field ∂zHā and the related nonlinear expression Vā; these will be particularly useful in order to prove the comparison Lemma 3.6. Lemma 2.4. Let ` ∈ N and let Vā : R` → R`, Hā : R` → R, and hā : [0,+∞)→ R be the functions defined in (2.20), (2.18), and (2.14) respectively. Then the following estimates hold for every z, z1, z2 ∈ R`: 1 c(p) ∣∣Vā(z1)− Vā(z2) ∣∣2 ≤ 〈∂zHā(z1)− ∂zHā(z2), z1 − z2〉 , (2.33) |∂zHā(z)| ≤ c(p)hā(|z|) , (2.34)∣∣Vā(z1)− Vā(z2) ∣∣2 ≥ 1 c(p) hā(|z1|+ |z2|) |z1|+ |z2| |z1 − z2|2 , (2.35) (|z1|+ |z2|)hā(|z1|+ |z2|) ≤ 2p+1 ( Hā(z1) + Hā(z2) ) , (2.36) |z1 − z2|hā(|z1|) ≤ c(p) ∣∣Vā(z1)− Vā(z2) ∣∣(√Hā(z1) + √ Hā(z2) ) , (2.37) under the further assumption |z1|+ |z2| > 0 in (2.35) and (2.37). Proof. Inequality (2.33) follows immediately by (2.19), rewriting (2.8) for the vector field Vā(·). Again by (2.19), we obtain |∂zHā(z)| = pϕ′ā(|z|) and inequality (2.34) follows from (2.15). Inequality (2.35) can be obtained rewriting (2.10) for Vā(·) together with (2.16). To prove (2.36), recalling (2.14), (2.11), (2.7) and (2.30), we calculate (|z1|+ |z2|)hā(|z1|+ |z2|) = pϕā(|z1|+ |z2|) ≤ p [ ϕā(2|z1|) + ϕā(2|z2|) ] = Hā(2z1) +Hā(2z2) ≤ 2p+1 [ Hā(z1) +Hā(z2) ] . Finally, let us prove inequality (2.37). Let us fix z1, z2 ∈ R` with |z1| + |z2| > 0, assuming without loss of generality that z1 6= 0; by (2.35), the monotonicity of hā(t), the monotonicity and the subadditivity of the function √ t on [0,+∞[, we estimate |z1 − z2| = |z1 − z2| √ hā(|z1|+ |z2|) |z1|+ |z2| √ |z1|+ |z2| hā(|z1|+ |z2|) ≤ c(p) ∣∣Vā(z1)− Vā(z2) ∣∣√ |z1|+ |z2| hā(|z1|+ |z2|) ≤ c(p) ∣∣Vā(z1)− Vā(z2) ∣∣√ |z2 − z1|+ 2|z1| hā(|z1|) ≤ c(p) ∣∣Vā(z1)− Vā(z2) ∣∣[√ |z1 − z2| hā(|z1|) + √ |z1| hā(|z1|) ] . Therefore, by (2.19) we obtain that |z1 − z2|hā(|z1|) ≤ c(p) ∣∣Vā(z1)− Vā(z2) ∣∣ [√hā(|z1|) |z1 − z2|+ √ Hā(z1) ] ≤ c(p) [∣∣Vā(z1)− Vā(z2) ∣∣√Hā(z1) + √ hā(|z1|) |z2| ] . EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 11 Using again monotonicity, (2.36), and subadditivity we can estimate√ hā(|z1|) |z2| ≤ √( |z1|+ |z2| ) hā(|z1|+ |z2|) ≤ c(p) (√ Hā(z1) + √ Hā(z2) ) and the proof of (2.37) is complete. � The following lemma will be useful to prove an excess-like decay estimate for our minimizer u, see Proposition 3.11. Lemma 2.5. For every a1, a2 ≥ 0 and every z ∈ R`, ` ∈ N the following estimate holds ∣∣Va1 (z)− Va2 (z) ∣∣ ≤ |a1 − a2||z|p/2 log(e+ |z|) . Proof. By the definition of the vector fields Va1 and Va2 , using the Lipschitz reg- ularity (with Lipschitz constant 1/2) of the function t ∈ [0,+∞) → √ 1 + t, we estimate |Va1(z)− Va2(z)| = ∣∣∣ √ ϕ′a1 (|z|) |z| z − √ ϕ′a2 (|z|) |z| z ∣∣∣ = |z|(p−1)/2|z|1/2 ∣∣∣√1 + a1 log(e+ |z|) + a1 p |z| e+ |z| − √ 1 + a2 log(e+ |z|) + a2 p |z| e+ |z| ∣∣∣ ≤ 1 2 |z|p/2|a1 − a2| [ log(e+ |z|) + 1 p |z| (e+ |z|) ] ≤ |z|p/2|a1 − a2| log(e+ |z|) . � We conclude this subsection by recalling the following approximation lemma (see [21, Lemma 3.2 and Remark 3.4]) which will be useful to specify the class of admissible test functions in the weak formulation of the Euler-Lagrange equation (Remark 3.3). Lemma 2.6. Let us consider a ball B = BR(x0) b Ω and φ ∈ W 1,p 0 (BR;RN ) a function such that H(x,Dφ) ∈ L1(BR), with H defined in (2.17) and the modulus of continuity ωa(·) of the function a(·) satisfying limR↘0+ ωa(R) log(1/R) < +∞. Then there exists a sequence {φk} ⊂ C∞0 (BR;RN ) such that Dφk → Dφ a.e., φk → φ strongly in W 1,p(BR;RN ) and H(x,Dφk)→ H(x,Dφ) strongly in L1(BR) . 2.5. Lp logγ L spaces. Given an open bounded set Ω ⊂ Rn and a Young function ϕ : [0,∞)→ [0,∞) (ϕ is convex, strictly monotone increasing, limt→0 ϕ(t) t = 0 and limt→∞ ϕ(t) t =∞) the Orlicz space Lϕ(Ω;R`), ` ∈ N is the set of measurable maps f : Ω → R` such that ∫ Ω ϕ(λ|f(x)|) dx < ∞ for some λ > 0. When ϕ(t) = tp/p, the previous quantity defines an averaged Lp norm; when ϕ(t) = (tp/p) logγ(e+ t), for p > 1, γ ∈ R or p = 1 and γ ≥ 0, the Orlicz space Lϕ(Ω;R`) is denoted by Lp logγ L(Ω;R`) and it consists of the measurable functions such that∫ Ω |f |p logγ(e+ |f |) dx <∞ . 12 P. BARONI, A. COSCIA EJDE-2022/80 In particular, we want to stress here that a classic inequality shows that for every q > 1 and f ∈ Lq(Ω;R`) we have − ∫ Ω |f | logγ ( e+ |f | (|f |)Ω ) dx ≤ c(n, `, γ, q) ( − ∫ Ω |f |q dx )1/q (2.38) (see [2, 3, 13]); being the quantity on the left-hand side equivalent to the Luxemburg norm in L logγ L, as a consequence of (2.38) we deduce that f ∈ Lq(Ω;R`) , q > p =⇒ f ∈ Lp logL(Ω;R`) . (2.39) 2.6. Basic estimates for minimizers of P. In this paragraph we describe a higher integrability result available for local minimizers of P that holds under the weak assumption lim sup R↘0+ ωa(R) log ( 1 R ) = lim sup R↘0+ ωlog(R) <∞ : (2.40) by Lemma 2.2 our assumption of log-Dini continuity on a(·) ensures the stronger lim R↘0+ ωa(R) log ( 1 R ) = 0 , (2.41) so we can assume that there exists a threshold R̄ > 0 such that sup R∈(0,R̄] ωa(R) log ( 1 R ) ≤ 1 . (2.42) As a consequence of (1.5) and (2.40) one has that local minimizers of P are higher