Partial Hölder regularity for elliptic systems with non-standard growth

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Partial Hölder regularity for elliptic systems with non-standard growth Jihoon Ok Department of Applied Mathematics and Institute of Natural Science, Kyung Hee University, Yongin 17104, Republic of Korea

a r t i c l e

i n f o

Article history: Received 21 February 2017 Accepted 23 November 2017 Available online xxxx Communicated by H. Brezis MSC: primary 35J60 secondary 35B65

a b s t r a c t We study partial Hölder regularity for elliptic systems with non-standard growth. We consider general systems with Orlicz growth and discontinuous coefficient factor, for which we prove that their weak solutions are partially Hölder continuous for any Hölder exponents. In addition, we also obtain a similar result for systems with double phase growth. © 2017 Published by Elsevier Inc.

Keywords: Partial regularity Elliptic system Orlicz function Double phase problem

1. Introduction In this paper, we study the partial Hölder continuity for elliptic systems with nonstandard growth. Partial regularity of weak solutions to elliptic systems with p-growth, which is usually called ‘standard growth’, has been extensively studied since the basic works of Campanato [6,7] after the pioneering works of Giusti & Miranda and Morrey [20,38]. In particular, the main question was that with respect to conditions of the coef-

E-mail address: [email protected]. https://doi.org/10.1016/j.jfa.2017.11.014 0022-1236/© 2017 Published by Elsevier Inc.

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ficient factor we can obtain partial regularity which are the natural counter parts of ones for single equations. For Hölder continuous coefficients, it has been well known that the gradient of weak solutions are Hölder continuous except a Lebesgue measure zero set. We refer to for instance a overview paper [37] of Mingione, in which one can find a lot of references and an historical account of the problem. On the other hand, the problem of establishing partial regularity results in the case coefficients are only continuous has remained open for a long time. In this situation, in analogy to the scalar case, the type of partial regularity expected is the Hölder continuity of solutions with any Hölder exponent outside a Lebesgue measure zero set, called indeed the singular set; no pointwise regularity of the gradient is expected even in the scalar case. After various attempts, this conjecture has been finally proved by Foss & Mingione [17] when p ≥ 2; in this paper the authors treated both the case of elliptic systems and the one of minimizers of quasi-convex functionals. We also refer to [27–29] for partial regularity in the functional setting and related estimates on singular sets Hausdorff dimension. We point out that no singular set dimension estimate is known in this case. Problems with non-standard growth conditions were first systematically studied by Marcellini in series of fundamental papers [33–36]. In addition, various non-standard growth conditions have been presented. Amongst them, let us introduce three significant non-standard growth conditions. The first one is the so-called Orlicz growth condition which is related to the so-called Orlicz function G(t) described below. Orlicz spaces have been studied for a long time, and Lieberman first applied this to partial differential equations and functionals in [30–32]. We refer to for instance [2,9,14] and references therein for regularity results about problems with Orlicz growth. The second one is the p(x)-growth condition which is related to the function t → tp(x) . Problems with p(x)-growth are motivated by modeling of electrorheological fluids [42] and image restoration [8], and so, for last two decades, there have been intensive research activities about problems with p(x)-growth, see [1,5,13,23] for regularity results and related references. The final one, that has recently attracted a lot of attention, is the so-called double phase growth condition which is related to the function H(x, t) described below. Double phase problems are motivated by the modeling of the mixtures of two different materials in the context of Homogenization theory [44] and one of typical examples of Lavrentiev’s phenomenon [43]. Recently, Baroni & Colombo & Mingione have obtained various regularity results for these problems [3,4,10–12]. We also refer to [18,39] for a borderline growth condition related to the function tp(x) log(e + t), and [22,24,25] for the so-called generalized Orlicz growth condition. The non-standard growth conditions we consider in this paper are the Orlicz growth condition and the double phase condition. Let us first introduce our result for systems with Orlicz growth. Suppose that G : [0, ∞) → [0, ∞) with G(0) = 0 is C 2 and satisfies tG (t) tG (t) ≤ sup  ≤ g2 − 1  t>0 G (t) t>0 G (t)

1 < g1 − 1 ≤ inf

(1.1)

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for some 2 < g1 ≤ g2 < ∞. The previous condition implies that G is convex, strictly increasing and, generally speaking, the exponent of G is bounded from below by g1 and from above by g2 . With this G, we consider the following system: div a(x, u, Du) = 0 in Ω.

(1.2)

Here, a : Ω × RN × RN n → RN n , N ≥ 1, basically satisfies 

|a(x, ζ, ξ)| + |∂a(x, ζ, ξ)|(1 + |ξ|) ≤ LG1 (1 + |ξ|), ∂a(x, ζ, ξ)η · η ≥ νG2 (1 + |ξ|)|η|2

(1.3)

for all x ∈ Ω, ζ ∈ RN and ξ, η ∈ RnN and for some 0 < ν ≤ L, where ∂a(x, ζ, ξ) := Dξ a(x, ζ, ξ), G1 (t) := t−1 G(t) and G2 (t) := t−2 G(t).

(1.4)

We note from the second inequality in (1.3) that (a(x, ζ, ξ1 ) − a(x, ζ, ξ2 )) : (ξ1 − ξ2 ) ≥ ν˜ G2 (1 + |ξ1 | + |ξ2 |)|ξ1 − ξ2 |2  ν˜  G2 (1 + |ξ1 |)|ξ1 − ξ2 |2 + G(|ξ1 − ξ2 |) . (1.5) ≥ 2 Then, we say u ∈ W 1,1 (Ω, RN ) with G(|Du|) ∈ L1 (Ω) is a weak solution to (1.2) if ˆ a(x, u, Du) : Dϕ dx = 0 ∀ϕ ∈ W01,1 (Ω, RN ) with G(|Dϕ|) ∈ L1 (Ω).

(1.6)

Ω

Now, to obtain the desired regularity, we impose assumptions on the nonlinearity a as follows. For the first variable x we suppose that lim V(ρ) = 0, where V(ρ) := sup sup

ρ→0

ˆ −

0
V (x, Br (x0 )) dx

(1.7)

and V (x, U ) :=

sup ζ∈RN ,ξ∈RnN

|a(x, ζ, ξ) − (a(·, ζ, ξ))U | ≤ 2L. G1 (1 + |ξ|)

(1.8)

Here we note that the condition (1.7) implies that the coefficient factor of a is VMO uniformly for the ζ and ξ variables. For the other variables we assume that there exist nondecreasing and concave function μ : [0, ∞) → [0, 1] with μ(0) = 0 such that |a(x, ζ1 , ξ) − a(x, ζ2 , ξ)| ≤ L μ(|ζ1 − ζ2 |) G1 (1 + |ξ|)

(1.9)

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and  |∂a(x, ζ, ξ1 ) − ∂a(x, ζ, ξ2 )| ≤ L μ

|ξ1 − ξ2 | 1 + |ξ1 | + |ξ2 |

 G2 (1 + |ξ|)

(1.10)

for all x ∈ Ω, ζ, ζ1 , ζ2 ∈ RN and ξ, ξ1 , ξ2 ∈ RnN . Then, under these assumptions we have the following result. Theorem 1.1. Suppose G : [0, ∞) → [0, ∞) is C 2 and satisfies (1.1), and a : Ω × RN × RnN → RN satisfies (1.3), (1.7), (1.9) and (1.10). Let u ∈ W 1,1 (Ω, RN ) with G(|Du|) ∈ L1 (Ω) be a weak solution to (1.2). Then there exists Ωu ⊂ Ω such that |Ω \ Ωu | = 0 and u ∈ C β (Ωu , RN ) for every β ∈ (0, 1). Moreover,

Ω \ Ωu ⊂

∪

⎧ ⎪ ⎨

ˆ x0 ∈ Ω : lim inf −

⎪ ⎩

r↓0

⎫ ⎪ ⎬

|Du − (Du)x0 ,r | dx > 0

⎪ ⎭

Br (x0 )

⎧ ⎪ ⎨ ⎪ ⎩

ˆ x0 ∈ Ω : lim sup −

G(|Du|) dx = ∞

r↓0 Br (x0 )

⎫ ⎪ ⎬ ⎪ ⎭

.

The second problem we are interested in this paper is the partial Hölder regularity for systems with double phase growth. We define H : Ω × [0, ∞) → [0, ∞) by H(x, t) := tp + a(x)tq ,

(1.11)

0 ≤ a(·) ≤ aL∞ < ∞ and a ∈ C α for some α ∈ (0, 1],

(1.12)

where a : Ω → R satisfies

and 2
and

α q <1+ . p n

(1.13)

We remark that on the region {a(x) = 0}, H(x, t) = tp hence H has p-phase, and that on the region {a(x) > 0}, H has q-phase. Here we note that for each x ∈ Ω with a(x) > 0 the function t → H(x, t) satisfies (1.1) with G(t) = H(x, t), g1 = p and g2 = q, hence the constants g1 , g2 are independent of the choice of x ∈ Ω with a(x) > 0. With this H, we consider the following system: div b(x, Du) = 0 in Ω. Here, b : Ω × RN n → RN n , N ≥ 1, basically satisfies

(1.14)

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|b(x, ξ)| + |∂b(x, ξ)|(1 + |ξ|) ≤ LH1 (x, 1 + |ξ|), ∂b(x, ξ)η · η ≥ νH2 (x, 1 + |ξ|)|η|2

(1.15)

|b(x1 , ξ) − b(x2 , ξ)| ≤ L[a]C α |x1 − x2 |α (1 + |ξ|)q−1

(1.16)

and

for every x, x1 , x2 ∈ Ω and ξ, η ∈ RnN and for some 0 < ν ≤ L, where ∂b(x, ξ) := Dξ b(x, ξ), H1 (x, t) := t−1 H(x, t) and H2 (x, t) := t−2 H(x, t).

(1.17)

Similarly to (1.5), the second inequality in (1.15) implies (b(x, ξ1 ) − b(x, ξ2 )) : (ξ1 − ξ2 ) ≥ ν˜ H2 (x, 1 + |ξ1 | + |ξ2 |)|ξ1 − ξ2 |2  ν˜  H2 (x, 1 + |ξ1 |)|ξ1 − ξ2 |2 + H(x, |ξ1 − ξ2 |) . (1.18) ≥ 2 Then, we say u ∈ W 1,1 (Ω, RN ) with H(·, |Du|) ∈ L1 (Ω) is a weak solution to (1.14) if ˆ b(x, Du) : Dϕ dx = 0 ∀ϕ ∈ W01,1 (Ω, RN ) with H(·, |Dϕ|) ∈ L1 (Ω).

(1.19)

Ω

In addition, we further assume that there exists a nondecreasing and concave function μ : [0, ∞) → [0, 1] with μ(0) = 0 such that  |∂b(x, ξ1 ) − ∂b(x, ξ2 )| ≤ L μ

|ξ1 − ξ2 | 1 + |ξ1 | + |ξ2 |

 H2 (x, 1 + |ξ|).

(1.20)

The following is our second result for the double phase problems. Theorem 1.2. Suppose H : Ω × [0, ∞) → [0, ∞) is denoted by (1.11) with (1.12) and (1.13) and b : Ω × RnN → RnN satisfies (1.15), (1.16) and (1.10). Let u ∈ W 1,1 (Ω, RN ) with H(·, |Du|) ∈ L1 (Ω) be a weak solution to (1.14). Then there exists Ωu ⊂ Ω such that |Ω \ Ωu | = 0 and u ∈ C β (Ωu , RN ) for every β ∈ (0, 1). Moreover,

Ω \ Ωu ⊂

∪

⎧ ⎪ ⎨ ⎪ ⎩

ˆ x0 ∈ Ω : lim inf − r↓0

⎫ ⎪ ⎬

|Du − (Du)x0 ,r | dx > 0

⎪ ⎭

Br (x0 )

⎧ ⎪ ⎨ ⎪ ⎩

ˆ x0 ∈ Ω : lim sup − r↓0 Br (x0 )

H(x, |Du|) dx = ∞

⎫ ⎪ ⎬ ⎪ ⎭

.

