Weighted Orlicz estimates for general nonlinear parabolic equations over nonsmooth domains

Weighted Orlicz estimates for general nonlinear parabolic equations over nonsmooth domains

Journal of Functional Analysis 272 (2017) 4103–4121 Contents lists available at ScienceDirect Journal of Functional Analysis www.elsevier.com/locate...

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Journal of Functional Analysis 272 (2017) 4103–4121

Contents lists available at ScienceDirect

Journal of Functional Analysis www.elsevier.com/locate/jfa

Weighted Orlicz estimates for general nonlinear parabolic equations over nonsmooth domains ✩ Sun-Sig Byun a , Seungjin Ryu b,∗ a

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea b Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea

a r t i c l e

i n f o

Article history: Received 24 August 2015 Accepted 31 January 2017 Available online 8 February 2017 Communicated by Cédric Villani MSC: primary 35K55 secondary 35R05 Keywords: Calderón–Zygmund type estimate Weighted Orlicz space Nonlinear parabolic problem Parabolic Muckenhoupt weight

a b s t r a c t We introduce a parabolic analogue of Muckenhoupt weights to study optimal weighted regularity in Orlicz spaces for a general nonlinear parabolic problem of p-Laplacian-type in divergence form over a nonsmooth domain. Assuming that the nonlinearity is measurable with respect to the time variable and has a small bounded mean oscillation (BMO) with respect to the spatial variables, that the lateral boundary of the parabolic cylinder is δ-Reifenberg flat and that the associated weight belongs to a suitable parabolic Muckenhoupt class, we obtain a global gradient estimate for such a nonlinear parabolic problem by essentially proving that the gradient of the weak solution is as globally integrable as the nonhomogeneous term in the weighted Orlicz space. Our results extend the existing regularity estimates in Lebesgue spaces to weighted Orlicz spaces. © 2017 Elsevier Inc. All rights reserved.

✩ S. Byun was supported by NRF grant (No. 2015R1A2A1A15053024). S. Ryu was partially supported by NRF grant (No. 2015R1A4A1041675). * Corresponding author. E-mail addresses: [email protected] (S.-S. Byun), [email protected] (S. Ryu).

http://dx.doi.org/10.1016/j.jfa.2017.01.024 0022-1236/© 2017 Elsevier Inc. All rights reserved.

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1. Introduction In this paper, we establish an optimal Calderón–Zygmund theory for a general nonlinear parabolic equation of p-Laplacian type over a nonsmooth domain. Since the celebrated Calderón–Zygmund estimate was first introduced in [15], it has played an important role in the regularity theory of elliptic and parabolic equations. Indeed, much research has been devoted to the study of improving and generalizing this classical theory in the area of partial differential equations, functional analysis and geometric analysis, see [1,13,10,16,24–26,28,31,35] and references therein. A fundamental result for nonlinear gradient estimates was obtained by Iwaniec who obtained a local Calderón–Zygmund estimate for a nonhomogeneous elliptic p-Laplacian equation. This result was extended to the system by DiBenedetto and Manfredi in [18]. The estimate for the p-Laplacian equation in [23] was also extended by Caffarelli and Peral in [14] for a wider class of general elliptic equations of p-Laplacian type. The elliptic estimates in [23,18,14] were extended by Acerbi and Mingione in [2] to the parabolic case. More precisely, it was shown that |F | ∈ Lqloc

=⇒

|Du| ∈ Lqloc ,

∀q ≥ p,

(1.1)

holds for weak solutions of ut − div(|Du|p−2 Du) = div(|F |p−2 F ),

p>

2n . n+2

Since then, research has been focused on generalizing the Calderón–Zygmund estimate in the classical Lebesgue space to a wider class of nonlinear equations that are not necessarily of variational form in various function spaces such as the Lorentz space, Morrey space, Orlicz space, and weighted space (see [3,4,13,8,11,12,19,29,30,33,34,32,38,42,43] and the references therein). Recently, a non-weighted Calderón–Zygmund theory with nonhomogeneous boundary data has been established by Bögelein [5]. We are particularly interested in two recent papers [11,8]. In [11] the Calderón–Zygmund estimates in the weighted Lebesgue space were obtained for elliptic equations of non-variational form. In [8] the Calderón–Zygmund estimates in Orlicz spaces were obtained for parabolic equations of non-variational form. Note that the Orlicz space is a natural generalization of the Lebesgue space; for instance, the Lebesgue space is the special case when the associated Young function is given by Φ(ρ) = ρq , 1 < q < ∞. Here, our goal is to derive the global Calderón–Zygmund estimate in weighted Orlicz space for parabolic equations of nonvariational form under possibly optimal regularity requirements on the nonlinearity, the underlying boundary, the associated Young function and the parabolic weight, thereby finding a complete extension of [11,8] to the setting of weighted Orlicz spaces. A main point of this paper is that we deal with a nonlinear parabolic problem to find an optimal weighted estimate in Orlicz spaces. For the elliptic case, the authors of [11,30] obtained the Calderón–Zygmund type estimates when the nonhomogeneous term