integrable, that is H(·, Du) belongs to a Lebesgue space smaller than L1. The fact in (2.41) allows us to state such result in a slightly simpler form; in particular allows to get rid of the dependence of the constants on the energy and this will simplify later our approach. Theorem 2.7 (Gradient’s higher integrability). Let u ∈ W 1,p(Ω;RN ) be a local minimizer of the functional P defined in (1.4) and suppose that (1.5) and (2.41) hold. Then there exists a positive integrability exponent δ̄g > 0, depending only on n,N, p and L/ν such that H(·, Du) ∈ L1+δ̄g loc (Ω). Moreover if M ≥ 1 is such that M ≥ ‖Du‖Lp (2.43) then there exists a threshold R0 = min { 1 e+M , R̄ } (2.44) such that if r ≤ R0 and Br(x0) ⊂ Ω the reverse Hölder’s inequality( − ∫ Bϑr(x0) [H(x,Du)]1+δ dx ) 1 1+δ ≤ c− ∫ Br(x0) H(x,Du) dx (2.45) holds for every ϑ ∈ [1/2, 3/4] and every δ ∈ [0, δ̄g], for constant c depending only on n,N, p and L/ν. Proof. We outline the changes the proof in [4] requires, keeping the notation here employed. The starting point for showing (2.45) is the proof of a reverse Hölder inequality − ∫ Br/2(x̄) H(x,Du) dx ≤ c̃ ( − ∫ Br(x̄) [H(x,Du)]d dx )1/d (2.46) EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 13 for every ball Br(x̄) ⊂ Ω with r ≤ e−1 and with the exponent d ∈ (0, 1) depending on n,N, p, L/ν and the constant c̃ depending on n,N, p, L/ν, L̃ and ‖Du‖Lp , see (4.11) in [4]. However, the dependence of c̃ on L̃ can be avoided taking radii smaller than R̄ (see (2.42) and compare with [4, Equation (4.5)]) and the dependence on ‖Du‖Lp in [4] only comes from the estimate (4.13), in particular when estimating log ( e+ ‖Du‖pLp rn ) ≤ ( 1 + ‖Du‖pLp ) log ( e+ 1 rn ) ≤ 2n ( 1 + ‖Du‖pLp ) log (1 r ) using (2.2), also compare with [4, Remark 4.3]. If, on the other hand, we assume (2.44) we can estimate ‖Du‖Lp ≤ r−1 if r ≤ R0 and therefore log ( e+ ‖Du‖pLp rn ) ≤ log ( e+ 1 rn+p ) ≤ 2(n+ p) log (1 r ) ; the proof now continues as in [4] but with c̃ not anymore depending on L̃ and ‖Du‖Lp . In view of (2.46) a standard application of Gehring’s Lemma yields the result. Notice indeed that a simple scaling argument ensures that both the exponent and the constant in (2.45) do not depend on R0: we fix Br(x0) ⊂ Ω and we write (2.46) setting f(x) = H(x0 + rx,Du(x0 + rx)) as − ∫ Bρ/2(x̃) |f | dx ≤ c̃ ( − ∫ Bρ(x̃) |f |d dx )1/d for every x̃ ∈ B1(0) and ρ ≤ 1 such that Bρ(x̃) ⊂ B1(0); notice that now c̃ does not depend on ‖Du‖Lp . Using Gehring’s lemma (see for instance [4, Lemma 3.5]) gives( − ∫ B3/4(0) |f |1+δ̄g dx ) 1 1+δ̄g ≤ c− ∫ B1(0) |f | dx with δ̄g, c as in the statement, in particular not depending on R0; scaling back to H(·, Du) gives (2.45) when ϑ = 3/4. If ϑ < 3/4 the results follows simply enlarging the integral on the left-hand side: − ∫ Bϑr(x0) [H(x,Du)]1+δ dx ≤ c(n)− ∫ B3r/4(x0) [H(x,Du)]1+δ dx . � 2.7. Estimates for reference functionals. This paragraph concerns the frozen functional obtained by freezing the switching coefficients a(·), b(·) in P, defined in (1.4). In particular, for A ⊂ Rn bounded domain (that in our case will always be a ball inside Ω), we consider minimizers of functionals of the type Pā(w,A) := ∫ A [ |Dw|p + ā|Dw|p log(e+ |Dw|) ] dx = ∫ A Hā(Dw) dx , (2.47) where ā ≥ 0 is a constant. The result we want to recall is the following excess decay estimate, which encodes the local C1,α regularity of minimizers; it can be found in [32, Theorem 6.4]. Notice that the Hölder regularity Assumption 2.2 necessary in [32] is satisfied in our case, see [4, Section 6]. Theorem 2.8. Let v ∈ W 1,p(A;RN ) be a local minimizer of the functional Pā defined in (2.47) such that Hā(Dv) ∈ L1(A), and let Br ≡ Br(x0) ⊂ A. The 14 P. BARONI, A. COSCIA EJDE-2022/80 excess-decay estimate − ∫ B% ∣∣Vā(Dv)− ( Vā(Dv) ) Bρ ∣∣2 dx ≤ c(ρ r )2α −∫ Br ∣∣Vā(Dv)− ( Vā(Dv) ) Br ∣∣2 dx holds for every couple of concentric balls Bρ ⊂ Br, for a constant c ≥ 1 and an exponent α ∈ (0, 1) both depending only on n,N and p. 3. Comparison estimates and excess decay In this section u ∈W 1,p(Ω;RN ) will always be a local minimizer of the functional P defined in (1.4). We are going to define two more regular comparison maps and first deduce an integral comparison estimate; then we shall show how this does imply an excess-decay estimate with a correction term. We choose in (2.43)-(2.44) M = (L ν ‖H(·, Du)‖L1 )1/p and R0 = min { 1 e+M , R̄ } , (3.1) R̄ being defined in (2.42), and we will work on a ball BR ≡ BR(x0) such that B2R(x0) ⊂ Ω, with radius 0 < R ≤ R0/2. 