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We shall write some comments of the above result for the double phase problems. Firstly, the function H depends on the x-variable, in particular, the function a(·). Hence the regularity of weak solutions is also influenced by the condition of a(·). In the scalar case, N = 1, if a(·) satisfies (1.13) we see that the weak solutions are Hölder continuous with any Hölder exponent, see [10,12]. On the other hand, if a(·) satisfies (3.2) below it is known that u is Hölder continuous for some Hölder exponent, see [3]. Hence it is natural that we assume (1.13). Secondly, it is possible to consider a discontinuous coefficient factor of b, like (1.7) for systems with Orlicz growth a. However, we do not consider any regularity assumption on the coefficient factor, in order to concentrate on the control of the effect of a(·). Finally, in contrast with the Orlicz growth condition, we do not consider lower order term. Because, the self improving property described in Lemma 3.6, which plays a crucial role in the proof, is not clear if we consider b(x, u, Du) instead of b(x, Du) in (1.14). We would like note that for systems with variable growth conditions, partial Hölder continuity has been proved in [21] for the p(x)-growth condition and in [40] for the borderline growth condition related to the function tp(x) log(e + t). In particular, in the second result, if p(·) ≡ p then the function tp log(e +t) is a special case of Orlicz functions. On the other hand, for systems with general Orlicz growth, there has been no any result about partial regularity. Hence in this paper we give a first partial regularity result even for systems with Orlicz growth. Finally, let us mention about a method in this paper. Our approach is based on the one in [17]. We compare our weak solutions with the A-harmonic maps via the A-harmonic approximation introduced in [16], and measure the decay of the so-called re-normalized oscillation functionals C(x0 , ρ), see (4.7) and (5.9), making use of hybrid excess functionals E(x0 , ρ), see (4.14) and (5.16). However, we point out that many of analyses in there are not directly applied in our case. The main reason is difficulty of controlling the re-normalized oscillation functions, which originates the radical difference between the p power function: tp and the Orlicz function: G(t). For example, for Orlicz function G, we cannot expect the Hölder type inequality in the integral sense and the relation that G(ts) ≈ G(t)G(s). We overcome these difficulties by using Jensen’s inequality and least concave majorant along with more delicate analyses. Moreover, for the double phase problems we have to consider both p-phase and q-phase in the same time, and to overcome this we employ the self improving property in Lemma 3.6. Then we can analyze the problem (1.14) in certain Orlicz classes. The rest of this paper is organized as follows. In the next section, Section 2, we present notation and auxiliary results. In Section 3, we introduce various self improving properties. In Section 4, we prove Theorem 1.1. In the final section, Section 5, we prove Theorem 1.2.

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2. Preliminaries 2.1. Notation and Orlicz function For x0 ∈ Rn , r > 0 and a locally integrable vector valued function f in Rn , we define Br (x0 ) by the standard ball centered at x0 with radius r and (f )x0 ,r by the integral average of f in Br (x0 ) such that (f )x0 ,r =

ˆ −

f dx =

Br (x0 )

ˆ

1 |Br (x0 )|

f dx. Br (x0 )

For matrix values A = (aij ), B = (bij ) ∈ RnN , we define the inner product by A : B :=  aij bij . P : Rn → RN is always an affine function, that is, P (x) = Ax + b for some A ∈ RnN and b ∈ RN . For a given u ∈ L2 (Br (x0 ), RN ), we define an affine function Px0 ,r by the minimizer of the functional P →

ˆ −

|u − P |2 dx.

Br (x0 )

Then one can see that Px0 ,r = DPx0 ,r (x − x0 ) + (u)x0 ,r with DPx0 ,r =

n+2 r2

ˆ −

u ⊗ (x − x0 ) dx.

Br (x0 )

If the center point of ball is clear or not important, we shall omit it in our notation, for example, Br = Br (x0 ) and (f )x0 ,r = (f )r . We next introduce so-called Orlicz functions. Suppose that G : [0, ∞) → [0, ∞) is a N -function, that is, G is differentiable and its derivative G is right continuous, nondecreasing and satisfies that G (0) = 0 and G (t) > 0 for all t > 0. In addition, G also satisfies that tG (t) tG (t) ≤ sup ≤ g2 t>0 G(t) t>0 G(t)

g1 ≤ inf

(2.1)

for some 1 < g1 ≤ g2 < ∞. Of course, the N -function is convex and G(t) = tp , 1 < p < ∞, is an N -function and satisfies (2.1) with g1 = g2 = p. Moreover, we easily see that if G is C 2 and satisfies (1.1) without the left-most inequality “2 <”, it is a N -function and satisfies (2.1). We also define the conjugate function G∗ : [0, ∞) → [0, ∞) of the N -function G satisfying (2.1), by G∗ (τ ) := sup (τ t − G(t)). t≥0

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Note that G∗ is also an N -function and satisfies (2.1) with g1 and g2 replaced by g2 /(g2 − 1) and g1 /(g1 − 1), respectively. The following is basic properties of the Orlicz functions G and G∗ (see for instance [15]). Proposition 2.1. Suppose G : [0, ∞) → [0, ∞) is convex and satisfies (2.1) 1 < g1 ≤ g2 < ∞. Let t, τ > 0, 0 < a < 1 < b < ∞. (1) G(t)t−g1 is increasing and G(t)t−g2 is decreasing. Hence we have G(at) ≤ ag1 G(t), G(bt) ≤ bg2 G(t).

(2.2)

Moreover, g2

g1

G∗ (at) ≤ a g2 −1 G∗ (t), and G∗ (bt) ≤ b g1 −1 G∗ (t).

(2.3)

(2) G(t + τ ) ≤ 2−1 (G(2t) + G(2τ )) ≤ 2g2 −1 (G(t) + G(τ )). (3) (Young’s inequality) For any κ ∈ (0, 1], we have tτ ≤ G(κ g1 t) + G∗ (κ− g1 τ ) ≤ κ G(t) + κ− g1 −1 G∗ (τ ) 1

1

1

(2.4)

and tτ ≤ G(κ−

g2 −1 g2

t) + G∗ (κ

g2 −1 g2

τ ) ≤ κ−g2 +1 G(t) + κ G∗ (τ )

(2.5)

(4) There exists c = c(g1 , g2 ) > 1 such that   c−1 G(t) ≤ G∗ G(t)t−1 ≤ c G(t).

(2.6)

The next lemma describes an almost concave condition and will be useful in this paper. A similar result can be found in [24, Lemma 2.2], from which we can reduce the next lemma. However, for the sake of readability, we present an exact proof. Lemma 2.2. Suppose that Ψ : [0, ∞) → [0, ∞) is non-decreasing and satisfies that the map ˜ : [0, ∞) → [0, ∞) t → Ψ(t)/t is non-increasing. Then there exists a concave function Ψ such that 1˜ ˜ Ψ(t) ≤ Ψ(t) ≤ Ψ(t) for all t ≥ 0. 2 Proof. Fix any τ > 0. Then, by the assumption of ψ(t)/t we have Ψ(t) ≤

t Ψ(τ ) ≤ τ



 t + 1 Ψ(τ ) for all t ≥ τ. τ

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  On the other hand, it is clear that Ψ(t) < Ψ(τ ) ≤ τt + 1 Ψ(τ ) for all t > τ . Therefore   we see that the map t → Pτ (t) := τt + 1 Ψ(τ ) is a concave majorant of Ψ. Now we ˜ : [0, ∞) → [0, ∞) by the least concave majorant of Ψ. Then by its definition we define Ψ have   t ˜ + 1 Ψ(τ ) for all t ≥ 0. Ψ(t) ≤ Ψ(t) ≤ Pτ (t) = τ ˜ ) ≤ 2Ψ(τ ). Since τ > 0 is arbitrary we obtain the Therefore, choosing t = τ we have Ψ(τ conclusion. 2 2.2. Basic inequalities We first recall Jensen’s inequality such that for any convex function G : [0, ∞) → [0, ∞), ⎛

⎞ ˆ ˆ G ⎝ − |f | dx⎠ ≤ − G(|f |) dx, U

U

which will be used frequently in this paper. For any vector valued function f , constant vector A and 1 < p < ∞, we have ˆ ˆ g2 − G(|f − (f )y,r |) dx ≤ 2 − G(|f − A|) dx, Br (y)

(2.7)

Br (y)

where G is an N -function and satisfies (2.1). This inequality follows from (2) of Proposition 2.1 and Jensen’s inequality. Lemma 2.3. Let G : [0, ∞) → [0, ∞) be convex and satisfy (2.1) for some 2 < g1 < g2 < ∞. (1) For any u ∈ L1 (Br (x0 ), RN ) with G(|u|) ∈ L1 (Br (x0 )) and θ ∈ (0, 1), we have G(|DPx0 ,r − DPx0 ,θr |) ≤ c



ˆ −

G

|u − Px0 ,r | θr

 dx.

(2.8)

Bθr (x0 )

(2) For any u ∈ W 1,1 (Br (x0 )) with G(|Du|) ∈ L1 (Br (x0 )), we have   G |DPx0 ,r − (Du)Br (x0 ) | ≤ c

ˆ −

Br (x0 )

Here, the constants c depend only on n, N, g1 , g2 .

G(|Du − (Du)Br (x0 ) |) dx.

(2.9)

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Proof. By [26, Lemma 2], we have |DPx0 ,r − DPx0 ,θr |2 ≤ c

ˆ −

|u − Px0 ,r |2 dx, (θr)2

Bθr (x0 )

ˆ |DPx0 ,r − (Du)Br (x0 ) | ≤ c −

|Du − (Du)Br (x0 ) |2 dx.

2

Br (x0 )

√ Using these and Jensen’s inequality for the convex map t → G( t), we obtain ⎛  ˆ  ⎜ G(|DPx0 ,r − DPx0 ,θr |) ≤ c G ⎝ −

≤c

ˆ −

⎞ |u − Px0 ,r |2 ⎟ dx⎠ (θr)2

Bθr (x0 )



G

|u − Px0 ,r | θr

 dx.

Bθr (x0 )

This shows (2.8). The same argument implies the inequality (2.9).

2

We finally state the following iteration lemma, see [19, Lemma 7.3] and also [17, Lemma 2.3]. Lemma 2.4. Let φ : (0, ρ] → R be a positive and nondecreasing function satisfying φ(θk+1 ρ) ≤ θλ φ(θk ρ) + c˜ (θk ρ)n

for every k = 0, 1, 2, . . . ,

where θ ∈ (0, 1), λ ∈ (0, n) and c˜ > 0. Then there exists c = c(n, θ, λ) > 0 such that    λ t λ φ(t) ≤ c˜ φ(ρ) + c˜ t ρ

for every t ∈ (0, ρ].

2.3. A-harmonic approximation Let A be a bilinear form in RnN and satisfy the Legendre–Hadamard condition, that is, there exist 0 < ν ≤ L such that ν|ξ|2 |η|2 ≤ A(ξ ⊗ η) : ξ ⊗ η ≤ L|ξ|2 |η|2 for every ξ ∈ Rn , η ∈ RN . We say h ∈ W 1,2 (Ω, RN ) is A-harmonic if ˆ ADh : Dϕ = 0 Ω

(2.10)

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for every ϕ ∈ C01 (Ω, RN ). We then state the A-harmonic approximation lemma, see [16, Lemma 3.3]. Lemma 2.5. Let  > 0 and bilinear form A on RnN satisfy (2.10). Then there exists small δ = δ(n, N, ν, L, ) > 0 such that if w ∈ W 1,2 (Br , RN ) satisfies that ˆ − |Dw|2 dx ≤ 1, Br

and   ˆ     − ADw : Dϕ dx ≤ δDϕL∞ (B ) for all ϕ ∈ C01 (Br , RN ), r     Br

then there exists A-harmonic map h ∈ W 1,2 (Br , RN ) such that ˆ − |Dh|2 dx ≤ 1, and

ˆ r−2 − |w − h|2 dx ≤ .

Br

Br

3. Self improving properties We introduce the self improving properties of gradient of weak solutions to (1.2) and (1.14). Similar results for problems with non-standard growth can be found in various literatures, from which we can deduce the versions stated below. However, for the shake of readability and clearness, we provide their detail proofs. We shall start with Sobolev–Poincaré type inequalities for the functions G and H. The result for the function G easily follows from [14, Theorem 7]. Lemma 3.1. Suppose G : [0, ∞) → [0, ∞) is an N -function and satisfies (2.1) for some 1 < g1 ≤ g2 < ∞ and that f ∈ W 1,1 (Br , RN ). Then there exist 0 < d1 < 1 < d2 depending only on n, N, g1 , g2 such that ⎞ d1 ⎛ ⎞ d1 2 1 d2 ˆ ˆ   | |f − (f ) r d1 ⎠ ⎝ ⎠ ⎝− G dx ≤ c − [G(|Df |)] dx , r ⎛

Br

(3.1)

Br

for some c = c(n, N, g1 , g2 ) > 0. We next consider the function H. Theorem 3.2. Suppose H : Ω × [0, ∞) → [0, ∞) is denoted in (1.11) with (1.12) and 1
and

α q ≤1+ . p n

(3.2)

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Then there exist 0 < d3 < 1 < d4 depending only on n, N, p, q such that for any f ∈ W 1,1 (Ω, RN ) and Br = Br (x0 )  Ω, we have ⎞ d1 4 d4 ˆ   ⎠ ⎝ − H x, |f − (f )r | dx r ⎛

Br

⎞ d1 ⎛ 3   ˆ (q−p)n d3 α− ⎠ ⎝ p ≤ c 1 + [a]C α Df q−p r [H(x, |Df |)] dx − Lp (Ω)

(3.3)

Br

for some c = c(n, N, p, q) ≥ 1. Proof. We follow the argument in [41, Theorem 2.13]. Suppose that sup a(·) ≤ 4[a]C α rα .