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belongs to a weighted Lebesgue space with the weight in Muckenhoupt’s class. Recently, we extended the relevant weighted estimates in Lebesgue spaces to those in the weighted Orlicz spaces, see [12]. The results in [11,30] were extended to the parabolic case in [9], where the weight depends only on the spatial variables but it is independent of the time variable. Here we allow the weight to be time dependent. In this case, it is not easy to find a correct parabolic version of the elliptic one, since compared to the elliptic case, the parabolic problem of p-Laplacian type does not satisfy the scaling invariance property, namely, the solution is no longer invariant under multiplication by a constant. Here we want to find a parabolic version of the elliptic one in [12]. Another point of the paper is that we consider a general nonlinearity and a very irregular domain. The nonlinearity is assumed to be merely measurable in the time variable and have a small bounded mean oscillation (BMO) semi-norm in the spatial variables. Regarding the bounded domain, we suppose that its boundary is Reifenberg flat [36]. This flatness condition is a sort of minimal geometric requirement on the boundary for some natural properties in geometric analysis and potential theory [40]. We will specify a minimal regularity assumption on the nonlinearity and a very general geometric assumption on the boundary later in Definition 2.2. Our paper is organized as follows. In Section 2, we introduce the parabolic problem, weighted Orlicz space and the main assumptions for the weighted Orlicz regularity. Some auxiliary lemmas are presented in Section 3. The last section is devoted to deriving the desired Calderón–Zygmund type estimate in the weighted Orlicz space. 2. Assumptions and main result Let

2n n+2

< p < ∞. We consider the following nonlinear parabolic problem: 

ut − div a(Du, x, t) = div (|F |p−2 F ) u =0

in ΩT , on ∂p ΩT .

(2.1)

Here, Ω is a bounded domain in Rn , n ≥ 2, with nonsmooth boundary ∂Ω, ΩT = Ω × (0, T ) is a cylindrical domain in Rn ×R for some T > 0. As usual, ∂p ΩT = (∂Ω × [0, T ])∪ (Ω × {t = 0}) denotes the parabolic boundary of ΩT . We let ut = ∂u ∂t and Du = Dx u denote the spatial gradient vector of u = u(x, t). F : ΩT −→ Rn is a given vector-valued function. Note that the divergence operator, div, acts on the spatial variables x only. Throughout this paper, the nonlinearity   a = a(ξ, x, t) = a1 (ξ, x, t), · · · , an (ξ, x, t) : Rn × Rn × R → Rn , is assumed to be of Carathéodory type; that is, it is measurable in (x, t) for every ξ ∈ Rn and continuous with respect to ξ for each fixed (x, t). We further assume that

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γ|ξ|p−2 |η|2 ≤ Dξ a(ξ, x, t)η, η ,

(2.2)

|a(ξ, x, t)| + |ξ| |Dξ a(ξ, x, t)| ≤ Λ|ξ|p−1

(2.3)

hold for almost every (x, t) ∈ Rn × R, for all ξ, η ∈ Rn and for constants γ, Λ > 0. As usual, we consider u in the classical parabolic Sobolev space   weak solutions   C 0 [0, T ]; L2 (Ω) ∩ Lp 0, T ; W01,p (Ω) satisfying the following weak integral formula: ˆ

ˆ uϕt dxdt − ΩT

ˆ

a(Du, x, t), Dϕ dxdt =

ΩT

|F |p−2 F, Dϕ dxdt, ΩT

for every test function ϕ ∈ C0∞ (ΩT ), where ϕ = 0 when t = T . According to classical theory, the problem (2.1) has a unique weak solution if F ∈ Lp (ΩT , Rn ) with the estimate sup u(·, t) L2 (Ω) + Du Lp (ΩT ) ≤ c F Lp (ΩT ) ,

(2.4)

0
where the constant c depends only γ, Λ, n, and p, see [7,37]. Because the weak solutions considered herein are not differentiable with respect to t, we use the following Steklov average. Given a function f ∈ L1 (ΩT ), the Steklov average fl of f is defined by  ´ t+l 1 fl (x, t) =

l

t

f (x, s)ds,

t ∈ (0, T − l) t>T −l

0,

for 0 < l < T . Therefore, we take the test function ϕ = u, which is possible in light of the Steklov average, since u = 0 on ∂p ΩT . We refer to [17, Chapter 1] for more details regarding the concept and properties of the Steklov average. Our goal is to obtain a global gradient estimate of the weak solution of (2.1) in weighted Orlicz spaces. To do this, we first introduce Muchenhoupt’s weight, which is adapted for the nonlinear parabolic setting. For every point z = (y, s) ∈ Rn+1 and for every scales ρ and θ, Q(ρ,θ) (z) = Bρ (y) × (s − θ, s + θ)

(2.5)

is the circular cylinder of radius r, height θ, and center point z = (y, s). Note that 2Q(ρ,θ) (z) = B2ρ (y) × (s − 4θ, s + 4θ). A positive locally integrable function w on Rn+1 is called a weight. We say that this weight w = w(x, t) belongs to a parabolic Aq class for some q > 1, denoted by w ∈ Aq , if ⎛ sup ⎝

1 |Q|

ˆ Q

⎞⎛ w(x, t) dxdt⎠ ⎝

1 |Q|

⎞q−1

ˆ w Q

−1 q−1

(x, t) dxdt⎠

≤ A < +∞

(2.6)