3.1. Comparison lemma. In this paragraph we prove a comparison lemma (see Lemma 3.6), where we estimate the distance between a minimizer of P and a minimizer of a frozen functional obtained by considering the case in which the coefficients a(·) and b(·) are constant. Let us denote ā := inf B2R a(·) and bav := (b)BR/2(x0) = − ∫ BR/2(x0) b(x) dx ; (3.2) then, recalling (2.18), let v̄ ∈ W 1,p(BR;RN ) and v ∈ W 1,p(BR/2;RN ) be the solu- tions of the following Dirichlet problems v̄ 7→ min w ∫ BR b(x)Hā(Dw) dx , w ∈ u+W 1,p 0 (BR) (3.3) and v 7→ min w ∫ BR/2 bav Hā(Dw) dx , w ∈ v̄ +W 1,p 0 (BR/2). (3.4) Remark 3.1. Problems (3.3) and (3.4) are well-posed. Thanks to the definition (3.2) of ā and to the bounds (1.5) on b(·) we obtain∫ BR b(x)Hā(Du) dx ≤ L ∫ BR H(x,Du) dx <∞ ; (3.5) further, by the minimality of v̄ and (3.5) we obtain∫ BR/2 Hā(Dv̄) dx ≤ ∫ BR Hā(Dv̄) dx ≤ 1 ν ∫ BR b(x)Hā(Dv̄) dx ≤ 1 ν ∫ BR b(x)Hā(Du) dx ≤ L ν ∫ BR H(x,Du) dx <∞ , (3.6) thus the Direct Methods of the Calculus of Variations guarantees that problems (3.3) and (3.4) have a minimizer. Finally, the minimality of v and (3.6) imply that∫ BR/2 Hā(Dv) dx ≤ ∫ BR Hā(Dv̄) dx ≤ L ν ∫ BR H(x,Du) dx <∞ , (3.7) EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 15 so both energies of v and v̄ can be estimated by the energy of the minimizer u. Remark 3.2. Notice that the bound M defined in (3.1) guarantees that M ≥ ‖Du‖Lp , thus the higher integrability result of Theorem 2.7 together with the local estimate (2.45) is available. Moreover, by (3.6) we obtain also that M ≥ ‖Dv̄‖Lp(BR), thus the threshold R0 for which the local higher integrability esti- mate for Dv̄ holds can be made independent of ‖Dv̄‖Lp(BR) but depending on ‖H(·, Du)‖L1 . Remark 3.3. We can show that the Euler-Lagrange equations − ∫ BR/2 〈∂zHā(Dv), Dφ〉 dx = 0 , (3.8) − ∫ BR b(x)〈∂zHā(Dv̄), Dφ〉 dx = 0 , (3.9) − ∫ BR b(x)〈∂zH(x,Du), Dφ〉 dx = 0 (3.10) are valid for every φ ∈ W 1,p 0 (BR/2;RN ) with Hā(Dφ) ∈ L1(BR/2) for (3.8), for every φ ∈ W 1,p 0 (BR;RN ) with Hā(Dφ) ∈ L1(BR) for (3.9) and for every φ ∈ W 1,p 0 (BR;RN ) with H(·, Dφ) ∈ L1(BR) for (3.10). Let us prove (3.10) since with exactly the same a rguments we can prove (3.8) and (3.9). We argue by approximation since it is well known that the equation holds for every φ ∈ C∞c (BR;RN ). Let φ ∈ W 1,p 0 (BR;RN ) such that H(·, Dφ) ∈ L1(BR): by Lemma 2.6 there exists a sequence {φk} ⊂ C∞c (BR;RN ) such that Dφk → Dφ a.e. and H(·, Dφk)→ H(·, Dφ) strongly in L1(BR) . Using the analogous of (2.32) for the function H(·, ·), with z = Du and λ = Dφk, we estimate on BR,∣∣b(x)〈∂zH(x,Du), Dφk〉 ∣∣ ≤ Lc(p)(H(x,Du) +H(x,Dφk) ) and we can conclude the strong convergence in L1(BR) of b(·)〈∂zH(·, Du), Dφk(·)〉 → b(·)〈∂zH(·, Du), Dφ(·)〉 by a well-known variant of the Lebesgue dominated convergence theorem. There- fore, since every φk satisfies (3.10) also φ does. To prove the comparison Lemma 3.6, we need to test the Euler equation (3.10) with the function φ = u − v̄ ∈ W 1,p 0 (BR;RN ), so by (2.31) again for the function H(·, ·), we must show that H(·, Dv̄) ∈ L1(BR). Indeed, from Hā(Dv̄) ∈ L1(BR) we deduce that Dv̄ ∈ Lp logL(BR;RnN ) , thus ∫ BR b(x)H(x,Dv̄) dx <∞ . This is immediate if ā > 0, while if ā = 0 the functional reduces to the classic p-Dirichlet functional, apart from the coefficient b(·), and the result follows for instance from the Theorem 3.4 and (2.39), since v̄ ∈ u+W 1,p 0 (BR;RN ) with Du ∈ Lp(1+δ̄g)(BR;RnN ) by Theorem 2.7. For the solutions v̄ and v of the Dirichlet problems (3.3)-(3.4) the following up to the boundary higher integrability result holds; for the proof, see [33, Theorem B.1] 16 P. BARONI, A. COSCIA EJDE-2022/80 for the scalar case and [23, Lemma 4.3] for the vectorial one. Notice that actually the result in [33] is stronger; anyway, the version we quote here will be sufficient for our purposes. Theorem 3.4. Let Br(x0) ⊂ Ω be a ball, Hā(·) as in (2.18), w̄ ∈ W 1,p(Br;RN ) with Hā(Dw̄) ∈ L1(Br) and let w ∈W 1,p(Br;RN ) be the minimizer in the Dirichlet class w̄ +W 1,p 0 (Br;RN ) of the functional w 7→ ∫ Br b(x)Hā(Dw) dx with b(·) as in (1.5). Suppose moreover that Hā(Dw̄) ∈ L1+δ̄g (Br(x0)) for some δ̄g > 0. Then there exists δg ≡ δg(n,N, p, L/ν) ∈ (0, δ̄g) such that Hā(Dw) ∈ L1+δg (Br(x0)) and the estimate − ∫ Br(x0) [ Hā(Dw) ]1+δ dx ≤ c− ∫ Br(x0) [ Hā(Dw̄) ]1+δ dx (3.11) holds for a constant c ≡ c(n,N, p, L/ν) and for every δ ∈ [0, δg]. Remark 3.5. To simplify the proofs, we apply the interior estimate (2.45) to the three functions u (local minimizer of (1.4)), v̄ (solution of (3.3)) and v (solution of (3.4)) and the boundary estimate (3.11) to both v̄ and v with the same common value for δ: we choose δg appearing above in Theorem 3.4. Now we can state and prove our comparison lemma. Lemma 3.6 (Comparison). If v ∈W 1,p(BR/2;RN ) is the solution of the Dirichlet problem (3.4), then there exists an exponent q = q(n,N, p, L/ν) < n such that the inequality − ∫ BR/2 ∣∣Vā(Du)− Vā(Dv) ∣∣2 dx ≤ c [[ ωa(R) log ( 1 R )]2 + R2 ν2 ( − ∫ BR |Db|q dx )2/q] − ∫ B3R/2 H(x,Du) dx (3.12) holds for a constant c ≡ c(n,N, p, L/ν). Proof. The proof of the comparison lemma consists of two steps, by freezing the coefficients a(·) and b(·) one at a time: more precisely we consider the minimizer v̄ of the Dirichlet problem (3.3) and we prove two comparison estimates, the first one between u and v̄ and the second one between v̄ and v. Step 1. Let v̄ ∈ W 1,p(BR,RN ) be the minimizer of the Dirichlet problem (3.3): the following comparison estimate