(3.4)

Br

We take d4 (n, p, q) > 1 sufficiently close to 1, so that !

(qd4 )∗ := max

nqd4 ,1 n + qd4

< p.

It is possible since q∗ := max{nq/(n + q), 1} < p by (3.2). Then, from (3.4) and the classical Sobolev–Poincaré’s inequality for the exponent qd4 , we have qd4 ⎞ (qd ⎛ 4 )∗  ˆ  ˆ q d4 |f − (f )r | α d4 ⎝ (qd4 )∗ ⎠ − a(x) dx ≤ [4[a]C α r ] dx . − |Df | rq

Br

Br

Therefore, Hölder’s inequality and (3.2) yield d ˆ  |f − (f )r |q 4 − a(x) dx rq Br pd4 4 ⎛ ⎞ (q−p)d ⎞ (qd p 4 )∗ ˆ ˆ d ⎝ − |Df |(qd4 )∗ dx⎠ ≤ c ([a]C α rα ) 4 ⎝ − |Df |p dx⎠

⎛

Br



α− ≤ c [a]C α Df q−p Lp (Ω) r

Br

d4 (q−p)n p

pd4 ⎞ (qd 4 )∗ ˆ ⎝ − |Df |(qd4 )∗ dx⎠ .

⎛

Br

From this estimate and the classical Sobolev–Poincaré’s inequality for the exponent pd4 , we have

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d4 ˆ   |f − (f )r | − H x, dx r Br



≤ c 1 + [a]C α Df q−p Lp (Ω) r

α− (q−p)n p

d4

pd4 ⎞ (qd 4 )∗ ˆ (qd ) 4 ∗ ⎝ − |Df | dx⎠ .

⎛

Br

Therefore choosing d3 = (qd4 )∗ /p < 1, we obtain (3.3). We next assume that (3.4) is not true. Then, by (1.12) we see that inf x∈Br a(x) > 3[a]C α rα and so 34 a(x) ≤ a(x0 ) ≤ 43 a(x) for all x ∈ Br . Hence we have and 4 3 H(x, t) ≤ H(x0 , t) ≤ H(x, t) for all x ∈ Br . 4 3 Consequently, applying Lemma 3.1 to G(t) = H(x0 , t), g1 = p and g2 = q, we have (3.3). 2 Remark 3.3. A similar result of Theorem 3.2 can be found in [10, Theorem 1.6]. Comparing with that result, our one is more general since we assume that p and q satisfy (3.2), instead of (1.13), and the exponents d3 and d4 are independent of α. In addition, our proof is more simple and elementary than the one of [10, Theorem 1.6]. Now we state the self improving properties. Theorem 3.4 (Self improving properties). (1) Suppose G : [0, ∞) → [0, ∞) is an N -function and satisfies (2.1) for some 1 < g1 ≤ g2 < ∞, and a : Ω × RN × RnN → RnN satisfies |a(x, ζ, ξ)| ≤ L G1 (s + |ξ|)

and

a(x, ζ, ξ) : ξ ≥ ν G(|ξ|) − ν0 G(s)

(3.5)

for all x ∈ Ω, ζ ∈ RN and ξ ∈ RnN , and for some 0 < ν ≤ L < ∞, ν0 and s ∈ [0, 1]. Let u ∈ W 1,1 (Ω) with G(|Du|) ∈ L1 (Ω) be a weak solution to (1.2). Then there exists 1 σ1 = σ1 (n, N, ν, L, g1 , g2 ) > 0 such that G(|Du|) ∈ L1+σ loc (Ω) with the estimate that for any σ ∈ [0, σ1 ] and B2r (x0 )  Ω, ⎛

1 ⎞ 1+σ ˆ ⎜ ⎟ 1+σ dx⎠ ≤c ⎝ − [G(s + |Du|)]

Br (x0 )

ˆ −

G(s + |Du|) dx

(3.6)

B2r (x0 )

for some c = c(n, N, ν, L, g1 , g2 ) > 0. (2) Suppose H : [0, ∞) → [0, ∞) is denoted by (1.11) with (1.12) and (3.2), and b : Ω × RnN → RnN satisfies

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|b(x, ξ)| ≤ L H1 (x, s + |ξ|)

b(x, ξ) : ξ ≥ ν H(x, |ξ|) − ν0 H(x, s)

and

(3.7)

for all x ∈ Ω and ξ ∈ RnN , and for some 0 < ν ≤ L < ∞, ν0 and s ∈ [0, 1]. Let u ∈ W 1,1 (Ω) with H(·, |Du|) ∈ L1 (Ω) be a weak solution to (1.2). Then there exists 2 σ2 = σ2 (n, N, ν, L, p, q, [a]C α , DuLp ) > 0 such that H(·, |Du|) ∈ L1+σ loc (Ω) with the estimate that for any σ ∈ [0, σ2 ] and B2r (x0 )  Ω with r < 1, ⎛

1 ⎞ 1+σ ˆ ⎜ ⎟ 1+σ dx⎠ ≤c ⎝ − [H(x, s + |Du|)]

Br (x0 )

ˆ −

H(x, s + |Du|) dx

(3.8)

B2r (x0 )

for some c = c(n, N, ν, L, p, q, [a]C α , DuLp (Ω) ) > 0. Moreover, if H satisfies 1
and

α q <1+ , p n

(3.9)

instead of (3.2). Then there exists σ3 = σ3 (n, N, ν, L, p, q) > 0 independent of [a]C α 3 and DuLp such that H(·, |Du|) ∈ L1+σ loc (Ω) with the estimate (3.8), whenever σ ∈ [0, σ3 ] and B2r (x0 )  Ω with  r < min

[a]C α Duq−p Lq (Ω)

1 − α−(q−p)n/p

! ,1 .

(3.10)

Here, the constant c in (3.8) depends only on n, N, ν, L, p, q. Proof. For simplicity, we write Bρ = Bρ (x0 ) for ρ > 0. (1) Let ψ ∈ C0∞ (B2r ) be a cut-off function so that 0 ≤ ψ ≤ 1, ψ ≡ 1 in Br and |Dψ| ≤ c(n)/r. By taking ϕ = ψ g2 (u − (u)2r ) in (1.6) and using (3.5), we have ˆ ˆ − ψ g2 G(|Du|) dx ≤ c − ψ g2 a(x, u, Du) : Du dx + c G(s) B2r

B2 r

ˆ |u − (u)2r | ≤ c − ψ g2 −1 G1 (s + |Du|) dx + c G(s) 2r B2r

Using (2.5) with (2.6) and (2.3), ˆ ˆ 1 − ψ g2 G(s + |Du|) dx ≤ − ψ g2 G(s + |Du|) dx 2 B2r

B2r

  ˆ |u − (u)2r | dx + c G(s). +c − G 2r B2r

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Therefore, by (3.1) we have ⎛ ⎞ d1 2 ˆ ˆ − G(s + |Du|) dx ≤ +c ⎝ − [G(s + |Du|)]d2 dx⎠ . Br

B2r

Finally, applying Gehring’s lemma, see for instance [19, Chapter 6], we have (3.6). (2) Let ψ ∈ C0∞ (B2r ) be a cut-off function as in above. Then taking ϕ = ψ q (u − (u)2r ) in (1.6) and using (3.7), we have ˆ ˆ |u − (u)2r | q dx + c H(x, s). − ψ H(x, |Du|) dx ≤ c − ψ q−1 H1 (x, s + |Du|) 2r B2r

B2r

At this point, we use Young’s inequality such that for any κ ∈ (0, 1), ψ

q−1

p−1 |u

|Du|

p  (q−1)p − (u)2r | |u − (u)2r | p p−1 ≤ κψ |Du| + c(κ) 2r 2r p  |u − (u)2r | ≤ κψ q |Du|p + c(κ) 2r

and ψ

q−1

q−1 |u

|Du|

− (u)2r | ≤ κψ q |Du|q + c(κ) 2r



|u − (u)2r | 2r

q .

Hence, we have   ˆ ˆ |u − (u)2r | dx + c H(s). − H(s + |Du|) dx ≤ c − H x, 2r Br

B2r

Now we apply Sobolev–Poincaré estimate for the functions H in (3.3). First, if (3.2) holds and r ≤ 1, then we have ⎞ d1 ⎛ 4 ˆ   ˆ d q−p 4 ⎠ ⎝ − H(s + |Du|) dx ≤ c 1 + DuLp (Ω) , − [H (x, s + |Du|)] dx Br

B2r

from which and Gehring’s lemma, one can find positive constants σ2 and c depending on n, N, ν, L, p, q, [a]C α , DuLp (Ω) satisfying (3.8) for all σ ∈ (0, σ2 ] and r < (0, 1]. Second, if (3.9) holds and r > 0 satisfies (3.10), we have ⎛ ⎞ d1 4 ˆ ˆ d − H(s + |Du|) dx ≤ c ⎝ − [H (x, s + |Du|)] 4 dx⎠ . Br

B2r

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Again, applying Gehring’s lemma to the previous estimate, we can find positive constants σ3 and c depending on n, N, ν, L, p, q, for which the estimate (3.8) holds for all σ ∈ (0, σ3 ] and r > 0 satisfying. 2 Remark 3.5. In (2) of the previous theorem, we see that the exponent σ2 and the constant c in (3.8) depend on the solution itself, precisely DuLp (Ω) . However, if we exclude the case that q/p = 1 + α/n, we can find the exponent σ3 and the constant c in (3.8) that are independent of DuLp (Ω) . Instead, in this case, the estimate (3.8) can be obtained in sufficiently small regions, see (3.10). We end this section stating the self improving result of the double phase problem on a region near the p-phase. We refer to [12, Theorem 4.1 and Remark 3.1] for, in particular, the vectorial case of the double phase problems. Note that this lemma is a key of comparison approach when treating the double phase problems. Lemma 3.6. Suppose H : [0, ∞) → [0, ∞) is denoted by (1.11) with (1.12) and (3.9), and b : Ω × RnN → RnN satisfies (1.15). Let u ∈ W 1,1 (Ω) with H(·, |Du|) ∈ L1 (Ω) be a weak solution to (1.2). Let Br = Br (x0 )  Ω with r ≤ 1 and satisfying that sup a(x) ≤ K[a]α rα

x∈Br

for some K ≤ 1. Then for every q˜ < np(n − 2α) (= ∞ when α = 1 and n = 2), there exists h(K, q˜) > 0 depending only on n, N, ν, L, p, q, [a]α , DuLp , K and q˜, such that the following holds: ⎛

⎞ q1˜ ⎛ ⎞ p1 ˆ ˆ ⎜ ⎟ q˜ p ⎝ − [1 + |Du|] dx⎠ ≤ h(K, q˜) ⎝ − [1 + |Du|] dx⎠ . Br/2

(3.11)

Br

4. Partial Hölder regularity for systems with Orlicz growth In this section we prove Theorem 1.1. Hence Let us suppose that G : [0, ∞) → [0, ∞) is C 2 and satisfies (1.1), a : Ω × RN × RnN → RN satisfies (1.3), (1.7), (1.9) and (1.10), and u ∈ W 1,1 (Ω, RN ) with G(|Du|) ∈ L1 (Ω) is a weak solution to (1.2). Let B4ρ (x0 )  Ω with x0 ∈ Ω and ρ < 1. We first derive a Caccioppoli type inequality. Lemma 4.1. We have  ˆ  G(|Du − DP |) |Du − DP |2 dx − + (1 + |DP |)2 G(1 + |DP |) Bρ (x0 )

≤c

ˆ −

B2ρ (x0 )



 G(|u − P |/(2ρ)) |u − P |2 dx + (2ρ)2 (1 + |DP |)2 G(1 + |DP |)

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⎛ ⎜ + cμ⎝

ˆ −

17

⎞ ⎟ |u − P (x0 )|2 dx⎠ + c V(2ρ),

(4.1)

B2ρ (x0 )

for some c = c(n, N, ν, L, g1 , g2 ) > 0, where P : Rn → RN is any affine function and V is denoted in (1.7). Proof. Set ϕ = η g2 (u − P ), where η ∈ C0∞ (B2ρ ) with 0 ≤ η ≤ 1, η ≡ 1 on Bρ and |Dη| ≤ c(n)/ρ. Taking ϕ as a test function in (1.2), we have ˆ ˆ − η g2 a(x, u, Du) : D(u − P ) dx = −g2 − η g2 −1 a(x, u, Du) : Dη ⊗ (u − P ) dx. B2ρ

B2ρ

Then this and the trivial identity ˆ − a(P (x0 ), DP ) : Dϕ dx = 0, B2ρ

where ˆ a(ζ, ξ) := (a(·, ζ, ξ))B2ρ = − a(x, ζ, ξ) dx, B2ρ

imply that ˆ I1 := − η g2 (a(x, u, Du) − a(x, u, DP )) : (Du − DP ) dx B2ρ

ˆ = − (a(x, u, Du) − a(x, u, DP )) : Dϕ dx B2ρ

ˆ − g2 − η(a(x, u, Du) − a(x, u, DP )) : Dη g2 −1 ⊗ (u − P ) dx B2ρ

ˆ = − − (a(x, u, DP ) − a(x, P (x0 ), DP )) : Dϕ dx B2ρ

ˆ − − (a(x, P (x0 ), DP ) − a(P (x0 ), DP )) : Dϕ dx B2ρ

ˆ − g2 − η g2 −1 (a(x, u, Du) − a(x, u, DP )) : Dη ⊗ (u − P ) dx B2ρ

=: −I2 − I3 − I4 .