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where the supremum runs over all circular cylinders Q formed by Q = Q(ρ,θ) (z), 0 < ρ < ∞, 0 < θ < ∞, and |Q| is the Lebesgue measure of Q. For example, we may take w(x, t) = w1 (x)w2 (t), where both w1 (x) and w2 (t) are standard Muchenhoupt’s Aq weights, in Rn and R, respectively. Remark 2.1. In (2.6), we are allowed to pick more cylinders Q(ρ,θ) (z) than the standard parabolic cylinders Q(ρ,ρ2 ) (z), which is natural in order to treat our situation. (In fact, we only need to Q = Q(ρ,ρ2 ) (z) for (2.6), when the problem (2.1) has linear growth, p = 2.) The main reason for using (2.6) is that the problem (2.1) does not have the invariance property under normalization except for p = 2; that is, multiplying a solution by a nonzero constant does not yield another solution, even in the simplest case when ut − div (|Du|p−2 Du) = 0. To overcome this scaling deficit, we consider the super-level set defined by {z = (y, s) ∈ ΩT : |Du(y, s)| > λ}, for each positive λ, and λ-intrinsic parabolic cylinders Qλρ (z) defined by ⎧ ⎨ Q(ρ,λ2−p ρ2 ) (z),  (z), Qλρ (z) = Q p−2 ⎩ 2 λ

2

for p ≥ 2, 2n for 2+n < p < 2.

ρ,ρ

(2.7)

With the presence of the new parameter λ, (2.6) is natural and works effectively for nonlinear parabolic problems of p-Laplacian type such as (1.1). We denote the smallest constant A for which (2.6) holds by [w]q . The weighted Lebesgue measure w(·) is defined by ˆ w(E) =

w(x, t) dxdt, E

for any bounded measurable set E ⊂ Rn+1 . Alternatively, the parabolic Aq weights can be defined as follows: a weight w belongs to Aq , 1 < q < ∞, if and only if ⎛ ⎝ 1 |Q|

ˆ Q

⎞q f (x, t)dxdt⎠ ≤

c w(Q)

ˆ



q f (x, t) w(x, t)dxdt

(2.8)

Q

holds for all positive, locally integrable functions f and all cylinders Q = Qρ,θ (z). The smallest constant c for which (2.8) is valid is [w]q . It is a direct consequence of (2.8) that

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 q |E| w(E) a parabolic Aq -weight w has a doubling property; |Q| ≤ c w(Q) for E ⊂ Q, and so for any α, 0 < α < 1, one can find β, 0 < β < 1, such that for all circular cylinder Q and all measurable subset E of Q, |E| ≥ α|Q| =⇒ w(E) ≥ βw(Q). Thus, the parabolic Aq weights have a remarkable feature which is the so-called reverse Hölder property; that is, for w ∈ Aq with 1 < q < ∞, there exists a small positive constant 0 , depending only on n, q, and [w]q , such that w ∈ Aq−0 with the estimate [w]q−0 ≤ c[w]q for some c = c(n, q, [w]q ) > 0 (see [39, Chapter 5] or [21, Chapter 9]). Combining the doubling and reverse Hölder properties, we see that 1 c1



|E| |Q|

q

w(E) ≤ c1 ≤ w(Q)



|E| |Q|

 τ1



 E ⊂ Q = Q(ρ,θ) (z) ,

(2.9)

for some positive constants c1 and τ1 . Note that these constants depend only on n, q, and [w]q and not on E and Q. We now turn to Orlicz spaces. A function Φ : [0, ∞) → [0, ∞) is said to be a Young function, if Φ is increasing and convex and satisfies Φ(ρ) = 0, ρ→0+ ρ

Φ(0) = 0, Φ(∞) = lim Φ(ρ) = +∞, lim ρ→+∞

Φ(ρ) = +∞. ρ→+∞ ρ lim

Throughout this paper, the Young function Φ is assumed to satisfy the following conditions: • (Δ2 -condition) There exists μ > 1 such that Φ(2ρ) ≤ μΦ(ρ) for all ρ ≥ 0. • (∇2 -condition) There exists ν > 1 such that 2νΦ(ρ) ≤ Φ(νρ) for all ρ ≥ 0. Note that if Φ satisfies Δ2 -condition, then for every λ > 1 there exists a positive constant μ1 = μ1 (λ) such that Φ(λρ) ≤ μ1 Φ(ρ) for all ρ ≥ 0. Moreover, satisfying both conditions ensures that the Young function grows moderately; that is, there are constants q0 and q1 with 1 < q0 ≤ q1 < ∞ such that 1 min{λq0 , λq1 }Φ(ρ) ≤ Φ(λρ) ≤ c max{λq0 , λq1 }Φ(ρ), c

λ, ρ ≥ 0,

(2.10)

where the constant c > 1 is independent of λ and ρ. Also note that satisfying both conditions is unavoidable for the type of regularity we consider (see [41]). If Φ satisfies both Δ2 and ∇2 conditions, we denote Φ ∈ Δ2 ∩ ∇2 . Next, we introduce the lower index of Φ ∈ Δ2 ∩ ∇2 , denoted by i(Φ): i(Φ) =

lim

λ→0+

log(hΦ (λ)) = log λ

sup 0<λ<1

log(hΦ (λ)) , log λ

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where hΦ (λ) = sup ρ>0

Φ(λρ) Φ(ρ)

(λ > 0).