between u and v̄ − ∫ BR ∣∣Vā(Du)− Vā(Dv̄) ∣∣2 dx ≤ c[ωa(R) log ( 1 R )]2 −∫ B3R/2 H(x,Du) dx (3.13) holds for a constant c = c(n,N, p, L/ν). First of all, let us show the following estimate, of which we will make frequent use in the sequel. Let δg = δg(n,N, p, L/ν) ∈ (0, 1) be the higher integrability exponent of Theorem 2.7-Remark 3.5 applied to u; by Lemma 2.1, applied with EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 17 f = |Du|p, σ = 1/p, β = 2, θ = p, s = 1 + δg, the reverse Hölder’s inequality (2.45) with θ = 2/3 and r = 3R/2, (3.1) together with Remark 3.2 we can estimate − ∫ BR |Du|p log2(e+ |Du|) dx ≤ c(n, p, δg) ( 1 +Rp‖Du‖pLp(Ω) )2 log2 ( 1 R ) ( − ∫ BR |Du|p(1+δg) dx )1/(1+δg) ≤ c ( 1 +Rp0‖Du‖ p Lp(Ω) )2 log2 ( 1 R ) ( − ∫ BR H(x,Du)1+δg dx )1/(1+δg) ≤ c log2 ( 1 R ) − ∫ B3R/2 H(x,Du) dx . (3.14) for c ultimately depending only on n,N, p and L/ν. Since both u and v̄ are min- imizers, we can use the corresponding Euler-Lagrange equations (3.10) and (3.9). By Remark 3.3 we can test with φ = u− v̄ ∈W 1,p 0 (BR;RN ): − ∫ BR b(x)〈∂zH(x,Du), Du−Dv̄〉 dx−− ∫ BR b(x)〈∂zHā(Dv̄), Du−Dv̄〉 dx = 0 . (3.15) Using (2.32), with z = Du(x) and λ = Du(x)−Dv̄(x), and (2.31) we may estimate∣∣〈∂zHā(Du), Du−Dv̄〉 ∣∣ ≤ c(p)(Hā(Du) +Hā(Dv̄) ) , (3.16) thus in (3.15) we can add and substract the integral − ∫ BR b(x)〈∂zHā(Du), Du−Dv̄〉 dx , which is finite by (3.16), (3.5) and (3.7), obtaining D1 := − ∫ BR b(x)〈∂zHā(Du)− ∂zHā(Dv̄), Du−Dv̄〉 dx = − ∫ BR b(x)〈∂zHā(Du)− ∂zH(x,Du), Du−Dv̄〉 dx := D2 . By applying (2.33) with z1 = Du(x) and z2 = Dv̄(x), and again (1.5), we can estimate D1 from below obtaining that ν c(p) − ∫ BR |Vā(Du)− Vā(Dv̄)|2 dx ≤ D1 = |D2| . (3.17) Using Cauchy-Schwarz inequality, (1.5), (2.18), (2.17), and (2.12) we have the esti- mate |D2| ≤ L− ∫ BR |∂zHā(Du)− ∂zH(x,Du)| |Du−Dv̄| dx = pL− ∫ BR |a(x)− ā|ϕ′log(|Du|) |Du−Dv̄| dx ≤ c(p)L− ∫ BR |a(x)− ā| |Du|p−1 log(e+ |Du|) |Du−Dv̄| dx = c(p)L− ∫ BR |a(x)− ā| |Du| p−2 2 + p 2 log(e+ |Du|)|Du−Dv̄| dx = I1 . (3.18) Now we need to distinguish two cases. 18 P. BARONI, A. COSCIA EJDE-2022/80 Case p ≥ 2. In this case, using Young’s inequality with conjugate exponents (2, 2) and ε ∈ (0, 1) to be chosen, we can estimate I1 ≤ c(p)Lε− ∫ BR |Du|p−2|Du−Dv̄|2 dx + c(p)L 4ε − ∫ BR |a(x)− ā|2|Du|p log2(e+ |Du|) dx = I2 + I3 . (3.19) As p ≥ 2, by (2.23) and (2.25) we have |Du|p−2|Du−Dv̄|2 ≤ ( |Du|+ |Dv̄| )p−2|Du−Dv̄|2 ≤ c(p) ∣∣Vp(Du)− Vp(Dv̄) ∣∣2 ≤ c(p)∣∣Vā(Du)− Vā(Dv̄) ∣∣2 , so the term I2 can be estimated by I2 ≤ c(p)Lε− ∫ BR ∣∣Vā(Du)− Vā(Dv̄) ∣∣2 dx and it can be reabsorbed in the left-hand side of (3.17) for ε sufficiently small depending only on p and L/ν. Since, using monotonicity and concavity of ωa(·), it is easy to prove that |a(x)− ā| ≤ 3ωa(R) , (3.20) by (3.14) we have the estimate I3 ≤ c(p) ε L [ ωa(R) ]2 −∫ BR |Du|p log2(e+ |Du|) dx ≤ c(n,N, p, L/ν)L [ ωa(R) log ( 1 R )]2 −∫ B3R/2 H(x,Du) dx . Case p < 2. In this case to estimate the integral in (3.18) we use (2.24) to obtain I1 = c(p)L− ∫ BR |a(x)− ā| |Du|p−1 log(e+ |Du|) |Du−Dv̄| dx ≤ c(p)L [ − ∫ BR |a(x)− ā| |Du|p−1 log(e+ |Du|)|Vp(Du)− Vp(Dv̄)|2/p dx +− ∫ BR |a(x)− ā| |Du|p/2 log(e+ |Du|)|Vp(Du)− Vp(Dv̄)| dx ] . Estimating both integrals with Young’s inequality, the first with conjugate expo- nents (p, p′), the second with (2, 2), both with ε ∈ (0, 1) to be chosen, and using (2.25) we obtain I1 ≤ 2c(p)Lε− ∫ BR ∣∣Vā(Du)− Vā(Dv̄) ∣∣2 dx + c(p) ε L− ∫ BR |a(x)− ā|p ′ |Du|p logp ′ (e+ |Du|) dx + c(p) ε L− ∫ BR |a(x)− ā|2|Du|p log2(e+ |Du|) dx =: I4 + I5 + I6 . The term I4 can be reabsorbed in the left-hand side of (3.17) for ε sufficiently small depending only on p and L/ν, the term I6 is estimated exactly as the term I3 in the case p ≥ 2 (note that the value of p is irrelevant). To estimate the remaining term I5, we follow the reasoning in (3.14), using again (3.20), Lemma 2.1 with EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 19 f = |Du|p, σ = 1/p, β = p′, θ = p, s = 1 + δg (δg from Remark 3.5), assumption (2.42) together with the fact that p′ > 2, and the reverse Hölder’s inequality (2.45) with θ = 2/3 and r = 3R/2, obtaining that I5 ≤ c(p) ε L [ ωa(R) ]p′ −∫ BR |Du|p logp ′ (e+ |Du|) dx ≤ c(n,N, p, L/ν)L [ ωa(R) log ( 1 R )]p′(−∫ BR |Du|p(1+δg) dx ) 1 1+δg ≤ c(n,N, p, L/ν)L [ ωa(R) log ( 1 R )]2 −∫ B3R/2 H(x,Du) dx . We conclude that in both cases (1 < p < 2 and p ≥ 2) the comparison estimate (3.13) holds for a constant c ≡ c(n,N, p, L/ν). Step 2. There exists an exponent q = q(n,N, p, L/ν) < n such that the following comparison estimate between v̄ and v holds: − ∫ BR/2 |Vā(Dv̄)−Vā(Dv)|2 dx ≤ cR 2 ν2 ( − ∫ BR |Db|q dx )2/q − ∫ B3R/2 H(x,Du) dx , (3.21) for a constant c depending on n,N, p and L/ν. Since both v̄ and v are minimizers, we can use the corresponding Euler-Lagrange equations (3.9) and (3.8). We can test with φ = v̄−v ∈W 1,p 0 (BR/2;RN ) (extended to 0 on BR \BR/2) as Hā(Dφ) ∈ L1(BR/2) by (3.7): − ∫ BR/2 b(x)〈∂zHā(Dv̄), Dv̄−Dv〉 dx−− ∫ BR/2 bav〈∂zHā(Dv), Dv̄−Dv〉 dx = 0 . (3.22) Using (2.32), with z = Dv̄(x) and λ = Dv̄(x)−Dv(x), and (2.31) we may estimate |〈∂zHā(Dv̄), Dv̄ −Dv〉| ≤ c(p) ( Hā(Dv̄) +Hā(Dv) ) , (3.23) thus in (3.22) we can add and substract the integral − ∫ BR/2 bav〈∂zHā(Dv̄), Dv̄ −Dv〉 dx , which is finite by (3.23) and (3.7), obtaining D1 := − ∫ BR/2 bav〈∂zHā(Dv̄)− ∂zHā(Dv), Dv̄ −Dv〉 dx = − ∫ BR/2 ( bav − b(x) ) 〈∂zHā(Dv̄), Dv̄ −Dv〉 dx := D2 . By applying (2.33) with z1 = Dv̄(x) and z2 = Dv(x), and noticing that ν ≤ bav ≤ L, we can estimate D1 from below obtaining that ν c(p) − ∫ BR/2 ∣∣Vā(Dv̄)− Vā(Dv) ∣∣2 dx ≤ D1 = |D2| . (3.24) 20 P. BARONI, A. COSCIA EJDE-2022/80 By Cauchy-Schwarz inequality, (2.34) with z = Dv̄(x), and (2.37) with z1 = Dv̄(x) and z2 = Dv(x), we can estimate |D2| as |D2| ≤ − ∫ BR/2 |b(x)− bav||∂zHā(Dv̄)| |Dv̄ −Dv| dx ≤ c(p)− ∫ BR/2 |b(x)− bav|hā(|Dv̄|)|Dv̄ −Dv| dx ≤ c(p)− ∫ BR/2 |b(x)− bav| ∣∣Vā(Dv̄)− Vā(Dv) ∣∣√Hā ( Dv̄ ) dx + c(p)− ∫ BR/2 |b(x)− bav| ∣∣Vā(Dv̄)− Vā(Dv) ∣∣√Hā ( Dv ) dx := I1 + I2 . (3.25) Using Young’s inequality with conjugate exponents (2, 2) and ε ∈ (0, 1) to be chosen, we can estimate I1 ≤ c(p) ε− ∫ BR/2 ∣∣Vā(Dv̄)− Vā(Dv) ∣∣2 dx + c(p) 4ε − ∫ BR/2 |b(x)− bav|2Hā(Dv̄) dx = I3 + I4 , (3.26) where the term I3 can be reabsorbed in the left-hand side of (3.24) for ε sufficiently small depending only on p, ν. To estimate I4 in (3.26), let δg ∈ (0, 1) be the higher integrability exponent from Theorem 2.7-Remarks 3.2 & 3.5 which holds for both v̄ and v, depending on n,N, p and L/ν, and let us choose q = q(n, δg) < n such that q∗ = nq n− q = 2 ( 1 + 1 δg ) ⇐⇒ q = q∗ n+ q∗ n = 2(1 + δg) nδg + 2(1 + δg) n . (3.27) Now, by applying first Hölder’s inequality with conjugate exponents (1 + 1/δg, 1 + δg), then the Sobolev-Poincaré and the reverse Hölder’s (2.45) inequalities with θ = 1/2 and r = R, and finally (3.7), we obtain I4 ≤ c(p) ε ( − ∫ BR/2 |b(x)− bav| 2(1+ 1 δg ) dx ) δg 1+δg ( − ∫ BR/2 [ Hā(Dv̄) ]1+δg dx ) 1 1+δg = c(p) ν [( − ∫ BR/2 |b(x)− bav|q ∗ dx )1/q∗]2( − ∫ BR/2 [ Hā(Dv̄) ]1+δg dx ) 1 1+δg ≤ c(n,N, p, L/ν) ν [ R ( − ∫ BR/2 |Db|q dx )1/q]2 − ∫ BR Hā(Dv̄) dx ≤ c(n,N, p, L/ν) ν R2 ( − ∫ BR |Db|q dx )2/q − ∫ B3R/2 H(x,Du) dx . (3.28) It remains to estimate the term I2 in (3.25): arguing as in (3.26) we obtain first I2 ≤ c(p) ε− ∫ BR/2 |Vā(Dv̄)− Vā(Dv)|2 dx+ c(p) 4ε − ∫ BR/2 |b(x)− bav|2Hā(Dv) dx = I5 + I6 , EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 21 with the term I5 which can be reabsorbed in the left-hand side of (3.24) for ε sufficiently small depending only on p, ν. Then, with the same exponent q defined in (3.27) and the same computations in (3.28), using (3.11) we can estimate the term I6 as I6 ≤ c R2 ν ( − ∫ BR |Db|q dx )2/q − ∫ B3R/2 H(x,Du) dx , for a constant c ≡ c(n,N, p, L/ν). Putting together all the estimates into (3.24) and (3.25), we obtain (3.21). From (3.13) and (3.21), we deduce immediately the comparison estimate (3.12). � Next, an inequality allowing to replace the energy on the right-hand side with a quantity more appropriate for the forthcoming iteration proof. Proposition 3.7. There exists a constant c ≡ c(n,N, p, L/ν) such that − ∫ B3R/2 H(x,Du) dx ≤ c− ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx holds for every ball B2R ≡ B2R(x0) ⊂ Ω with R smaller than R0/2. Proof. The self-improving character of reverse-Hölder inequality yields that, as a consequence of the higher integrability estimate (2.45), for every σ > 0 it holds − ∫ B3R/2 H(x,Du) dx ≤ c(n,N, p, L/ν, σ) ( − ∫ B2R [ H(x,Du) ]σ dx )1/σ . (3.29) Observing that for every x0 ∈ Ω the function H(x0, Du(·)) ∈ L1(B2R) due to the higher integrability result of Theorem 2.7 and (2.39), we choose σ = 1/2 and we use sub-additivity to estimate the right-hand side: − ∫ B2R [ H(x,Du) ]σ dx ≤ − ∫ B2R [ H(x0, Du) ]1/2 dx+− ∫ B2R ∣∣a(x)− a(x0) ∣∣1/2|Du|p/2 log1/2(e+ |Du|) dx . Since the first integral is bounded by ( − ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx)1/2 , using Hölder’s inequality and (2.22), we focus on the second one. To estimate it, we use first Lemma 2.1 with f = |Du|p/2, σ = 2/p, β = 1/2, θ = p, s = 2, then (2.42) and the fact that R ≤ R0 < 1, and finally (2.22): − ∫ B2R ∣∣a(x)− a(x0) ∣∣1/2|Du|p/2 log1/2(e+ |Du|) dx ≤ [ ωa(2R) ]1/2 −∫ B2R |Du|p/2 log1/2(e+ |Du|) dx ≤ c(n, p) [ ωa(R) log ( 1 2R )]1/2[ 1 +Rp‖Du‖p/2 Lp/2(B2R) )1/2(−∫ B2R |Du|p dx )1/2 ≤ c [ ωa(R) log ( 1 R )]1/2[ 1 +Rp ∫ B2R (1 + |Du|p) dx ]1/2( − ∫ B2R |Du|p dx )1/2 ≤ c [ ωa(R) log ( 1 R )]1/2[ 1 + c(n)Rn+p 0 +Rp0‖Du‖ p Lp(Ω) ]1/2(−∫ B2R |Du|p dx )1/2 22 P. BARONI, A. COSCIA EJDE-2022/80 ≤ c ( − ∫ B2R H(x0, Du) dx )1/2 ≤ c(n, p) ( − ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx)1/2 . The proof is concluded in view of (3.29) with σ = 1/2. � Remark 3.8. Note that we can prove in a similar way that if B4R(x0) ⊂ Ω has radius R ≤ R0/4, then − ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx ≤ c(n,N, p, L/ν) − ∫ B4R H(x,Du) dx . Indeed, using (2.22), Lemma 2.5, (3.14) from B2R to B4R (in the higher integrability estimate (2.45) choose θ = 1/2 and r = 4R), the concavity of ωa(·) together with the monotonicity of the logarithm function, and (2.42) we have − ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx ≤ 2− ∫ B2R ∣∣Va(x)(Du) ∣∣2 dx+ 2− ∫ B2R ∣∣Va(x)(Du)− Va(x0)(Du) ∣∣2 dx ≤ 4− ∫ B2R H(x,Du) dx+ 2− ∫ B2R |a(x)− a(x0)|2|Du|p log2(e+ |Du|) dx ≤ c(n)− ∫ B4R H(x,Du) dx+ c(n,N, p, L/ν) [ ωa(R) log ( 1 R )]2 − ∫ B4R H(x,Du) dx . 