(4.2)

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Estimation I1 : We have from (1.5) that   ˆ |Du − DP |2 g2 G(1 + |DP |) − η + G(|Du − DP |) dx ≤ cI1 . (1 + |DP |)2

(4.3)

B2ρ

Estimation I2 : Applying (1.9),   ˆ   |u − P | dx. |I2 | ≤ c − μ |u − P (x0 )|2 G1 (1 + |DP |) η g2 |Du − DP | + ρ B2ρ

Then, using (2.4) and Jensen’s inequality, we see that   ˆ  1 |u − P | g2 dx |I2 | ≤ − η G(|Du − DP |) + G 4 ρ B2ρ

ˆ   + c G(1 + |DP |) − μ |u − P (x0 )|2 dx B2ρ

  ˆ ˆ 1 |u − P | dx ≤ − η g2 G(|Du − DP |) dx + c − G 4 ρ B2ρ

⎛

⎞

B2ρ

ˆ ⎜ ⎟ + cμ ⎝ − |u − P (x0 )|2 dx⎠ G(1 + |DP |).

(4.4)

B2ρ

Estimation I3 : Using (1.7), (1.8) and (2.4) with (2.3) and (2.6), we have   ˆ |u − P | g2 dx |I3 | ≤ c − V (x, B2ρ ) G1 (1 + |DP |) η |Du − DP | + ρ B2ρ

ˆ 1 ≤ − η g2 G(|Du − DP |) dx 4 B2ρ

  ˆ  |u − P | ∗ dx + c − G (V (x, B2ρ ) G1 (1 + |DP |)) + G ρ B2ρ

ˆ 1 ≤ − η g2 G(|Du − DP |) dx 4 B2ρ

  ˆ 1 |u − P | dx. + c(2L + 1) g1 −1 G(1 + |DP |) V(2ρ) + c − G ρ B2ρ

(4.5)

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Estimation I4 : We have from the first inequality in (1.3), Young’s inequalities with (2.5) that ⎞ ⎛ 1 ˆ ˆ |u − P | dx |I4 | ≤ c − η g2 −1 ⎝ |∂a(x, u, t(Du − DP ) + DP )| dt⎠ |Du − DP | ρ 0

B2ρ

ˆ |u − P | G(1 + |DP | + |Du − DP |) ≤ c − η g2 −1 dx |Du − DP | 2 (1 + |DP | + |Du − DP |) ρ B2ρ

ˆ |u − P | G(1 + |DP |) g2 −1 ≤c − dx η |Du − DP | (1 + |DP |)2 ρ B2ρ

ˆ G(|Du − DP |) |u − P | dx + c − η g2 −1 |Du − DP | ρ B2ρ

 ˆ  1 g2 G(1 + |DP |) 2 g2 ≤ − η |Du − DP | + η G(|Du − DP |) dx 4 (1 + |DP |)2 B2ρ

⎛

⎞   ˆ ˆ 2 |u − P | |u − P | ⎜ G(1 + |DP |) ⎟ + c⎝ dx⎠ . − dx + − G 2 2 (1 + |DP |) ρ ρ B2ρ

(4.6)

B2ρ

Consequently, inserting (4.3)–(4.6) into (4.2), we get the estimate (4.1). 2 Now we define  ˆ  G(|Du − DP |) |Du − DP |2 dx, C(x0 , ρ, P ) := − + (1 + |DP |)2 G(1 + |DP |)

(4.7)

Bρ

⎡ ⎛

⎞⎤ 12 ˆ 1 ⎢ ⎜ ⎟⎥ E(x0 , ρ, P ) := C(x0 , ρ, P ) + ⎣μ ⎝ − |u − P (x0 )|2 dx⎠⎦ + [V(ρ)] 2g2 −1 Bρ

and A :=

∂a(x0 , P (x0 ), DP ) G2 (1 + |DP |)

and w :=

u−P ( . (1 + |DP |) E(x0 , ρ, P )

(4.8)

Then we see from (1.3) that A satisfies (2.10). The next lemma implies that if E(x0 , ρ, P ) is sufficiently small then one can apply the harmonic approximation lemma, Lemma 4.2, to A and w denoted above.

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Lemma 4.2. Suppose that C(x0 , ρ, P ) ≤ 1.

(4.9)

Then, there exists c = c(n, N, ν, L, g1 , g2 ) > 0 such that     ˆ   ) ( * 12   sup |Dϕ|  − ADw : Dϕ dx ≤ c μ( E(x0 , ρ, P )) + E(x0 , ρ, P )   Bρ (x0 )  Bρ (x0 )

(4.10)

for every ϕ ∈ C0∞ (Bρ (x0 )). Proof. We first consider ϕ ∈ C0∞ (Bρ (x0 )) with supBρ (x0 ) |Dϕ| ≤ 1, and set v := u − P = ( (1 + |DP |) E(x0 , ρ, P ) w. Then (

G1 (1 + |DP |)

ˆ E(x0 , ρ, P ) − ADw : Dϕ dx Bρ

ˆ = G2 (1 + |DP |) − ADv : Dϕ dx Bρ

ˆ ˆ1 = − (∂a(x0 , P (x0 ), DP ) − ∂a(x0 , P (x0 ), DP + sDv))Dv : Dϕ ds dx Bρ 0

ˆ ˆ1 + − ∂a(x0 , P (x0 ), DP + sDv)Dv : Dϕ ds dx Bρ 0

=: I5 + I6 .

(4.11)

Estimation I5 : Applying (1.10), we have  ˆ ˆ1  s|Dv| G2 (1 + |DP | + |DP + sDv|)|Dv| ds dx |I5 | ≤ c − μ 1 + |DP | Bρ

 ˆ  |Du − DP | (G2 (1 + |DP |) + G2 (|Du − DP |)) |Du − DP | dx ≤c−μ 1 + |DP | Bρ

  ˆ  |Du − DP | G1 (|Du − DP |) |Du − DP | = c G1 (1 + |DP |) − μ + dx. 1 + |DP | 1 + |DP | G1 (1 + |DP |) Bρ

Set Bρ+ := {x ∈ Bρ : |Du(x) − DP | > 1 + |DP |} and Bρ− := Bρ \ Bρ+ . Then, since μ(·) ≤ 1, we have that in Bρ+ ,

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 μ

|Du − DP | 1 + |DP |

≤



|Du − DP | G1 (|Du − DP |) + 1 + |DP | G1 (1 + |DP |)

21



|Du − DP |2 G(|Du − DP |) . + 2 (1 + |DP |) G(1 + |DP |)

On the other hand, in Bρ− we have   |Du − DP | G1 (|Du − DP |) |Du − DP | + 1 + |DP | 1 + |DP | G1 (1 + |DP |)    |Du − DP | G(|Du − DP |) 1 + |DP | |Du − DP | + ≤μ 1 + |DP | 1 + |DP | G(1 + |DP |) |Du − DP | +  1 ,  |Du − DP | G(|Du − DP |) 2 |Du − DP | + ≤μ . 1 + |DP | 1 + |DP | G(1 + |DP |)

 μ

1

In the last inequality we have used the fact that the map t → [G(t)] 2 /t is non-decreasing by (2.1), which follows from (1.1), so that 1 + |DP | ≤ |Du − DP |



G(1 + |DP |) G(|Du − DP |)

 12

in Bρ− .

Then, combining previous results and using Hölder’s inequality, the fact that μ(·) ≤ 1 and Jensen’s inequality for the concave function μ, we have |I5 | G1 (1 + |DP |)  ˆ  |Du − DP |2 G(|Du − DP |) ≤c− dx + (1 + |DP |)2 G(1 + |DP |) Bρ

⎡ ⎛

⎞⎤ 12 ⎛ ⎞ 12  ˆ ˆ  |Du − DP |2 G(|Du − DP |) ⎢ ⎜ |Du − DP | ⎟⎥ ⎜ ⎟ + c ⎣μ ⎝ − dx⎠⎦ ⎝ − dx⎠ . + 1 + |DP | 1 + |DP |2 G(1 + |DP |) Bρ

Bρ

Therefore, recalling the definitions of C(x0 , ρ, P ) and E(x0 , ρ, P ), and using Young’s inequality, we obtain ! ) ( * 12 ( |I5 | ≤ c C(x0 , ρ, P ) + μ( C(x0 , ρ, P )) C(x0 , ρ, P ) G1 (1 + |DP |) ! ) ( * 12 ( ≤ c E(x0 , ρ, P ) + μ( E(x0 , ρ, P )) E(x0 , ρ, P ) . We next estimate I6 . Applying (1.7) and (1.9), we have

(4.12)

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ˆ I6 = − (a(x0 , P (x0 ), Du) − a(x0 , P (x0 ), DP )) : Dϕ dx Bρ

ˆ = − (a(x0 , P (x0 ), Du) − a(x, P (x0 ), Du)) : Dϕ dx Bρ

ˆ + − (a(x, P (x0 ), Du) − a(x, u, Du)) : Dϕ dx Bρ

ˆ ˆ ≤ c − V (x, Bρ ) G1 (1 + |Du|) dx + c − μ(|u − P (x0 )|2 ) G1 (1 + |Du|) dx Bρ

Bρ

=: c(I6,1 + I6,2 ). For I6,1 , using Hölder’s inequality and recalling the definitions of V and V in (1.8) and (1.7), respectively, we have 2 −2 ⎞ 2g 1−1 ⎛ ⎞ 2g 2g2 −1 2 ˆ ˆ 2g2 −1 ⎜ ⎜ ⎟ ⎟ ≤ ⎝ − [V (x, Bρ )]2g2 −1 dx⎠ ⎝ − [G1 (1 + |Du|)] 2g2 −2 dx⎠

⎛

I6,1

Bρ

Bρ

⎛ 2g2 −2

1

≤ (2L) 2g2 −1 [V(ρ)] 2g2 −1

2 −2 ⎞ 2g 2g2 −1 ˆ   2g2 −1 ⎜ ⎟ −1 , ⎝ − (Ψ ◦ Ψ ) [G1 (1 + |Du|)] 2g2 −2 dx⎠

Bρ

  2g2 −1 where Ψ(t) := t[G−1 (t)]−1 2g2 −2 . At this stage, we see that the map t → Ψ(t)/t is ˜ such that 1 Ψ(t) ˜ ˜ non-increasing, and so there exists a concave function Ψ ≤ Ψ(t) ≤ Ψ(t), 2 ˜ in view of Lemma 2.2. Therefore, using this inequality, Jensen’s inequality to Ψ and the fact that ˆ ˆ − G(1 + |Du|) dx ≤ c − G(|Du − DP |) dx + c G(1 + |DP |) ≤ c G(1 + |DP |), Bρ

Bρ

by (4.9), we have ⎛

2 −2 ⎞ 2g 2g2 −1 ˆ   2g2 −1 ⎜ ⎟ −1 ⎝ − (Ψ ◦ Ψ ) [G1 (1 + |Du|)] 2g2 −2 dx⎠

Bρ

⎡ ⎛

2 −2 ⎞⎤ 2g 2g2 −1 ˆ ⎢˜ ⎜ ⎟⎥ ≤ ⎣Ψ ⎝ − G(1 + |Du|) dx⎠⎦

Bρ 2g2 −2

2g2 −2

≤ [2Ψ (G(1 + |DP |))] 2g2 −1 = 2 2g2 −1 G1 (1 + |DP |).