When Φ(ρ) = ρq with q > 1, i(Φ) = q. Note from (2.10) that 1 < i(Φ) < ∞. In fact, the index number i(Φ) is equal to the supremum of q0 which satisfies (2.10); more precisely, (2.10) implies that for a given small > 0, there exists c = c( , n, Φ) > 0 such that λi(Φ)− Φ(ρ) ≤ c Φ(λρ),

λ ≥ 1, ρ ≥ 0.

(2.11)

The condition that w ∈ Ai(Φ) is the main assumption on the parabolic Muckenhoupt weight w(x, t). Recall the self-improved property of w ∈ Ai(Φ) : there exists a small positive constant 0 = 0 (n, Φ, w) such that for some positive constant c = c(n, w, Φ), w ∈ Ai(Φ)−20

with

[w]i(Φ)−20 ≤ c[w]i(Φ) .

Because the classes Aq increase as q increases, we have w ∈ Ai(Φ)−0 with [w]i(Φ)−0 ≤ [w]i(Φ)−20 . We then apply (2.11) to 0 , which yields λi(Φ)−0 Φ(ρ) ≤ c Φ(λρ),

λ ≥ 1, ρ ≥ 0.

Consequently, we obtain [w]i(Φ) ≤ [w]i(Φ)−0 ≤ [w]i(Φ)−20 ≤ c[w]i(Φ)

(2.12)

and λi(Φ)−20 Φ(ρ) ≤ λi(Φ)−0 Φ(ρ) ≤ c Φ(λρ),

λ ≥ 1, ρ ≥ 0,

(2.13)

where the constants c and 0 depend only on w, Φ, and n. We refer to [20,22,27] for a more detailed discussion of the conditions on w ∈ Ai(Φ) and its relevant properties. Hereafter, we fix 0 according to (2.12) and (2.13). We are now ready to introduce the weighted Orlicz space. For a Young function Φ ∈ Δ ∩ ∇2 and a parabolic weight w = w(x, t) ∈ Ai(Φ) , the weighted Orlicz space LΦ w (ΩT ) consists of all measurable functions v : ΩT → R satisfying ˆ Φ(|v(x, t)|)w(x, t)dxdt < +∞. ΩT

This is a Banach space equipped with the following weighted Luxemburg norm

v LΦw (ΩT ) = inf

⎧ ⎨



ˆ λ>0:

Φ



ΩT

|v(x, t)| λ

⎫ ⎬

 w(x, t)dxdt ≤ 1



.

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1 One can show that LΦ w (ΩT ) ⊂ L (ΩT ) as follows:

ˆ

ˆ

|f (x, t)|w(x, t) i(Φ)−0 w(x, t)− i(Φ)−0 dxdt 1

|f (x, t)| dxdt = ΩT

1

ΩT

⎛ ≤⎝

1 ⎞ i(Φ)−

ˆ

0

|f (x, t)|

i(Φ)−0

w(x, t)dxdt⎠

ΩT

⎛ ×⎝

0 −1 ⎞ i(Φ)− i(Φ)−

ˆ

0

w(x, t)

−1 i(Φ)−0 −1

dxdt⎠

ΩT

⎛ ≤ c⎝

1 ⎞ i(Φ)−

ˆ

0

|f (x, t)|i(Φ)−0 w(x, t)dxdt⎠

,

ΩT

for some constant c = c(n, Φ, w, ΩT ). The last inequality follows from (2.6). If |f (x, t)| ≥ 1 and if λ = |f (x, t)| and ρ = 1, (2.13) implies |f (x, t)|i(Φ)−0 ≤

c Φ(|f (x, t)|); Φ(1)

therefore, we conclude ⎛

ˆ

|f (x, t)| dxdt ≤ c ⎝

ΩT

ˆ

1 ⎞ i(Φ)− 0   . Φ(|f (x, t)|) + 1 w(x, t)dxdt⎠

(2.14)

ΩT

We now discuss the regularity assumptions on the nonlinearity and the lateral boundary. Given positive numbers ρ and θ, and a point z = (y, s) ∈ Rn+1 , to measure the oscillation of a(ξ, x, t) in Q(ρ,θ) (z) from being the integral average of a(ξ, ·, t) over Bρ (y) in the x-variable, uniformly in ξ, we introduce the following function: 



Θ a; Q(ρ,θ) (z) (x, t) =

sup

 a(ξ, x, t) − aB |ξ|p−1

ξ∈Rn \{0}

ρ (y)

 (ξ, t)

,

  where aBρ (y) (ξ, t) = a1 Bρ (y) , · · · , an Bρ (y) is the component-wise integral average of a(ξ, ·, t) in x over Bρ (y) for a fixed ξ ∈ Rn and t ∈ (s − θ, s + θ); more precisely, ˆ aBρ (y) (ξ, t) = − a(ξ, x, t) dx = Bρ (y)

1 |Bρ (y)|

ˆ a(ξ, x, t) dx. Bρ (y)

From (2.3), this function is uniformly bounded with Θ ≤ 2Λ.