3.2. Excess decay estimate. From now on, let us denote for radii r ≤ R and for the exponent q ∈ (1, n) defined in Lemma 3.6, Br,q ≡ Br,q(x0) := r ν ( − ∫ Br(x0) |Db|q dx )1/q and we recall that ωlog has been defined in (1.6). Also in this paragraph the center of all the balls will be x0 and therefore we shall omit it. The forthcoming Lemma is a preliminary decay estimate for the L2-excess of a certain nonlinear function of the gradient. The function ξ 7→ Vā(ξ) with ā as in (3.2) reflects the growth of the comparison problems but it is not appropriate for the iteration procedures we are going to perform; it will be replaced later in order to get the final excess-decay estimate (3.31). Lemma 3.9. There exist an exponent α ∈ (0, 1) depending on n,N and p, and a constant c ≡ c(n,N, p, L/ν), such that for every pair of concentric balls Bρ ≡ Bρ(x0) ⊂ B2R ≡ B2R(x0) ⊂ Ω with R ≤ R0/2, it holds − ∫ Bρ ∣∣Vā(Du)− ( Vā(Du) ) Bρ ∣∣2 dx ≤ c ( ρ R )2α −∫ B2R ∣∣Vā(Du)− ( Vā(Du) ) B2R ∣∣2 dx + c [(R ρ )n + ( ρ R )2α][ ω2 log(R) + B2 R,q ] − ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx . (3.30) Proof. It suffices to prove the lemma for 0 < ρ ≤ R/2, indeed for R/2 < ρ ≤ 2R the estimate follows immediately using (2.1), enlarging the integral, and observing that 2(ρ/R) ≥ 1. Now, noticing that the function v is also a local minimizer of EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 23 the functional Pā defined in (2.47) on A = BR/2, thus Theorem 2.8 applies to the minimizer v, and using also (2.1), we can estimate − ∫ Bρ ∣∣Vā(Du)− ( Vā(Du) ) Bρ ∣∣2 dx ≤ 8 [ − ∫ Bρ ∣∣Vā(Du)− Vā(Dv) ∣∣2 dx+− ∫ Bρ ∣∣Vā(Dv)− ( Vā(Dv) ) Bρ ∣∣2 dx] ≤ c(n,N, p) [(R ρ )n −∫ BR/2 ∣∣Vā(Du)− Vā(Dv) ∣∣2 dx + ( ρ R )2α −∫ BR/2 ∣∣Vā(Dv)− ( Vā(Dv) ) BR/2 ∣∣2 dx] . As − ∫ BR/2 ∣∣Vā(Dv)− ( Vā(Dv) ) BR/2 ∣∣2 dx ≤ c(n) [ − ∫ BR/2 ∣∣Vā(Du)− Vā(Dv) ∣∣2 dx+− ∫ B2R ∣∣Vā(Du)− ( Vā(Du) ) B2R ∣∣2 dx] , by the comparison Lemma 3.6 and Proposition 3.7 we conclude that − ∫ Bρ ∣∣Vā(Du)− ( Vā(Du) ) Bρ ∣∣2 dx ≤ c(n,N, p) [(R ρ )n + ( ρ R )2α]−∫ BR/2 ∣∣Vā(Du)− Vā(Dv) ∣∣2 dx + c(n,N, p) ( ρ R )2α −∫ B2R ∣∣Vā(Du)− ( Vā(Du) ) B2R ∣∣2 dx ≤ c ( ρ R )2α −∫ B2R ∣∣Vā(Du)− ( Vā(Du) ) B2R ∣∣2 dx + c [(R ρ )n + ( ρ R )2α][ ω2 log(R) + B2 R,q ] − ∫ B2R |Va(x0)(Du)|2 dx with c ≡ c(n,N, p, L/ν). � The next Lemma is necessary to perform the final iteration more smoothly; in particular we need to uniformize the nonlinear expression of the gradient appearing in the left- and right-hand sides of (3.30), replacing Vā(·) with Va(x0)(·). Lemma 3.10. There exists a constant c ≡ c(n,N, p, L/ν) such that for every pair of concentric balls Bρ ≡ Bρ(x0) ⊂ B2R ≡ B2R(x0) ⊂ Ω with R ≤ R0/2, it holds − ∫ Bρ ∣∣Vā(Du)− Va(x0)(Du) ∣∣2 dx ≤ c(R ρ )n[ ωlog(R) ]2 −∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx , where ā is the infimum of the continuous function a(·) on B2R as defined in (3.2). Proof. Using Lemma 2.5, with a1 = ā and a2 = a(x0), the fact that |ā − a(x0)| ≤ 2ωa(R), (3.14) and Proposition 3.7, we can estimate − ∫ Bρ ∣∣Vā(Du)− Va(x0)(Du) ∣∣2 dx ≤ (R ρ )n −∫ BR |ā− a(x0)|2|Du|p log2(e+ |Du|) dx 24 P. BARONI, A. COSCIA EJDE-2022/80 ≤ c(n,N, p, L/ν) (R ρ )n[ ωlog(R) ]2 −∫ B2R |Va(x0)(Du)|2 dx . � Finally, we have the decay estimate. Proposition 3.11. There exist an exponent α ∈ (0, 1) depending on n,N and p, and a constant c̄ ≡ c̄(n,N, p, L/ν), such that for every pair of concentric balls Bρ ≡ Bρ(x0) ⊂ B2R ≡ B2R(x0) ⊂ Ω with R ≤ R0/2, it holds − ∫ Bρ ∣∣Va(x0)(Du)− ( Va(x0)(Du) ) Bρ ∣∣2 dx ≤ c̄ ( ρ R )2α −∫ B2R ∣∣Va(x0)(Du)− ( Va(x0)(Du) ) B2R ∣∣2 dx + c̄ [(R ρ )n + ( ρ R )2α][ ω2 log(2R) + B2 2R,q ] − ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx . (3.31) Proof. Putting together Lemmata 3.10 and 3.9, we obtain for a constant c depend- ing on n,N, p and L/ν: − ∫ Bρ ∣∣Va(x0)(Du)− ( Va(x0)(Du) ) Bρ ∣∣2 dx ≤ c ( ρ R )2α −∫ B2R ∣∣Vā(Du)− ( Vā(Du) ) B2R ∣∣2 dx + c [(R ρ )n + ( ρ R )2α][ ω2 log(R) + B2 R,q ] − ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx . Using again Lemma 3.10 we estimate − ∫ B2R ∣∣Vā(Du)− ( Vā(Du) ) B2R ∣∣2 ≤ 8 [ − ∫ B2R ∣∣Vā(Du)− Va(x0)(Du) ∣∣2 dx+− ∫ B2R ∣∣Va(x0)(Du)− ( Va(x0)(Du) ) B2R ∣∣2 dx] ≤ c [ − ∫ B2R ∣∣Va(x0)(Du)− ( Va(x0)(Du) ) B2R ∣∣2 dx+ ω2 log(R)− ∫ B2R ∣∣Va(x0)(Du) ∣∣2 dx] for a constant c ≡ c(n,N, p, L/ν); the conclusion follows from BR,q ≤ c(n, q)B2R,q and ωlog(R) ≤ ωlog(2R), with the second one obtained by the monotonicity of ωa(·) and the inequality log ( 1 R ) ≤ 4 log ( 1 2R ) which holds for every R ≤ 1/e. � 4. Iteration and conclusion Once having at hand the excess decay estimate of Proposition 3.11, the conclu- sion is quite standard (see [38, 2, 3, 40] for instance). We sketch