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Therefore, we obtain 1 I6,1 ≤ c [V(ρ)] 2g2 −1 ≤ c E(x0 , ρ, P ). G1 (1 + |DP |)

As for I6,2 , using the fact that G(|Du − DP |) G1 (|Du − DP |) ≤ + 1, G1 (1 + |DP |) G(1 + |DP |) μ(·) ≤ 1 and Jensen’s inequality, we have   ˆ G(|Du − DP |) I6,2 ≤ c − μ(|u − P (x0 )|2 ) 1 + dx G1 (1 + |DP |) G(1 + |DP |) Bρ

⎛ ⎞ ˆ ˆ G(|Du − DP |) ⎜ ⎟ dx + c μ ⎝ − |u − P (x0 )|2 dx⎠ ≤c− G(1 + |DP |) Bρ

Bρ

≤ c E(x0 , ρ, P ). Consequently, we obtain |I6 | ≤ c E(x0 , ρ, P ). G1 (1 + |DP |)

(4.13)

Finally, inserting (4.12) and (4.13) into (4.11) and using the definition of (4.8), we have (4.10) for every ϕ ∈ C0∞ (Bρ (x0 )) with supBρ |Dϕ| ≤ 1. This implies (4.10) for every ϕ ∈ C0∞ (Bρ (x0 )) by the standard normalization argument. 2 Now, we consider the affine function P = (Du)x0 ,ρ (x − x0 ) + (u)x0 ,ρ , and set C(x0 , ρ) := C(x0 , ρ, (Du)x0 ,ρ (x − x0 ) + (u)x0 ,ρ )  ˆ  |Du − (Du)x0 ,ρ |2 G(|Du − (Du)x0 ,ρ |) dx, + = − (1 + |(Du)x0 ,ρ |)2 G(1 + |(Du)x0 ,ρ |) Bρ

˜ 0 , ρ) := E(x0 , ρ, (Du)x ,ρ (x − x0 ) + (u)x ,ρ ) E(x 0 0 ⎡ ⎛ ⎞⎤ 12 ˆ 1 ⎢ ⎜ ⎟⎥ = C(x0 , ρ) + ⎣μ ⎝ − |u − (u)x0 ,ρ |2 dx⎠⎦ + [V(ρ)] 2g2 −1 , Bρ

and

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⎡ ⎛

⎞⎤ 12 ˆ 1 ⎢ ⎜ ⎟⎥ E(x0 , ρ) := C(x0 , ρ) + ⎣μ ⎝ρ − |Du|2 dx⎠⎦ + [V(ρ)] 2g2 −1 .

(4.14)

Bρ

Then, by Poincaré’s inequality along with the fact that ρ < 1, we see that ˜ 0 , ρ) ≤ c E(x0 , ρ) E(x

(4.15)

for some c = c(n, N ) ≥ 1. Lemma 4.3. For θ ∈ (0, 1/8), there exists small 1 = 1 (n, N, ν, L, g1 , g2 , μ(·), θ) ∈ (0, 1) such that if ρ ≤ θn and E(x0 , ρ) ≤ 1 ,

(4.16)

C(x0 , θρ) ≤ c1 θ2 E(x0 , ρ)

(4.17)

then

for some c1 = c1 (n, N, ν, L, g1 , g2 ) ≥ 1. Proof. Step 1. We first estimate the following integrals ˆ |u(x) − P2θρ |2 − dx and (2θρ)2 B2θρ

  ˆ |u − P2θρ | dx, − G 2θρ

(4.18)

B2θρ

where the affine function P2θρ = Px0 ,2θρ is denoted in Section 2.1. Recall A and w denoted in (4.8) with P = (Du)ρ (x − x0 ) + (u)ρ . Then we see that w :=

ˆ u − (Du)ρ (x − x0 ) − (u)ρ and − |Dw|2 dx ≤ 1. ˜ 0 , ρ) (1 + |(Du)ρ |) E(x Bρ

Let us take  ∈ (0, 1) such that  = θn+4 , for which we consider δ = δ(n, N, ν, L, ) > 0 determined in Lemma 2.5. Then by Lemma 4.2 together with (4.16), we have    ˆ      − ADw : Dϕ dx ≤ δ sup |Dϕ|,   Bρ  Bρ by taking sufficiently small 1 = 1 (n, N, ν, L, g1 , g2 , μ(·), θ) ∈ (0, 1). Therefore, in view of Lemma 2.5, there exists an A-harmonic map h such that

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ˆ − |Dh|2 dx ≤ 1 and Bρ

ˆ − |w − h|2 dx ≤ θn+4 ρ2 .

25

(4.19)

Bρ

We notice by a basic regularity theory for the A-harmonic maps, see for instance [19, Chapter 10], that ˆ ρ−2 sup |Dh|2 + sup |D2 h| ≤ cρ−2 − |Dh|2 dx ≤ cρ−2 . Bρ/2

Bρ/2

Bρ

Moreover, the Taylor expansion of h implies that for θ ∈ (0, 1/4), sup |h(x) − h(x0 ) − Dh(x0 )(x − x0 )|2 ≤ c(2θρ)4 sup |D2 h|2 ≤ cθ4 ρ2 .

x∈B2θρ

B2θρ

This and the second inequality in (4.19) imply that ˆ |w − h(x0 ) − Dh(x0 )(x − x0 )|2 − dx ≤ cθ2 , (2θρ)2 B2θρ

hence, by the definitions of the affine function P2θρ := Px0 ,2θρ and w and (4.15), we obtain ˆ |u − P2θρ |2 − dx (2θρ)2 B2θρ

ˆ 2 ˜ 0 , ρ) − |w − h(x0 ) − Dh(x0 )(x − x0 )| dx ≤ (1 + |(Du)ρ |)2 E(x 2 (2θρ) B2θρ

≤ c θ (1 + |(Du)ρ |) E(x0 , ρ). 2

2

(4.20)

We next estimate the second integral in (4.18). Let t ∈ (0, 1) be a number satisfying t 1 = (1 − t) + , g2 g2 d2 where d2 > 1 is given in Lemma 3.1. Then by Hölder’s inequality, Jensen’s inequality to ˜ with 1 Ψ(t) ˜ ˜ the concave map Ψ ≤ Ψ(t) := [G(t1/2 )]1/g2 ≤ Ψ(t) (see Lemma 2.2), (4.20), 2 (3.1) and (2.2), we have   ˆ |u − P2θρ | dx − G 2θρ B2θρ

⎞(1−t)g2 ⎛ ⎞ dt 2      ˆ ˆ d 2 |u − P2θρ | |u − P2θρ |2 ⎟ ⎜ ⎜ ⎟ ˜ dx⎠ ≤⎝ − Ψ dx⎠ ⎝ − G (2θρ)2 2θρ ⎛

B2θρ

B2θρ

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.  2 / ˜ θ (1 + |(Du)ρ |)2 E(x0 , ρ) (1−t)g2 ≤c Ψ

 ˆ t − G(|Du − DP2θρ |) dx B2θρ

t )  *1−t  ˆ ( − G(|Du − DP2θρ |) dx ≤ c G θ(1 + |(Du)ρ |) E(x0 , ρ) B2θρ

⎛

⎞t ˆ ( ⎟ 1−t ⎜ ≤ c [θ E(x0 , ρ)]g1 (1−t) [G(1 + |(Du)ρ |)] ⎝ − G(|Du − DP2θρ |) dx⎠ . B2θρ

In addition, we have from (2.8), (2.2), (3.1), (2.9) and the definition of E that ˆ ˆ − G(|Du − DP2θρ |) dx ≤ c − G(|Du − DPρ |) dx + c G(|DPρ − DP2θρ |) B2θρ

B2θρ

≤ cθ

−n

  ˆ ˆ |u − Pρ | dx − G(|Du − DPρ |) dx + c − G 2θρ

Bρ

ˆ −n−g2 ≤ cθ − G(|Du − DPρ |) dx

B2θρ

Bρ

ˆ ≤ c θ−n−g2 − [G(|Du − (Du)ρ |) + G(|DPρ − (Du)ρ |)] dx Bρ

≤ cθ

−n−g2

≤ cθ

−n−g2

ˆ − G(|Du − (Du)ρ |) dx

Bρ

G(1 + |(Du)ρ |)E(x0 , ρ).

Combining the two above estimates, we obtain   ˆ g1 |u − P2θρ | dx ≤ c θg1 −(n+g1 +g2 )t G(1 + |(Du)ρ |)[E(x0 , ρ)]( 2 −1)(1−t)+1 . − G 2θρ B2θρ

Therefore, taking 1 > 0 sufficiently small so that E(x0 , ρ)(

g1 2

−1)(1−t)

(

g1

≤ 1 2

−1)(1−t)

≤ θ−g1 +(n+g1 +g2 )t+2 ,

we obtain   ˆ |u − P2θρ | dx ≤ c θ2 G(1 + |(Du)ρ |)E(x0 , ρ). − G 2θρ B2θρ

(4.21)

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Moreover, by further assuming that (

E(x0 , ρ) ≤

√

1 ≤

θn , 8

we have 1 + |(Du)ρ | ≤ 2(1 + |(Du)θρ |)

and 1 + |(Du)2θρ | ≤ 2(1 + |(Du)θρ |).

(4.22)

Indeed, 1 + |(Du)ρ | ≤ 1 + |(Du)θρ | + |(Du)θρ − (Du)ρ | ( ≤ 1 + |(Du)θρ | + θ−n E(x0 , ρ)(1 + |(Du)ρ |) 1 ≤ 1 + |(Du)θρ | + (1 + |(Du)ρ |), 8 which implies the first inequality in (4.22). The second inequality in (4.22) can be obtained by the same argument along with the first inequality in (4.22) with θ replaced by 2θ such that 1 + |(Du)2θρ | ≤ 1 + |(Du)θρ | + |(Du)θρ − (Du)ρ | + |(Du)2θρ − (Du)ρ | ( ≤ 1 + |(Du)θρ | + (θ−n + (2θ)−n ) E(x0 , ρ)(1 + |(Du)ρ |) 1 ≤ 1 + |(Du)θρ | + (1 + |(Du)2θρ |). 2 Therefore, inserting the first inequality in (4.22) into (4.20) and (4.21), we obtain ˆ |u − P2θρ |2 − dx ≤ cθ2 (1 + |(Du)θρ |)2 E(x0 , ρ), (2θρ)2

(4.23)

B2θρ

  ˆ |u − P2θρ | dx ≤ cθ2 G(1 + |(Du)θρ |)E(x0 , ρ). − G 2θρ

(4.24)

B2θρ

Step 2. Now we prove (4.17). Suppose that E(x0 , ρ) ≤ 1 ≤ θn .

(4.25)

Then, in view of Lemma 4.1 with ρ replaced by θρ and P = P2θρ , we have ˆ ˆ − G2 (1 + |DP2θρ |)|Du − DP2θρ |2 dx + − G(|Du − DP2θρ |) dx Bθρ

Bθρ

  ˆ ˆ |u − P2θρ | |u − P2θρ |2 dx dx + c − G ≤ c G2 (1 + |DP2θρ |) − (2θρ)2 2θρ B2θρ

B2θρ

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⎧ ⎛ ⎫ ⎞ ⎪ ⎪ ˆ ⎨ ⎬ ⎟ ⎜ 2 + c G(1 + |DP2θρ |) μ ⎝ − |u − (u)2θρ | dx⎠ + V(2θρ) . ⎪ ⎪ ⎩ ⎭ B2θρ

1 ˜ Here, we note that G(t) := G(t 2 ) is also an N -function and satisfies (2.1) with g1 and g2 replaced by g21 and g22 which are larger than 1. Therefore, in view of (3) and (4) of ˜ Proposition 2.1 with G(t) = G(t), we have G2 (t)τ 2 ≤ c(G(t) + G(τ )). From this, (2.7) and Lemma 2.3 with (ρ, θ) replaced by (θρ, 1/2), we have

ˆ − G2 (1 + |(Du)θρ |)|Du − (Du)θρ |2 dx Bθρ

ˆ ≤ c G2 (1 + |DP2θρ |) − |Du − (Du)θρ |2 dx Bθρ

ˆ + c − G2 (|(Du)θρ − DPθρ |)|Du − (Du)θρ |2 dx Bθρ

ˆ + c − G2 (|DPθρ − DP2θρ |)|Du − (Du)θρ |2 dx Bθρ

ˆ ˆ ≤ c − G2 (1 + |DP2θρ |)|Du − DP2θρ |2 dx + c − G(|Du − (Du)θρ |) dx Bθρ

Bθρ

+ c (G(|(Du)θρ − DPθρ |) + G(|DPθρ − DP2θρ |)) ˆ ˆ ≤ c − G2 (1 + |DP2θρ |)|Du − DP2θρ |2 dx + c − G(|Du − (Du)θρ |) dx Bθρ

Bθρ

  ˆ |u − P2θρ | dx. +c − G 2θρ B2θρ

Using the above two estimates along (2.7), we obtain G(1 + |(Du)θρ |) C(x0 , θρ) ˆ ˆ = − G2 (1 + |(Du)θρ |)|Du − (Du)θρ |2 dx + − G(|Du − (Du)θρ |) dx Bθρ

Bθρ

  ˆ ˆ |u − P2θρ |2 |u − P2θρ | dx dx + c − G ≤ c G2 (1 + |DP2θρ |) − (2θρ)2 2θρ B2θρ

B2θρ

⎧ ⎛ ⎫ ⎞ ⎪ ⎪ ˆ ⎨ ⎬ ⎟ ⎜ 2 + c G(1 + |DP2θρ |) μ ⎝ − |u − (u)2θρ | dx⎠ + V(2θρ) . ⎪ ⎪ ⎩ ⎭ B2θρ

(4.26)

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We further estimate the right hand side on the above inequality. Applying (2.9), (4.25) and (4.22), we see G(|DP2θρ |) ≤ c G(|DP2θρ − (Du)2θρ |) + c G(|(Du)2θρ |) ˆ ≤ c θ−n − G(|Du − (Du)ρ |) dx + c G(|(Du)2θρ |) Bρ

≤ c (θ−n E(x0 , ρ) + 1)G(1 + |(Du)θρ |) ≤ c G(1 + |(Du)θρ |), which implies that |DP2θρ | ≤ c (1 + |(Du)θρ |). Moreover, using Poincaré’s inequality and the fact that ρ ≤ θn , we have ˆ ˆ ˆ − |u − (u)2θρ |2 dx ≤ c θ−n − |u − (u)ρ |2 dx ≤ c ρ − |Du|2 dx. B2θρ

Bρ

Bρ

Therefore, inserting the previous two inequalities, (4.23) and (4.24) into (4.26), we obtain C(x0 , θρ) ≤ c θ2 E(x0 , ρ) + c[E(x0 , ρ)]2 Finally, assuming E(x0 , ρ) ≤ 1 ≤ θ2 , we prove (4.17).