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Definition 2.2. We say that (a, Ω × R) is weakly (δ, R)-vanishing, if   (i) for every point (y, s) ∈ Ω × R and for every numbers ρ ∈ (0, R] and θ ∈ 0, R2 with dist(y, ∂Ω) = min dist(y, y0 ) > ρ, y0 ∈∂Ω

there exists a coordinate system depending on (y, s) and ρ, whose variables are still denoted by (x, t), such that in the new coordinate system, (y, s) is the origin and ˆ     − Θ a; Q(ρ,θ) (x, t) dxdt ≤ δ.

(2.15)

Q(ρ,θ)

  (ii) for every point (y, s) ∈ Ω × R and for every numbers ρ ∈ (0, R] and θ ∈ 0, R2 with dist(y, ∂Ω) = dist(y, y0 ) ≤ ρ for some y0 ∈ ∂Ω, there exists a coordinate system depending on (y, s) and ρ, whose variables are still denoted by (x, t), such that in the new coordinate system, B3ρ ∩ {xn > 0} ⊂ B3ρ ∩ Ω ⊂ B3ρ ∩ {xn > −6δρ} ,

(2.16)

and ˆ −

      (x, t) Θ a; Q+  dxdt ≤ δ, (3ρ,9θ)

(2.17)

Q+ (3ρ,9θ)

where Q+ (3ρ,9θ) = Q(3ρ,9θ) ∩ {(x, t) : xn > 0}. Remark 2.3. The conditions (2.15)–(2.17) suggest that for the interior case, the nonlinearity a has a small BMO semi-norm in x; there is no regularity assumption for the time variable, which results in large jumps along the time direction. On the other hand, for the boundary case, under a suitable rotation of spatial variables, the boundary ∂Ω of the domain Ω can be trapped between two hyperplanes. Moreover, the nonlinearity has a small weak BMO semi-norm similar to the interior case. Note that there is no smoothness assumption on the boundary, but it is locally approximated by hyperplanes (see (2.16)). This is the so called the (δ, R)-Reifenberg flatness, and it guarantees a kind of measure 1 density (see (2.20)). This flatness is only significant for small δ, where 0 < δ ≤ 2n+1 , see [40]. We now state the main result of this paper. Theorem 2.4. Let Φ ∈ Δ2 ∩ ∇2 and let w ∈ Ai(Φ) . Assume that |F |p ∈ LΦ w (ΩT ). Then there exists a constant δ = δ(γ, Λ, n, p, Φ, w) > 0 such that if (a, Ω × R) is weakly

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(δ, R0 )-vanishing for some R0 > 0, then the weak solution of (2.1) satisfies |Du|p ∈ LΦ w (ΩT ) and we have the estimate ˆ







Φ |Du|p w(x, t)dxdt ≤ c ⎝

ΩT

ˆ

  Φ |F |p w(x, t)dxdt

ΩT

⎤d ⎞ ˆ ⎟ ⎜ + Φ ⎝⎣ − |F |p dxdt + 1⎦ ⎠ , ⎛⎡

(2.18)

ΩT

where the constant c depends on γ, Λ, n, p, ΩT , R0 , Φ and w. The constant d is defined by  d=

p 2,

if p ≥ 2,

2p 2p+np−2n ,

if

2n n+2

(2.19)

< p < 2.

We have a few comments in order. We recall (2.14) to see that |F |p ∈ LΦ w (ΩT ) with w ∈ Ai(Φ) implies |F |p ∈ L1 (ΩT ), which ensures the existence and uniqueness of weak 2n solutions. The lower bound n+2 of p is unavoidable for the type of regularity considered herein, see [6,7]. Since the problem (2.1) has a scaling deficit, the presence of the number d is natural, and the intrinsic geometry must be carefully analyzed (see [2,6] for more details). Since (a, Ω ×R) is weakly (δ, R0 )-vanishing, the geometric settings in (2.16) give rise to the following measure density property: |Bρ (y)| ≤ |Bρ (y) ∩ Ω|



2 1−δ

n

 ≤

16 7

n ,

(2.20)

for every y ∈ Ω and ρ ∈ (0, R0 ]. Finally, note that the present work is a natural extension of [8], where the same equation (2.1) is studied in unweighted Lebesgue and Orlicz spaces. We improve the work in [8] in two ways: (1) the nonlinearity is allowed to be merely measurable in the time variable, and (2) the weighted estimates for the parabolic problem with the polynomial growth are considered. 3. Auxiliary lemmas In this section, we assume |F |p ∈ LΦ w (ΩT ), Φ ∈ Δ2 ∩ ∇2 , and w ∈ Ai(Φ) . Hereafter, we use c to refer to any constant that can be explicitly computed in terms of the known data n, p, γ, Λ, Φ and w. Moreover, recall the number d from (2.19) and the number 0 from (2.12) and (2.13). For the unique weak solution u of (2.1) and any positive number λ, we introduce the following notation: E(λ) = {(x, t) ∈ ΩT : |Du(x, t)| > λ}

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and Kρλ (z) = Qλρ (z) ∩ ΩT , where Qλr (z) is given by (2.5) and (2.7). We define the constant λ0 by  ˆ  p 1 λ0d = − |Du|p + p |F |p + 1 dxdt, δ ΩT 1 where δ ∈ (0, 2n+1 ) is temporally fixed. Note that δ will be determined independently of u and F in the next section. Observe from the standard Lp estimate in (2.4) that

ˆ λ0 ≤ c − (|F |p + 1) dxdt. p d

(3.1)

ΩT

Next, we write c∗ = 2i(Φ)−20 −1 [w]i(Φ)−20

(3.2)

and  B = max

16 7

n

|ΩT | |B1 |R0n+2

 dp ,4 .