the proof for the reader’s convenience. We take x0 ∈ Ω and a radius R such that B2R(x0) ⊂ Ω and 2R is smaller than the threshold R0 as defined in (3.1); we shall further reduce the value of R0. We define the sequence of radii and corresponding balls R̃j = 2Rδj , `Bj = `Bj(x0) = B`R̃j (x0) , ` > 0, j ∈ N0 , (4.1) for δ ∈ (0, 1) that will be chosen later. The fundamental result all the forthcoming proofs are based upon is the following, whose proof can be found in [38, Lemma 1] or [3]. EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 25 Lemma 4.1. Let Ω ⊂ Rn and f ∈ Ln,1loc (Ω;Rn); let moreover δ ∈ (0, 1) and q ∈ (1, n) be fixed. For every K b Ω and ε > 0, there exists a radius Rε > 0 depending on n, q, δ, |f(·)| and ε such that if R ∈ (0, Rε] and R < dist(∂Ω,K), then sup x∈K ∞∑ j=0 R̃j ( − ∫ Bj(x) |f |q dy )1/q ≤ ε with R̃j and Bj(x) as in (4.1) (with x replacing x0). The analogous result regarding ωlog is based on a simple computation (see [13, Eq. (4.6)]) and ensures ∞∑ j=0 ωlog(R̃j) ≤ δ−1 ∫ 2R0 0 ωlog(ρ) dρ ρ ; (4.2) note that the right-hand side of the previous inequality tends to zero as R0 ↘ 0+. We immediately choose δ ≡ δ(n,N, p, L/ν) ∈ (0, 1/4) as the constant satisfying√ c̄ (2δ)α = 1/4, with c̄ and α the constants from Proposition 3.11. We then define the radii and the balls as in (4.1), set for j ∈ N0 ωj = ωlog(R̃j), Bj = BR̃j ,q and aj := ∣∣∣−∫ Bj Va(x0)(Du) dx ∣∣∣, Ej := ( − ∫ Bj ∣∣Va(x0)(Du)− ( Va(x0)(Du) ) Bj ∣∣2 dx)1/2 . (4.3) At this point we have for every j ∈ N0, using (3.31) with 2R = R̃j , ρ = R̃j+1, Ej+1 ≤ 1 4 Ej + c̃ [ ωj + Bj ]( − ∫ Bj |Va(x0)(Du)|2 dx )1/2 ≤ 1 4 Ej + c̃ [ ωj + Bj ] Ej + c̃ [ ωj + Bj ] ∣∣∣−∫ Bj Va(x0)(Du) dx ∣∣∣ with the constant c̃ depending only on n,N, p and L/ν; in the last line we used triangle’s inequality. Now we reduce, in view of (2.41) and Lemma 4.1, the value of R0 so that sup R≤R0 ωlog(R) + sup R≤R0 BR,q ≤ 1 4c̃ =⇒ c̃ [ ωj + Bj ] ≤ 1 4 ; R0 at this point depends also on ωa(·) and |Db(·)|/ν; this yields Ej+1 ≤ 1 2 Ej + c̃ [ ωj + Bj ] aj and in turn, summing for j ∈ {`, . . . , k} with `, k ∈ N0, ` ≤ k fixed, k+1∑ j=`+1 Ej ≤ 1 2 k∑ j=` Ej+ c̃ k∑ j=` [ ωj+Bj ] aj ⇒ k+1∑ j=` Ej ≤ 2E`+2c̃ k∑ j=` [ ωj+Bj ] aj (4.4) for c̃ depending on n,N, p and L/ν. 26 P. BARONI, A. COSCIA EJDE-2022/80 4.1. Gradient boundedness by induction. In this paragraph we suppose that x0 ∈ Ω is a Lebesgue point for Du and we finally are in the position to prove by induction that aj = ∣∣∣−∫ Bj Va(x0)(Du) dx ∣∣∣ ≤ 12δ−n ( − ∫ B2R(x0) ∣∣Va(x0)(Du) ∣∣2 dx)1/2 = λ0 for all j ∈ N0, with δ defined just after (4.2): thanks to the choice of x0, this leads to (1.10), using also Remark 3.8, after renaming R. The base case j = 0 is trivial by Hölder’s inequality; notice also that a0 + E0 ≤ 3 ( − ∫ B2R(x0) ∣∣Va(x0)(Du) ∣∣2 dx)1/2 ≤ δn 4 λ0 . Suppose now that aj ≤ λ0 holds for all j ∈ {0, 1, . . . , k} for some k ∈ N0 and further reduce R0, in a way depending on n,N, p, L/ν, ωa(·) and |Db(·)|/ν, so that ∞∑ j=0 [ ωj + Bj ] ≤ δn 8c̃ , this being possible thanks to Lemma 4.1 and (4.2). Since aj+1 − aj = ∣∣∣−∫ Bj+1 Va(x0)(Du) dx ∣∣∣− ∣∣∣−∫ Bj Va(x0)(Du) dx ∣∣∣ ≤ ∣∣∣−∫ Bj+1 Va(x0)(Du) dx−− ∫ Bj Va(x0)(Du) dx ∣∣∣ ≤ δ−n − ∫ Bj ∣∣Va(x0)(Du)− ( Va(x0)(Du) ) Bj ∣∣ dx ≤ δ−nEj , we have by telescopic summation ak+1 = a0 + k∑ j=0 [ aj+1 − aj ] ≤ a0 + δ−n k∑ j=0 Ej and using (4.4) for ` = 0 and our inductive assumption ak+1 ≤ a0 + 2δ−nE0 + 2c̃δ−n ∞∑ j=0 [ ωj + Bj ] λ0 ≤ λ0 4 + λ0 2 + λ0 4 = λ0 ; the proof of boundedness is complete. 4.2. Quantitative locally uniform VMO-type estimate. To be able to prove the gradient continuity in the next paragraph, we need as intermediate step a qualitative result of VMO-type. From the result of the previous paragraph, we know that the gradient is locally bounded in Ω and therefore, for K b Ω we fix an intermediate compact set K b K̃ b Ω such that dist(K, ∂K̃) = dist(K, ∂Ω)/2 and we set λ2 1 := ‖H(·, Du)‖L∞(K̃) . We prove here that for every ε > 0, there exists a radius Rε, depending on n,N, p, L/ν, ωa(·), |Db(·)|/ν, ‖H(·, Du)‖L1 and ε, such that if Rε ≤ dist(K, ∂Ω)/4 EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 27 then sup R∈(0,Rε] sup x∈K ( − ∫ BR(x) ∣∣Va(x)(Du)− ( Va(x)(Du) ) BR(x) ∣∣2 dy)1/2 ≤ ε λ1 . (4.5) We fix δ ≡ δ(n,N, p, L/ν, ε) such that 2 √ c̄ (2δ)α = ε/2, where c̄ and α are the constants from Proposition 3.11, and we define, for a starting radius R ≤ R1/2, with R1 ≤ R0/2 a threshold to be chosen appropriately, R̃j , Bj(x) as in (4.1) and Ej(x) the excess over Bj(x) as in (4.3). Using Proposition 3.11 and Remark 3.8 we have for x ∈ K, if B4R(x) ⊂ K̃ Ej+1(x) ≤ √ c̄ (2δ)αEj(x) + √ c̄2(2δ)−n/2 [ ωlog(R̃j) + BR̃j ,q ]( − ∫ 2Bj(x) H(y,Du) dy )1/2 ≤ √ c̄ [ 2(2δ)α + 2(2δ)−n/2 [ ωlog(R̃j) + BR̃j ,q ]] λ1 ≤ ε 2 λ1 + √ c̄ [ δ−n/2 [ ωlog(R̃j) + BR̃j ,q ]] λ1 for all j ∈ N0, due to our choice of δ. Now we can reduce the value of R1 so that supR∈(0,R1] ωlog(R) ≤ δ n2 ε/[4 √ c̄ ] (and this is possible in view of (2.41)) and so that supR∈(0,R1] BR,q ≤ δ n 2 ε/[4 √ c̄] (and this is possible in view of Lemma 4.1, uniformly in K) so that supx∈K Ej(x) ≤ ελ1 for all j ∈ N. We conclude the proof noticing that if we take Rε = δR1, then for every R ≤ Rε there exists a radius r ∈ (δR1, R1] such that R = δjr for some j ∈ N and thus Bj(x) = Bδjr(x) = BR(x); this concludes the proof of (4.5). 