2

Now, we are ready to prove Theorem 1.1 Proof of Theorem 1.1. Fix any β ∈ (0, 1). Let us determine several constants such that λ := n − 2(1 − β) ∈ (n − 2, n), θ = θ(n, N, ν, L, g1 , g2 , β) := min

1 1 1 ,√ , 1/(n−λ) 8 2c1 3

(4.27)

! ,

(4.28)

and 2 = 2 (n, N, ν, L, g1 , g2 , μ(·), β) := min

θ n 1 , 16 2

! ,

(4.29)

where c1 and 1 are determined in Lemma 4.3. Furthermore, by the definitions of μ(·) and V(·), we can find δ1 = δ1 (n, N, ν, L, g1 , g2 , μ(·), V(·), β) > 0 such that

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1

1

[μ(r)] 2 + [V(r)] 2g2 −1 ≤ 2 for every r ∈ (0, δ1 ].

(4.30)

Then we further denote ρ1 := min {θn , δ1 } < 1.

(4.31)

Step 1. In this step, we fix Bρ = Bρ (x0 )  Ω with x0 ∈ Ω and ρ ∈ (0, ρ1 ], and suppose that ˆ C(x0 , ρ) ≤ 2 and M (x0 , ρ) := ρ − |Du|2 dx ≤ δ1 . (4.32) Bρ

Then we claim that for any k = 0, 1, 2, . . . , ˆ C(x0 , θk ρ) ≤ 2 and M (x0 , θk ρ) := θk ρ − |Du|2 dx ≤ δ1 .

(4.33)

Bθ k ρ

For the sake of convenience, for k = 0, 1, 2, . . . , we write (4.33)k,1 (resp. (4.33)k,2 ) with the first (resp. second) inequality in (4.33). We prove the claim by induction. Hence we suppose that the inequalities in (4.33) hold for k, and then prove that (4.33) holds for k replaced by k + 1. We first observe from (4.33)k,1 and Hölder’s inequality that ˆ − |Du − (Du)θk ρ |2 dx ≤ (1 + |(Du)θk ρ |)2 C(x0 , θk ρ) Bθ k ρ

⎛

⎞ ˆ ⎜ ⎟ ≤ 22 ⎝1 + − |Du|2 dx⎠ , Bθ k ρ

and so, by (4.33)k,2 , ˆ θ ρ − |Du − (Du)θk ρ |2 dx ≤ 22 θk ρ + 22 δ1 . k

Bθ k ρ

This together with (4.28), (4.29) and (4.31) implies M (x0 , θk+1 ρ) ≤ 2θk+1 ρ

ˆ −

Bθk+1 ρ

≤ 2θ

|Du − (Du)θk ρ |2 dx + 2θk+1 ρ|(Du)θk ρ |2

ˆ θ ρ − |Du − (Du)θk ρ |2 dx + 2θM (x0 , θk ρ)

1−n k

Bθ k ρ

(4.34)

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≤ 4θk+1−n 2 ρ + 4θ1−n 2 δ1 + 2θδ1 ≤ 4θ−n 2 ρ + 4θ−n 2 δ1 + 2θδ1 ≤ δ1 , which shows (4.33)k+1,2 . It remains to prove (4.33)k+1,1 . We notice from (4.14), (4.33)k,1 , (4.33)k,2 , the fact that θk ρ ≤ ρ1 ≤ δ1 by (4.31) and (4.30) that 1

1

E(x0 , θk ρ) := 2 + [μ(δ1 )] 2 + [V(δ1 )] 2g2 −1 ≤ 22 . Therefore, applying Lemma 4.3 and (4.28), we have C(x0 , θk+1 ρ) ≤ 2c1 θ2 2 ≤ 2 . This shows (4.33)k+1,1 . Then, by the induction argument, we prove that (4.33) holds for every k = 0, 1, 2, . . . . Consequently, we see that (4.34) also holds for every k = 0, 1, 2, . . . , which together with (4.29) and (4.34) implies ˆ ˆ − |Du|2 dx ≤ 2 − |Du − (Du)θk ρ |2 dx + 2|(Du)θk ρ |2 Bθk+1 ρ

Bθk+1 ρ

ˆ ˆ ≤ 2θ−n − |Du − (Du)θk ρ |2 dx + 2 − |Du|2 dx Bθ k ρ

Bθ k ρ

ˆ ≤ 4θ−n 2 + (4θ−n 2 + 2) − |Du|2 dx Bθ k ρ

ˆ −n ≤ 4θ 2 + 3 − |Du|2 dx, Bθ k ρ

and so, by (4.28), ˆ

ˆ |Du|2 dx ≤ θλ

Bθk+1 ρ

Bθ k ρ

Finally, applying Lemma 2.4 with φ(r) = ˆ |Du|2 dx ≤ c Br (x0 )

|Du|2 dx + 4|B1 |(θk ρ)n . ´ Br (x0 )

|Du|2 dx, we have for every r ∈ (0, ρ],

⎧ ⎪ ⎨ r  λ ˆ ⎪ ⎩ ρ

Bρ (x0 )

|Du|2 dx + rλ

⎞ ⎛ ˆ c ⎝ |Du|2 dx + 1⎠ rλ . ≤ λ ρ Ω

⎫ ⎪ ⎬ ⎪ ⎭ (4.35)

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Step 2. Now, we complete the proof. Define   Ωu := x0 ∈ Ω : u ∈ C β (Ux0 , RN ) for every β ∈ (0, 1) and for some Ux0 ⊂ Ω , where Ux0 is an open neighborhood of x0 . Suppose that x0 ∈ Ω satisfies ˆ lim inf −

|Du − (Du)ρ | dx = 0,

ρ↓0

Mx0

ˆ := lim sup −

G(|Du|) dx < ∞.

(4.36)

ρ↓0

Bρ (x0 )

Bρ (x0 )

Then we show that x0 ∈ Ωu . Fix β ∈ (0, 1), and set t ∈ (0, 1) such that (1 − t) 1 , =t+ g2 g2 (1 + σ1 )

(4.37)

where σ1 is determined in (1) of Theorem 3.4. We further define 

+  ,     12  (M + 2)t−1  1t 2 x 0 s := min G−1 G , δ1 < 1, 2 c2

(4.38)

where c2 = c2 (n, N, ν, L, g1 , g2 ) > 0 will be determined later. Then, in view of (4.36), one can find ρ > 0 with 



−1



2n (Mx0 + 1) +1 ρ < min ρ1 , G(1)

δ1

(4.39)

and B3ρ (x0 )  Ω such that ˆ −

ˆ −

|Du − (Du)ρ | dx < s and

Bρ (x0 )

G(|Du|) dx < Mx0 + 1.

(4.40)

B2ρ (x0 )

We first observe from Hölder’s inequality with (4.37) that ˆ − Bρ (x0 )

⎛

⎞tg2 ˆ 1 ⎜ ⎟ G(|Du − (Du)ρ |) dx ≤ ⎝ − [G(|Du − (Du)ρ |)] g2 dx⎠ Bρ (x0 )

⎛

1−t ⎞ 1+σ 1 ˆ ⎜ ⎟ 1+σ1 × ⎝ − [G(|Du − (Du)ρ |)] dx⎠ .

Bρ (x0 )

˜ with Then using Jensen’s inequality for the concave function Ψ 1 ˜ g [G(t)] 2 ≤ Ψ(t) (see Lemma 2.2) and (4.40), we have

1˜ 2 Ψ(t)

≤ Ψ(t) :=

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⎛ ˆ ˆ 1 ⎜ ˜ g2 − [G(|Du − (Du)ρ |)] dx ≤ Ψ ⎝ − Bρ (x0 )

33

⎞ 1 ⎟ |Du − (Du)ρ | dx⎠ < 2[G(s)] g2 .

Bρ (x0 )

On the other hand, using Jensen’s inequality for the convex map t → [G(t)]1+σ1 , (3.6) and (4.40), we have ˆ − [G(|Du − (Du)ρ |)]1+σ1 dx Bρ (x0 )

ˆ ≤ c − [G(|Du|)]1+σ1 dx Bρ (x0 )

⎧⎛ ⎪ ⎨ ⎜ ≤c ⎝ ⎪ ⎩

⎞1+σ1

ˆ −

⎟ G(|Du|) dx⎠

B2ρ (x0 )

⎫ ⎪ ⎬ + 1 < c (Mx0 + 2)1+σ1 . ⎪ ⎭

Therefore, we have ˆ −

G(|Du − (Du)ρ |) dx < c2 [G(s)]t (Mx0 + 2)1−t ,

Bρ (x0 )

for some c2 = c2 (n, N, ν, L, g1 , g2 ) > 0, and so by (4.38), ˆ −

G(|Du − (Du)ρ |) dx < G

  1   2 2

2

,

Bρ (x0 )

√ from which together with Jensen’s inequality for the convex map t → G( t), we have ⎡

⎛

⎞⎤2

ˆ ⎢ −1 ⎜ C(x0 , ρ) ≤ ⎣G ⎝ −

⎟⎥ G(|Du − (Du)ρ |) dx⎠⎦

Bρ (x0 )

+ [G(1)]−1

ˆ −

G(|Du − (Du)ρ |) dx

Bρ (x0 )

≤

  1  g 2  2  21 2 2 2 + [G(1)]−1 G + ≤ ≤ 2 . 2 2 2 2

Moreover, by (4.39) and (4.40), we see that ˆ M (x0 , ρ) = ρ − Bρ (x0 )

⎛ ⎜ 2n |Du|2 dx ≤ ρ ⎝ G(1)

ˆ − B2ρ (x0 )

⎞ ⎟ G(|Du|) dx + 1⎠ < δ1 .

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In addition, by the continuity of integrals, there exists  > 0 such that (4.40) with x0 replaced by y holds for every y ∈ B (x0 ). With out loss of generality, we can assume that  ≤ ρ, which yields the previous two estimates with x0 replaced by y hold for every y ∈ B (x0 ). Therefore, in view of Step 1, we see that (4.35) with x0 replaced by y holds for every y ∈ B (x0 ) and t ≤ ρ. Finally, by Morrey–Campanato’s embedding theorem, we have u is C β (B (x0 ), RN ), that is, x0 ∈ Ωu . 2 5. Partial Hölder regularity for double phase problems We now prove Theorem 1.2, hence suppose that H : Ω ×[0, ∞) → [0, ∞) is denoted by (1.11) with (1.12) and (1.13), b : Ω × RnN → RnN satisfies (1.15), (1.16) and (1.10), and u ∈ W 1,1 (Ω, RN ) with H(·, |Du|) ∈ L1 (Ω) is a weak solution to (1.14). The procedure of the proof is almost same as the one in the previous section. We fix B4ρ (x0 )  Ω with x0 ∈ Ω and 4ρ ≤ 1. Let K ≥ 1 be a sufficiently large number which will be determined later, see (5.20). Before start the proof, we shall define h0 := max{h1 , h2 },

(5.1)

where  p(q−1)   p−1 p(q − 1) h1 := h K + 2, p−1

q−1

and h2 := [h (K + 2, q − 1)]

.

(5.2)

Here, the constant h(K, q˜) > 0 is given in Lemma 3.6. Lemma 5.1. Let P : Rn → RN ba any affine function. Suppose that ˆ −

H(x0 , |Du − DP |) dx ≤ H(x0 , 1 + |DP |).