(3.3)

A key element of our argument is the derivation of the following power decay estimate for the super-level sets of E(λ) for each sufficiently large λ. Lemma 3.1. There is a universal constant N = N (γ, Λ, n, p) > 1 such that for any fixed ∈ (0, 1), one can find a small positive constant δ = δ( , γ, Λ, n, p) such that if (a, Ω × R) is weakly (δ, R0 )-vanishing for some R0 > 0, then for every λ ≥ Bλ0 and for some positive constant c = c(γ, Λ, n, p, q, Φ, ΩT , w, R0 ), the following holds: ⎡   w E(2N λ) ≤ c

pτ1 λp(i(Φ)−20 )

ˆ

⎢ ⎢ ⎣

|Du|p(i(Φ)−20 ) wdxdt

ΩT ∩{|Du|> 4cλ∗ }

ˆ + ΩT ∩{| Fδ |> 4cλ∗ }

where τ1 is given by (2.9).

⎤  p(i(Φ)−20 ) F  ⎥   wdxdt⎥ δ ⎦,

(3.4)

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The proof of the lemma is rather long, so we divide it into several steps. STEP 1: Choose a class of cylinders whose side lengths depend on the solution from DiBenedetto’s intrinsic geometry method, localization method, and the following Vitali covering lemma. $ # Lemma 3.2. Let λ ≥ Bλ0 . Then there exists a family of disjoint cylinders Krλi (zi ) i≥1 , with zi = (yi , si ) ∈ E(λ) and ri = ri (zi ) ∈ (0, R0 ), such that   1 p p |Du| + p |F | dxdt = λp , δ

ˆ −

(3.5)

Krλ (zi ) i

  1 p p |Du| + p |F | dxdt < λp δ

ˆ −

for every r > ri ,

(3.6)

Krλ (zi )

and E(λ) ⊂

%

λ K5r (zi ). i

(3.7)

i≥1

Proof. We refer to [8, Lemma 4.1] and [2] for the proof. 2 STEP 2: Under the same hypothesis and results as in Lemma 3.2, we have ⎡ w(Krλi (zi ))



c λp(i(Φ)−20 )

ˆ

⎢ ⎢ ⎣

|Du|p(i(Φ)−20 ) wdxdt

Krλ (zi )∩{|Du|> 4cλ∗ } i



(3.8)

 p(i(Φ)−20 ) ⎥ F    wdxdt⎥ δ ⎦,

ˆ + δλ Krλ (zi )∩{|F |> 4c } ∗ i

for some positive constant c = c(n, Φ, w). Proof. It follows from (3.5) and the fact that i(Φ) − 2 0 > 1 that for some positive constant c = c(n, Φ, w),

λp(i(Φ)−20 )

⎡⎛ ˆ ⎢⎜ − ≤ c⎢ ⎝ ⎣ Krλ (zi ) i

⎞i(Φ)−20 ⎟ |Du|p dxdt⎠



ˆ ⎜ +⎝ −

⎞i(Φ)−20 ⎤  p F  ⎥   dxdt⎟ ⎥. ⎠ δ ⎦

Krλ (zi ) i

Then, we apply (2.8) and replace q by i(Φ) − 2 0 to discover that for some positive constant c = c(n, Φ, w),

S.-S. Byun, S. Ryu / Journal of Functional Analysis 272 (2017) 4103–4121

⎡ λ

p(i(Φ)−20 )

w(Krλi (zi ))

4115

ˆ

⎢ ≤ cc∗ ⎣

|Du|p(i(Φ)−20 ) w(x, t)dxdt

Krλ (zi ) i

⎤  p(i(Φ)−20 ) F  ⎥   w(x, t)dxdt⎦ . δ

ˆ +

(3.9)

Krλ (zi ) i

Moreover, direct calculation yields 

ˆ |Du|

p(i(Φ)−20 )

Krλ (zi ) i

w(x, t)dxdt ≤

λ 4c∗

p(i(Φ)−20 )   w Krλi (zi )

ˆ |Du|p(i(Φ)−20 ) w(x, t)dxdt.

+ Krλ (zi )∩{|Du|> 4cλ∗ i

(3.10)

}

  Combining (3.9), (3.10), and (3.10) with |Du| replaced by  Fδ  yields (3.8).