4.3. Gradient continuity by locally uniform convergence. Here we prove that the gradient Du is continuous in Ω by taking a generic but fixed compact set K b Ω and proving that Du is continuous in K; to do that, we show that the family of continuous maps MR : x ∈ K 7→ − ∫ BR(x) Va(x) ( Du(y) ) dy, R ≤ 1 4 dist(K, ∂Ω) , satisfy the Cauchy’s criterion uniformly in K. Since their limit coincides almost ev- erywhere with Va(·)(Du), the continuity of Va(·)(Du) is hence proven. Gradient con- tinuity follows easily thanks to triangle’s inequality, the estimate in Lemma 2.5 and the fact that Du is bounded, see [3, Last section]. We take K̃, λ1 as in the previous Paragraph 4.2 and we show that for every ε > 0 there exists a radius R2 ≤ R0/4 and a constant δ ∈ (0, 1), the first depending on n,N, p, L/ν, ωa(·), |Db(·)|/ν, dist(K, ∂Ω), ‖H(·, Du)‖L1 and ε, the latter on n,N, p and L/ν such that, setting again as in the previous Paragraph R̃j = δj(2R2), ωj = ωlog(R̃j),Bj(x) = BR̃j ,q(x), it holds Bj ⊂ B4R2 (x) ⊂ K̃ for all x ∈ K and sup x∈K ∣∣∣(Va(x)(Du) ) Bk(x) − ( Va(x)(Du) ) B`(x) ∣∣∣ < ελ1 for all 1 ≤ ` < k ; (4.6) to see how this does lead to the conclusion we refer for instance to [2, Paragraph 4.2], [13, After Step 3 in the proof of Theorem 4.3], [38, After eq. (124)], [39, After eq. (179)]. (4.6) indeed allows to prove that for any ε > 0 there exists a 28 P. BARONI, A. COSCIA EJDE-2022/80 threshold Rε, depending on n,N, p, L/ν, ωa(·), |Db(·)|/ν, dist(K, ∂Ω), ‖H(·, Du)‖L1 and ε such that sup x∈K ∣∣∣(Va(x)(Du) ) Br1 (x) − ( Va(x)(Du) ) Br2 (x) ∣∣∣ < ελ1 for all 0 < r1 ≤ r2 ≤ Rε . To prove (4.6) we define δ ≡ δ(n,N, p, L/ν) ∈ (0, 1/4) as after (4.1) and we notice that from (4.4) and for x ∈ K,∣∣(Va(x)(Du) ) Bk(x) − ( Va(x)(Du) ) B`(x) ∣∣ ≤ δ−n k−1∑ j=` Ej(x) ≤ 2δ−nE`(x) + 2c̃δ−n ∞∑ j=0 [ ωj + Bj(x) ] λ1 since now aj ≤ λ1 for all j ∈ N0. ForR2 sufficiently small, depending on n,N, p, L/ν, ωa(·), |Db(·)|/ν and ε, the second term is smaller than λ1ε/2, uniformly in x ∈ K, thanks to Lemma 4.1 and (4.2). The first one is also smaller than λ1ε/2 thanks to (4.5) again for R2 sufficiently small. The proof is concluded. 5. Dini continuous coefficients The proofs for b(·) Dini continuous are actually much simpler; therefore we are going only to highlight the necessary changes. We start defining the comparison maps v̄ ∈ W 1,p(BR;RN ) and v ∈ W 1,p(BR/2;RN ) exactly as in (3.3) and (3.4). Comparison estimate (3.12) is now replaced by − ∫ BR/2 ∣∣Vā(Du)−Vā(Dv) ∣∣2 dx ≤ c[[ωlog(R) ]2 + [ ωb(R) ]2]−∫ B3R/2 H(x,Du) dx , (5.1) the constant having the same dependencies and ωb(·) as defined in (1.8). The only change in the proof is in the estimate for D2 (3.25): now we can simply estimate |D2| ≤ c(p)ωb(R)− ∫ BR/2 ∣∣Vā(Dv̄)− Vā(Dv) ∣∣√Hā ( Dv̄ ) dx + c(p)ωb(R)− ∫ BR/2 ∣∣Vā(Dv̄)− Vā(Dv) ∣∣√Hā ( Dv ) dx ≤ 2c(p)ε− ∫ BR/2 ∣∣Vā(Dv̄)− Vā(Dv) ∣∣2 dx + c(p) ε [ ωb(R) ]2[−∫ BR/2 Hā(Dv̄) dx+− ∫ BR/2 Hā(Dv) dx ] for ε ∈ (0, 1) to be chosen, using Young’s inequality. At this point we use (3.6)-(3.7) and reabsorb the integral of |Vā(Dv̄) − Vā(Dv)|2 and the proof is concluded. The subsequent results have the same form except for the fact that ωb(R) replaces BR,q and the same with 2R. 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Marcellini; Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Rational Mech. Anal., 105 (1989), no. 3, 267–284. [43] P. Marcellini; Regularity and existence of solutions of elliptic equations with (p, q)-growth conditions. J. Diff. Equations, 90 (1991), no. 1, 1–30. [44] J. Ok; Regularity of ω-minimizers for a class of functionals with non-standard growth. Calc. Var., 56 (2017), article 48. [45] M. Schäffner; Higher integrability for variational integrals with non-standard growth. Calc. Var., 60 (2021), article 77. Paolo Baroni Department of Mathematical, Physical and Computer Sciences, University of Parma, I-43124 Parma, Italy Email address: paolo.baroni@unipr.it Alessandra Coscia Department of Mathematical, Physical and Computer Sciences, University of Parma, I-43124 Parma, Italy Email address: alessandra.coscia@unipr.it 1. Introduction and results 2. Preliminaries 2.1. Local and global minimizers 2.2. Notation and elementary properties 2.3. N-functions setting 2.4. Preliminary results 2.5. bold0mu mumu LplogLLplogLLplogLLplogLLplogLLplogL spaces 2.6. Basic estimates for minimizers of P 2.7. Estimates for reference functionals 3. Comparison estimates and excess decay 3.1. Comparison lemma 3.2. Excess decay estimate 4. Iteration and conclusion 4.1. Gradient boundedness by induction 4.2. Quantitative locally uniform VMO-type estimate 4.3. Gradient continuity by locally uniform convergence 5. Dini continuous coefficients References