B4ρ (x0 )

Then we have 

ˆ − Bρ (x0 )

≤c

 |Du − DP |2 H(x0 , |Du − DP |) dx + (1 + |DP |)2 H(x0 , 1 + |DP |) 

ˆ −

 |u − P |2 H(x0 , |u − P |/(2ρ)) dx dx + (2ρ)2 (1 + |DP |)2 H(x0 , 1 + |DP |)

B2ρ (x0 )

+c

  p(q−p) pα−n(q−p) 1 + h1 DuLp (Ω) + 1 p−1 ρ p−1 K

for some c = c(n, N, ν, L, p, q, [a]C α ) > 0.

!

(5.3)

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Proof. Set ϕ = η q (u − P ), where η ∈ C0∞ (B2ρ ) with 0 ≤ η ≤ 1, η ≡ 1 on Bρ and |Dη| ≤ c(n)/ρ. Taking ϕ as a test function in (1.14) and using (1.18), we have ˆ   1 − η q H2 (x0 , 1 + |DP |)|Du − DP |2 + H(x0 , |Du − DP |) dx c

(5.4)

B2ρ

ˆ ≤ − (b(x0 , Du) − b(x, Du)) : Dϕ dx B2ρ

ˆ − q − η q−1 (b(x0 , Du) − b(x0 , DP )) : Dη ⊗ (u − P ) dx B2ρ

=: Ia − Ib . Estimation Ia : Using (1.16), we have   ˆ |u − P | α q−1 q dx η |Du − DP | + |Ia | ≤ c − [a]C α ρ (1 + |Du|) ρ

(5.5)

B2ρ

We first assume that inf a(x) ≤ K[a]C α (4ρ)α ,

x∈B4ρ

(5.6)

which yields sup a(x) ≤ K[a]C α (4ρ)α + [a]C α (8ρ)α ≤ (K + 2)[a]C α (4ρ)α

x∈B4ρ

Then, in view of Lemma 3.6 with q˜ =

p(q−1) p−1

and (5.2), we have

q−1 ⎛ ⎞ p−1 ˆ ˆ p(q−1) ⎜ ⎟ − (1 + |Du|) p−1 dx ≤ h1 ⎝ − (1 + |Du|)p dx⎠ .

B2ρ

Here, we note that to (5.5), we have

B4ρ

p(q−1) p−1

< 2q − p since 2 < p ≤ q. Applying this and Young’s inequality

 ˆ  1 |u − P |p q p dx |Ia | ≤ − η |Du − DP | + 4 ρp B2ρ

ˆ p(q−1) pα + cρ p−1 − (1 + |Du|) p−1 dx B2ρ

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≤

  ˆ ˆ 1 |u − P | − η q H(x0 , |Du − DP |) dx + − H x0 , dx 4 ρ B2ρ

B2ρ

⎛ pα

+ c h1 ρ p−1

q−1 ⎞ p−1 ˆ ⎜ ⎟ p ⎝ − (1 + |Du|) dx⎠

B4ρ

Moreover, we see from (5.3) that ⎛ ⎞ ˆ ˆ ⎜ ⎟ − H(x0 , 1 + |Du|) dx ≤ c ⎝H(x0 , 1 + |DP |) + − H(x0 , |Du − DP |) dx⎠ B4ρ

B4ρ

≤ cH(x0 , 1 + |DP |),

(5.7)

and so ⎛

q−1 ⎞ p−1 ˆ ⎜ ⎟ p ⎝ − (1 + |Du|) dx⎠

B4ρ

≤ cρ−

n(q−p) p−1

ˆ   p(q−p) DuLp (Ω) + 1 p−1 − H(x0 , 1 + |Du|) dx B4ρ

≤ cρ−

n(q−p) p−1

  DuLp (Ω) + 1

p(q−p) p−1

H(x0 , 1 + |DP |).

Therefore, we obtain |Ia | ≤

  ˆ ˆ 1 |u − P | − η q H(x0 , |Du − DP |) dx + − H x0 , dx 4 ρ B2ρ

B2ρ

+ c h1 ρ

pα−n(q−p) p−1



 p(q−p) DuLp (Ω) + 1 p−1 H(x0 , 1 + |DP |).

Now, we assume that (5.6) is not true, that is, inf a(x) > K[a]C α (4ρ)α .

x∈B4ρ

(5.8)

Then, using this, (2.4) with G(t) = H(x0 , t) and (2.6), and (5.7), we have from (5.5) that   ˆ |u − P | dx |Ia | ≤ K −1 − H1 (x0 , 1 + |Du|) η q |Du − DP | + ρ B2ρ

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≤

37

  ˆ ˆ 1 |u − P | − η q H(x0 , |Du − DP |) dx + − H x0 , dx 4 ρ B2ρ

+

B2ρ

c H(x0 , 1 + |DP |). K

Consequently, in any case (5.6) and (5.8), we have   ˆ ˆ 1 |u − P | q dx |Ia | ≤ − η H(x0 , |Du − DP |) dx + − H x0 , 4 ρ B2ρ

+c

B2ρ

h1 ρ

pα−n(q−p) p−1



 p(q−p) p−1

DuLp (Ω) + 1

1 + K

! H(x0 , 1 + |DP |).

Estimation Ib : From the first inequality in (1.15) and (2.5) with G(t) = H(x0 , t), we have ˆ |u − P | dx |Ib | ≤ c − H2 (x0 , 1 + |DP | + |Du − DP |)|Du − DP |η q−1 ρ B2ρ

ˆ |u − P | dx ≤ cH2 (x0 , 1 + |DP |) − η q−1 |Du − DP | ρ B2ρ

ˆ |u − P | dx + c − H1 (x0 , |Du − DP |)η q−1 ρ B2ρ

ˆ / . 1 ≤ − η q H2 (x0 , 1 + |DP |)|Du − DP |2 + H(x0 , |Du − DP |) dx 4 B2ρ

⎞   ˆ ˆ 2 |u − P | |u − P | ⎟ ⎜ dx⎠ . + c ⎝H2 (x0 , 1 + |DP |) − dx + − H x0 , 2 ρ ρ ⎛

B2ρ

B2ρ

Finally, inserting the above two estimates into (5.4), we get the desired estimate.

2

Now we shall define C(x0 , ρ, P ) :=

ˆ −



 |Du − DP |2 H(x0 , |Du − DP |) dx + (1 + |DP |)2 H(x0 , 1 + |DP |)

Bρ (x0 )

and E(x0 , ρ, P ) := C(x0 , ρ, P ) + K −1 + h0 (DuLp (Ω) + 1) where

p(q−p) p−1

ργ ,

(5.9)

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pα − n(q − p) pα − n(q − p) > =: γ > 0. p−1 p

(5.10)

Furthermore, let us set A :=

u−P ∂b(x0 , DP ) ( and w := . H2 (x0 , 1 + |DP |) (1 + |DP |) E(x0 , ρ, P )

Then we see from (1.15) that A satisfies (2.10). Lemma 5.2. Suppose that (5.3) holds. There exists c = c(n, N, ν, Λ, p, q, [a]C α ) > 0 such that     ˆ   ) ( * 12   sup |Dϕ| (5.11)  − ADw : Dϕ dx ≤ c μ( E(x0 , ρ, P )) + E(x0 , ρ, P )   Bρ (x0 )  Bρ (x0 ) for every ϕ ∈ C0∞ (Bρ (x0 )). Proof. We first note that many part of the proof is same as the proof of Lemma 4.2. For any ϕ ∈ C0∞ (Bρ (x0 )) with |Dϕ| ≤ 1 we have H1 (x0 , 1 + |DP |)

(

ˆ E(x0 , ρ, P ) − ADw : Dϕ dx Bρ

ˆ ˆ1 = − (∂b(x0 , DP ) − ∂b(x0 , DP + sDv))Dv : Dϕ ds dx Bρ 0

ˆ ˆ1 + − ∂b(x0 , DP + sDv)Dv : Dϕ ds dx Bρ 0

=: Ic + Id ,

(5.12)

see (4.11). Moreover, from the same way to estimate |I5 | in (4.12) with G(t) = H(x0 , t) we see that ! ) ( * 12 ( |Ic | ≤ c E(x0 , ρ, P ) + μ( E(x0 , ρ, P )) E(x0 , ρ, P ) . (5.13) H1 (x0 , 1 + |DP |) Hence, it remains to estimate Id . Note that it is different from the estimation of I6 in the proof of Lemma 4.2. Applying (1.16), we have ˆ ˆ Id = − (b(x0 , Du) − b(x, Du)) : Dϕ dx ≤ c − [a]C α ρα (1 + |Du|)q−1 dx. Bρ

Bρ

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Then, we first assume that inf a(x) ≤ K[a]C α (2ρ)α .

(5.14)

x∈B2ρ

This yields sup a(x) ≤ (K + 2)[a]C α (2ρ)α ,

x∈B2ρ

and so, by Lemma 3.6 with q˜ = q − 1 and (5.2), we have ⎛ ⎞ q−1 p ˆ ˆ ⎜ ⎟ ρα − (1 + |Du|)q−1 dx ≤ h2 ρα ⎝ − (1 + |Du|)p dx⎠ Bρ

B2ρ

⎛

≤ c h2 ρα−

(q−p)n p

⎞ p−1 p ˆ q−p ⎜  ⎟ p . DuLp (Ω) + 1 ⎝ − (1 + |Du|) dx⎠ B2ρ q

Observe that from Jensen’s inequality for a convex function F (t) := t + a(x0 )t p and (5.3) with (5.7), ⎛ ⎞ ˆ ˆ ⎜ ⎟ − (1 + |Du|)p dx ≤ F −1 ⎝ − H(x0 , 1 + |Du|) dx⎠ B2ρ

B2ρ

≤ c F −1 (H(x0 , 1 + |DP |)) = c (1 + |DP |)p . Therefore, we have |Id | ≤ c h2 (DuLp (Ω) + 1)q−p ρα−

(q−p)n p

H1 (x0 , 1 + |DP |).

On the other hand, if (5.14) is not true, then we have [a]C α ρα ≤

a(x0 ) , K

from which we see that ˆ c − H1 (x0 , 1 + |Du|) dx |Id | ≤ K Bρ

≤

 ˆ  H1 (x0 , |Du − DP |) c H1 (x0 , 1 + |DP |) − 1+ dx K H1 (x0 , 1 + |DP |) Bρ

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⎛

⎞ ˆ c H1 (x0 , 1 + |DP |) ⎜ H(x0 , |Du − DP |) ⎟ dx⎠ . ≤ ⎝1 + − K H(x0 , 1 + |DP |) Bρ

Therefore, from the results with (5.10), we obtain 0 1 (q−p)n |Id | ≤ c C(x0 , ρ, P ) + K −1 + h2 (DuLp (Ω) + 1)q−p ρα− p H1 (x0 , 1 + |DP |) ≤ c E(x0 , ρ, P ).

(5.15)

Finally, inserting (5.13) and (5.15) into (5.12), we have    ˆ    ) ( * 12    − ADw : Dϕ dx ≤ c μ( E(x0 , ρ, P )) + E(x0 , ρ, P )   Bρ (x0 )  for all ϕ ∈ C0∞ (Bρ (x0 )) with |Dϕ| ≤ 1, which yields (5.11). 2 Now, putting P = (Du)x0 ,ρ (x − x0 ) + (u)x0 ,ρ , we set C(x0 , ρ) := C(x0 , ρ, (Du)x0 ,ρ (x − x0 ) + (u)x0 ,ρ )  ˆ  H(x0 , |Du − (Du)x0 ,ρ |) |Du − (Du)x0 ,ρ |2 dx, + = − (1 + |(Du)x0 ,ρ |)2 H(x0 , 1 + |(Du)x0 ,ρ |) Bρ

and E(x0 , ρ) := C(x0 , ρ) + K −1 + h0 (DuLp (Ω) + 1)

p(q−p) p−1

ργ .

(5.16)

Lemma 5.3. For θ ∈ (0, 1/8), there exists small 3 = 3 (n, N, ν, L, p, q, [a]C α , μ(·), θ) ∈ (0, 1) such that if E(x0 , ρ) ≤ 3 , then C(x0 , θρ) ≤ c3 θ2 E(x0 , ρ) for some c3 = c3 (n, N, ν, L, p, q, [a]C α , DuLp ) ≥ 1.