2

STEP 3: There is a constant N = N (γ, Λ, n, p) > 1 such that for any ∈ (0, 1), one can find a small positive constant δ = δ( , γ, Λ, n, p) such that if (a, Ω × R) is weakly (δ, R0 )-vanishing for such small δ and some R0 > 0, then under the same notation and results of Lemma 3.2, for some positive constant c = c(γ, Λ, n, p), # $ λ  (x, t) ∈ K5r (zi ) : |Du(x, t)| > 2N λ  i ≤ c p . |Krλi (zi )|

(3.11)

Proof. The proof is based on the regularity assumptions on a and ∂Ω, specially that (a, Ω × R) is weakly (δ, R0 )-vanishing for such small δ and some R0 > 0. Under these assumptions, we compare the solution u to solutions v of a suitable limiting problem with zero boundary conditions on the flat boundary where the associated nonlinearity depends on Dv and t, but is independent of x. Then the regularity of u depends on how close it is to v, whose regularity is Lipschitz continuous in x. In this way, we can derive the small density of the upper-level sets in (3.11). We refer to the proof of [8, Lemma 4.3] for more details. 2 STEP 4: The proof of Lemma 3.1. Proof. For some positive constant c = c(γ, Λ, n, p, q, Φ, ΩT , w) > 0,   w E(2N λ)



& # $ λ w (x, t) ∈ K5r (z ) : |Du(x, t)| > 2N λ i i

=

 &  w Qλ5ri (zi ) ∩ E(2N λ)

(3.7)

i≥1

i≥1

S.-S. Byun, S. Ryu / Journal of Functional Analysis 272 (2017) 4103–4121

4116

(2.9)



c

&

'

i≥1



c

& i≥1

(3.11)



c pτ1

'

|Qλ5ri (zi ) ∩ E(2N λ)| |Qλ5ri (zi )| |Qλ5ri (zi ) ∩ E(2N λ)| |Qλri (zi ) ∩ ΩT |

(τ1

( τ1

  w Qλ5ri (zi )   w Qλ5ri (zi )

 &  w Qλ5ri (zi ) i≥1

(2.9),(2.20)



c pτ1

 &  w Krλi (zi ) i≥1

(3.8)





c pτ1 λp(i(Φ)−20 )

ˆ

&⎢ ⎢ ⎣ i≥1

|Du|p(i(Φ)−20 ) wdxdt

Krλ (zi )∩{|Du|> 4cλ∗ } i

ˆ + δλ Krλ (zi )∩{|F |> 4c } ∗ i

⎡ ≤

c pτ1 λp(i(Φ)−20 )

⎤  p(i(Φ)−20 ) ⎥ F    wdxdt⎥ δ ⎦

ˆ

⎢ ⎢ ⎣

|Du|p(i(Φ)−20 ) wdxdt

ΩT ∩{|Du|> 4cλ∗ }

ˆ +

⎤  p(i(Φ)−20 ) ⎥ F    wdxdt⎥ δ ⎦,

δλ ΩT ∩{|F |> 4c } ∗

# $ where the disjointedness of Krλi (zi ) from Lemma 3.2 is applied for the last inequality. This completes the proof. 2 The following lemma will be used in the next section. Lemma 3.3. Assume Φ ∈ Δ2 ∩ ∇2 , w = w(x, t) ∈ Ai(Φ) and g ∈ LΦ w (ΩT ). Then ˆ ΩT

ˆ∞     Φ |g(x, t)| w(x, t)dxdt = w {(x, t) ∈ ΩT : |g(x, t)| > λ} d[Φ(λ)].

(3.12)

0

Furthermore, there is a small number 0 = 0 (n, Φ, w) > 0 such that for each a, b > 0, the following estimate holds:

S.-S. Byun, S. Ryu / Journal of Functional Analysis 272 (2017) 4103–4121

ˆ∞ I:= 0





ˆ

1 ⎢ ⎣ λi(Φ)−20

4117

⎥ |g|i(Φ)−20 w(x, t)dxdt⎦ d[Φ(bλ)]

ΩT ∩{|g|>aλ}

ˆ

≤c

Φ(|g|)w(x, t)dxdt,

(3.13)

ΩT

where the constant c depends on a, b, Φ and w. Proof. Identity (3.12) follows directly from Fubini’s theorem. To prove (3.13), we take 0 from (2.12) and (2.13) and recall that λi(Φ)−0 Φ(ρ) ≤ c Φ(λρ)

(λ ≥ 1, ρ ≥ 0) .

(3.14)

Next, we interchange the order of integration I, which yields ˆ I=

⎡ ⎢ ⎣



|g(x,t)| a

ˆ

1 λi(Φ)−20

⎥ d[Φ(bλ)]⎦ |g(x, t)|i(Φ)−20 w(x, t)dxdt.