(5.17)

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Proof. The proof is exactly same as the one of Lemma 4.3 for the case that G(t) = H(x0 , t). Note that if G(t) = H(x0 , t) one can take g1 = p and g2 = q in (1.1), and that the condition (5.17) implies (5.3). 2 Now, we are ready to prove Theorem 1.2 Proof of Theorem 1.2. We follows the same argument in the proof of Theorem 1.1. Hence, we shall omit the details of parts which are exactly same to ones in the proof of Theorem 1.1. Fix β ∈ (0, 1) and define λ = λ(β) ∈ (n − 2, n) as in (4.27), θ = θ(n, N, ν, L, p, q, [a]C α , DuLp , β) := min

1 1 1 ,√ , 1/(n−λ) 8 2c3 3 θ n 3 , 16 2

4 = 4 (n, N, ν, L, p, q, [a]C α , DuLp , μ(·), β) := min

! ,

(5.18)

,

(5.19)

!

and choose K = K(n, N, ν, L, p, q, [a]C α , DuLp , μ(·), β) =

2 . 3

(5.20)

Here, c3 and 3 is determined in Lemma 5.3. Moreover, one can find δ2 = δ2 (n, N, ν, L, p, q, μ(·), DuLp (Ω) , β) > 0 such that h0 (DuLp (Ω) + 1)

p(q−p) p−1

δ2γ ≤

4 , 2

(5.21)

where h0 and γ are dented in (5.1) and (5.10), respectively. Then we further denote ρ2 := min {4 , δ2 } . Now fix Bρ = Bρ (x0 )  Ω with x0 ∈ Ω and ρ ∈ (0, ρ2 ], and suppose that C(x0 , ρ) ≤ 4 .

(5.22)

Then we see by the induction argument that C(x0 , θk ρ) ≤ 4 for all k = 0, 1, 2, . . . .

(5.23)

Indeed, assume that (5.23) holds. Then by (5.16), (5.21) along with the fact that θk ρ ≤ ρ2 ≤ δ1 , (5.23) and (5.19) that E(x0 , θk ρ) ≤ 4 + K −1 + h0 (DuLp (Ω) + 1) Therefore, applying Lemma 5.3 along with (5.19), we have C(x0 , θk+1 ρ) ≤ 2c3 θ2 4 ≤ 4 .

p(q−p) p−1

δ2γ ≤ 24 .

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This shows (4.33) with k replaced by k + 1. Moreover, by Hölder’s inequality we have from (5.23) that ˆ − |Du − (Du)θk ρ |2 dx ≤ (1 + |(Du)θk ρ |)2 C(x0 , θk ρ) Bθ k ρ

⎛

⎞ ˆ ⎜ ⎟ ≤ 24 ⎝1 + − |Du|2 dx⎠ , Bθ k ρ

which together with (5.18) yields (4.35). Now we define   Ωu := x0 ∈ Ω : u ∈ C β (Ux0 , RN ) for every β ∈ (0, 1) and for some Ux0 ⊂ Ω , where Ux0 is an open neighborhood of x0 . Suppose that x0 ∈ Ω satisfies ˆ ˆ lim inf − |Du − (Du)ρ | dx = 0, Mx0 := lim sup − H(x, |Du|) dx < ∞. (5.24) ρ↓0

ρ↓0

Bρ (x0 )

Bρ (x0 )

We then show that x0 ∈ Ωu . We fix β ∈ (0, 1), and set t ∈ (0, 1) such that (1 − t) 1 =t+ , q q(1 + σ2 ) where σ2 is determined in (2) of Theorem 3.4. We further define    pt1    p2 1 4 s := min , δ1 < 1, 2 c4 (Mx0 + 2)1−t

(5.25)

(5.26)

where c4 > 0 will be determined later. Then in view of (4.36), one can find ρ > 0 with ρ < ρ2 and B3ρ (x0 )  Ω such that ˆ ˆ − |Du − (Du)ρ | dx < s and − H(x, |Du|) dx < Mx0 + 1. (5.27) Bρ (x0 )

B2ρ (x0 )

We then observe from Hölder’s inequality with (5.25) that ⎛ ⎞tq ˆ ˆ 1 ⎜ ⎟ − H(x0 , |Du − (Du)ρ |) dx ≤ ⎝ − [H(x0 , |Du − (Du)ρ |)] q dx⎠ Bρ (x0 )

Bρ (x0 )

⎛

1−t ⎞ 1+σ 2 ˆ ⎜ ⎟ 1+σ2 × ⎝ − [H(x0 , |Du − (Du)ρ |)] dx⎠ .

Bρ (x0 )

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˜ with 1 Ψ(t) ˜ Then, using Jensen’s inequality for the concave function Ψ ≤ Ψ(t) := 2 1 ˜ q [H(x0 , t)] ≤ Ψ(t), see Lemma 2.2, and the first inequality in (5.27), we have ˆ 1 1 1 − [H(x0 , |Du − (Du)ρ |)] q dx ≤ 2[H(x0 , s)] q ≤ 2[(1 + aL∞ )sp ] q . Bρ (x0 )

On the other hand, using Jensen’s inequality for the convex map t → [H(x0 , t)]1+σ2 , we have ˆ ˆ 1+σ2 − [H(x0 , |Du − (Du)ρ |)] dx ≤ c − [H(x0 , |Du|)]1+σ2 dx. Bρ (x0 )

Bρ (x0 )

At this point, if a(·) satisfies (5.6), we have from (3.11) that ˆ −

⎛ ⎜ |Du|p(1+σ2 ) dx ≤ c ⎝

Bρ (x0 )

⎞1+σ2

ˆ −

⎟ |Du|p dx + 1⎠

B2ρ (x0 )

and ˆ − [a(x0 )|Du|q ]1+σ2 dx Bρ (x0 )

⎛ ⎜ ≤ c ρα(1+σ2 ) ⎝

ˆ −

2) ⎞ q(1+σ p

⎟ |Du|p dx + 1⎠

B2ρ (x0 ) 

≤ cρ

⎛



α− (q−p)n (1+σ2 ) p

(q−p)(1+σ2 )

DuLp

⎜ ⎝

ˆ −

⎞1+σ2 ⎟ |Du|p dx + 1⎠

.

B2ρ (x0 )

Here, we may assume that q(1 + σ2 ) < 2q − p without loss of generality. Using these, the fact that ρ ≤ 1 and the second inequality in (5.27), ⎛ ˆ ⎜ − [H(x0 , |Du|)]1+σ2 dx ≤ c ⎝ Bρ (x0 )

ˆ −

⎞1+σ2 ⎟ H(x, |Du|) dx + 1⎠

B2ρ (x0 )

≤ c(Mx0 + 2)1+σ2 . In addition, if (5.6) is not true then we see that  H(x0 , t) ≤

1+

1 K

 H(x, t) ≤ 2H(x, t),

(5.28)

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hence by (3.8) and the second inequality in (5.27) we have (5.28). Consequently, combining the above results, we have ˆ −

H(x0 , |Du − (Du)ρ |) dx < c4 spt (Mx0 + 2)1−t

Bρ (x0 )

for some c4 > 0 depending only on n, N, ν, L, p, q, [a]C α , aL∞ and DuLp , which together with (5.26) implies ˆ −

H(x0 , |Du − (Du)ρ |) dx <

  1  4 2 . ≤ H x0 , 2

   p2 4

2

Bρ (x0 )

This together with Jensen’s inequality for the convex map t → H(x0 , C(x0 , ρ) ≤

ˆ −

ˆ |Du − (Du)ρ | dx + − 2

Bρ (x0 )

⎛

⎛

ˆ ⎜ ⎜ ≤ ⎝[H(x0 , ·)]−1 ⎝ − ˆ + −

√

t) implies

H(x0 , |Du − (Du)ρ |) dx

Bρ (x0 )

⎞⎞2

⎟⎟ H(x0 , |Du − (Du)ρ |) dx⎠⎠

Bρ (x0 )

H(x0 , |Du − (Du)ρ |) dx

Bρ (x0 )

 4  4  2 + ≤ 4 , 2 2 p

≤

which shows (5.22). Consequently, we obtain (4.35) for all r ∈ (0, ρ). In addition, due to the continuity of integrals, there exists  > 0 with  < ρ such that (5.27) with x0 replaced by y holds for every y ∈ B (x0 ). Therefore, we see that (4.35) with x0 replaced by y holds for every y ∈ B (x0 ) and r ≤ ρ, which implies u ∈ C β (B (x0 ), RN ) by Morrey–Campanato’s embedding theorem, and so x0 ∈ Ωu . 2 Acknowledgments J. Ok was supported by the National Research Foundation of Korea funded by Korean Government (NRF-2017R1C1B2010328). References [1] E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 (2) (2001) 121–140. [2] P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations 53 (3–4) (2015) 803–846.

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[3] P. Baroni, M. Colombo, G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015) 206–222. [4] P. Baroni, M. Colombo, G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (3) (2016) 347–379. [5] S. Byun, J. Ok, On W 1,q(·) -estimates for elliptic equations of p(x)-Laplacian type, J. Math. Pures Appl. (9) 106 (3) (2016) 512–545. [6] S. Campanato, Partial Hölder continuity of solutions of quasilinear parabolic systems of second order with linear growth, Rend. Semin. Mat. Univ. Padova 64 (1981) 59–75. [7] S. Campanato, On the nonlinear parabolic systems in divergence form Hölder continuity and partial Hölder continuity of the solutions, Ann. Mat. Pura Appl. (4) 137 (1984) 83–122. [8] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (4) (2006) 1383–1406. [9] A. Cianchi, V. Maz’ya, Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems, J. Eur. Math. Soc. (JEMS) 16 (3) (2014) 571–595. [10] M. Colombo, G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2) (2015) 443–496. [11] M. Colombo, G. Mingione, Bounded minimizers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (1) (2015) 219–273. [12] M. Colombo, G. Mingione, Calderón–Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal. 270 (4) (2016) 1416–1478. [13] A. Coscia, G. Mingione, Hölder continuity of the gradient of p(x)-harmonic mappings, C. R. Acad. Sci. Paris Sér. I Math. 328 (4) (1999) 363–368. [14] L. Diening, F. Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math. 20 (3) (2008) 523–556. [15] L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math., vol. 2017, Springer-Verlag, Berlin, 2011. [16] F. Duzaar, K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. 546 (2002) 73–138. [17] M. Foss, G. Mingione, Partial continuity for elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (3) (2008) 471–503. [18] F. Giannetti, A. Passarelli di Napoli, Regularity results for a new class of functionals with nonstandard growth conditions, J. Differential Equations 254 (3) (2013) 1280–1305. [19] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003. [20] E. Giusti, M. Miranda, Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasilineari, Arch. Ration. Mech. Anal. 31 (1968/1969) 173–184. [21] J. Habermann, Partial regularity for nonlinear elliptic systems with continuous growth exponent, Ann. Mat. Pura Appl. (4) 192 (3) (2013) 475–527. [22] P. Harjulehto, P. Hästö, Boundary regularity under generalized growth conditions, preprint. [23] P. Harjulehto, P. Hästö, Ú.V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (12) (2010) 4551–4574. [24] P. Harjulehto, P. Hästö, R. Klén, Generalized Orlicz spaces and related PDE, Nonlinear Anal. 143 (2016) 155–173. [25] P. Harjulehto, P. Hästö, O. Toivanen, Hölder regularity of quasiminimizers under generalized growth conditions, Calc. Var. Partial Differential Equations 56 (2) (2017), 55:22. [26] M. Kronz, Partial regularity results for minimizers of quasiconvex functionals of higher order, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2) (2002) 81–112. [27] J. Kristensen, G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (3) (2006) 331–398. [28] J. Kristensen, G. Mingione, The singular set of Lipschitzian minima of multiple integrals, Arch. Ration. Mech. Anal. 184 (2007) 341–369. [29] J. Kristensen, G. Mingione, Boundary regularity in variational problems, Arch. Ration. Mech. Anal. 198 (2) (2010) 369–455. [30] G.M. Lieberman, The natural generalization of the natural conditions of Ladyzenskaja and Uralâtzeva for elliptic equations, Comm. Partial Differential Equations 16 (2–3) (1991) 311–361. [31] G.M. Lieberman, On the regularity of the minimizer of a functional with exponential growth, Comment. Math. Univ. Carolin. 33 (1992) 45–49. [32] G.M. Lieberman, Gradient estimates for a new class of degenerate elliptic and parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 21 (1994) 497–522.

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[33] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions, Arch. Ration. Mech. Anal. 105 (1989) 267–284. [34] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations 90 (1991) 1–30. [35] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993) 296–333. [36] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996) 1–25. [37] G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math. 51 (4) (2006) 355–426. [38] C.B. Morrey, Partial regularity results for non-linear elliptic systems, J. Math. Mech. 17 (1967/1968) 649–670. [39] J. Ok, Gradient estimates for elliptic equations with Lp(·) log L growth, Calc. Var. Partial Differential Equations 55 (2) (2016), 55:26. [40] J. Ok, Partial continuity for a class of elliptic systems with non-standard growth, Electron. J. Differential Equations 2016 (323) (2016) 1–24. [41] J. Ok, Regularity of ω-minimizers for a class of functionals with non-standard growth, preprint. [42] M. Røužička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. [43] V.V. Zhikov, On Lavrentiev’s phenomenon, Russ. J. Math. Phys. 3 (2) (1995) 249–269. [44] V.V. Zhikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.