(3.15)

0

ΩT

Integrating by parts, we have 

|g(x,t)| a

ˆ

1 λi(Φ)−20

Φ d[Φ(bλ)] = 

0

b|g(x,t)| a

|g(x,t)| a



Φ(bλ) lim i(Φ)−2 i(Φ)−20 − λ→0 0 λ |g(x,t)| a

ˆ

+ (i(Φ) − 2 0 )

Φ(bλ) dλ. λi(Φ)−20 +1

(3.16)

0

But (3.14) implies  i(Φ)−0   1 1 = cΦ(b) Φ(bλ) ≤ cΦ bλ λ λ

(0 < λ < 1) ,

and so Φ(bλ) ≤ cλi(Φ)−0 Φ(b) for 0 < λ < 1. Consequently, lim

λ→0

On the other hand, for 0 < λ < Φ(bλ)  Φ

Φ(bλ) = Φ

b|g(x,t)| a



Φ(bλ) = 0. λi(Φ)−20

|g(x,t)| , a

b|g(x, t)| a



(3.17)

we have 

(3.14)



c

aλ |g(x, t)|

i(Φ)−0

 Φ

b|g(x, t)| a

 ,

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S.-S. Byun, S. Ryu / Journal of Functional Analysis 272 (2017) 4103–4121

which implies |g|

ˆa



Φ(bλ) λi(Φ)−20 +1

dλ ≤ c

0

 ≤c

1 |g| 1 |g|

i(Φ)−0

 Φ

|g|

b|g| a

 ˆa

λi(Φ)−0 λi(Φ)−20 +1



0

i(Φ)−20

 Φ

b|g| a

 .

(3.18)

Finally, we combine (3.15), (3.16), (3.17), and (3.18) to discover 

ˆ I≤c

Φ

b|g| a



ˆ w(x, t)dxdt ≤ c

ΩT

Φ (|g|) w(x, t)dxdt,

ΩT

where the constant c depends on a, b, Φ and w. This completes the proof. 2 4. Optimal weighted Orlicz regularity In this section, we prove Theorem 2.4 by essentially deriving the desired estimate (2.18) under the a priori assumption that ˆ

  Φ |Du(x, t)|p w(x, t)dxdt < +∞.

(4.1)

ΩT

To do this, we further assume that F , a, and ∂Ω are all regular enough to ensure that (4.1) holds. These assumptions, needless to say, can be removed by an approximation procedure based on the so-called variable domain technique (see [8, Section 5] or [11, Section 5]). We are now ready to obtain estimate (2.18). Proof. The proof proceeds with Lemma 3.3. According to the identity formula in (3.12) of Lemma 3.3, when g is replaced by |Du|p and λ is replaced by [2N λ]p , we have ˆ Φ(|Du|p ) w(x, t)dxdt ΩT

ˆ∞      = w {(x, t) ∈ ΩT : |Du|p > [2N λ]p } d Φ [2N λ]p 0 Bλ ˆ 0

     w {(x, t) ∈ ΩT : |Du|p > [2N λ]p } d Φ [2N λ]p

= 0

S.-S. Byun, S. Ryu / Journal of Functional Analysis 272 (2017) 4103–4121

ˆ∞ +

4119

     w {(x, t) ∈ ΩT : |Du|p > [2N λ]p } d Φ [2N λ]p

Bλ0

=: II 1 + II 2 . Estimate of II 1 : Recall that Φ ∈ Δ2 ∩ ∇2 and w ∈ Ai(Φ) . From (3.1), observe that     II 1 ≤ w ΩT · Φ [2N Bλ0 ]p ≤ cΦ (λp0 ) ⎛⎡ ⎤d ⎞ ˆ ⎟ ⎜ ≤ cΦ ⎝⎣ − |F |p dxdt + 1⎦ ⎠ , ΩT

where the constant c depends on γ, Λ, n, p, R0 , ΩT , Φ and w. Estimate of II 2 : In light of (3.4) in Lemma 3.1, for some positive constant c = c(γ, Λ, n, p, Φ, w), ˆ∞ II 2 =

     w {(x, t) ∈ ΩT : |Du|p > [2N λ]p } d Φ [2N λ]p

Bλ0

ˆ∞ ≤ c pτ1





ˆ

⎜ 1 ⎜ λp(i(Φ)−20 ) ⎝

⎟    p |Du|p(i(Φ)−20 ) wdxdt⎟ ⎠ d Φ [2N λ]

ΩT ∩{|Du|> 4cλ∗ }

0



ˆ∞ + c( )

1 λp(i(Φ)−20 )

⎜ ⎜ ⎝

ˆ

⎞  p(i(Φ)−20 ) F  ⎟    p   wdxdt⎟ δ ⎠ d Φ [2N λ] .

δλ ΩT ∩{|F |> 4c } ∗

0

  Applying the inequality (3.13) in Lemma 3.3 when g is replaced both by |Du|p and  Fδ , and λ is replaced by [2N λ]p , we find that ˆ II 2 ≤ c pτ1

ˆ     Φ |Du|p w(x, t)dxdt + c(δ, ) Φ |F |p w(x, t)dxdt,

ΩT

ΩT

for some positive constant c = c(γ, Λ, n, p, Φ, w). We combine the estimates of II 1 and II 2 , to conclude ˆ ΩT

ˆ     Φ |Du|p w(x, t)dxdt ≤ c1 pτ1 Φ |Du|p w(x, t)dxdt ΩT

ˆ

+ c2 (δ, ) ΩT

  Φ |F |p w(x, t)dxdt

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⎤d ⎞ ˆ ⎟ ⎜ + c3 Φ ⎝⎣ − |F |p dxdt + 1⎦ ⎠ , ⎛⎡

ΩT

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