Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities

Nonlinear Analysis xxx (xxxx) xxx

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Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities Youchan Kim a , Yeonghun Youn b ,∗ a b

Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

article

info

Article history: Received 30 November 2018 Accepted 1 February 2019 Communicated by Enzo Mitidieri MSC: primary 35J60 secondary 35B65 35R05

abstract We consider elliptic equations with measurable nonlinearities in a half cube Q2R ∩{(x1 , x′ ) ∈ Rn : x1 > 0} where the boundary data is given on Q2R ∩{(x1 , x′ ) ∈ Rn : x1 = 0}. We obtain a point-wise estimate of the gradient in terms of Riesz potential of the right-hand side data and the oscillation of the gradient of the boundary data under the assumption that the nonlinearity is only allowed to be measurable in x1 -variable. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear elliptic equations Riesz potentials

1. Introduction We study the elliptic equation with measurable nonlinearities and measure data: { div a(Du, x1 , x′ ) = µ in Q2R (0) ∩ Rn+ , u = ϕ on Q2R (0) ∩ Rn0 ,

(1.1)

where µ is a Radon measure with finite mass on Q2R (0) ∩ Rn+ and ϕ is a C 1 -function on Q2R (0) ∩ Rn0 . Here and in what follows, we denote x = (x1 , x′ ) = (x1 , x2 , . . . , xn ), Rn+ = {(x1 , x′ ) ∈ Rn : x1 > 0}, Rn0 = {(x1 , x′ ) ∈ Rn : x1 = 0} and by Qρ (x) the cube with center x and side-length 2ρ. We assume the following ellipticity and growth conditions: { a(ξ, x) is measurable in x for every ξ ∈ Rn , (1.2) a(ξ, x) is C 1 -regular in ξ for almost every x ∈ Rn , and

∗

⎧ ⎨

|a(ξ, x)| ≤ Λ|ξ|, |Dξ a(ξ, x)| ≤ Λ, ⎩ 2 ⟨Dξ a(ξ, x)ζ, ζ⟩ ≥ λ|ζ| ,

(1.3)

Corresponding author. E-mail addresses: [email protected] (Y. Kim), [email protected] (Y. Youn).

https://doi.org/10.1016/j.na.2019.02.001 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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for any x, ξ, ζ ∈ Rn and some constants 0 < λ ≤ Λ. Also we assume that a(ξ, x1 , x′ ) is Dini-continuous in x′ -variables: |a(ξ, x1 , x′ ) − a(ξ, x1 , y ′ )| ≤ ω(|x′ − y ′ |)|ξ| (1.4) for every (x1 , x′ ) ∈ Rn , ξ ∈ Rn and y ′ ∈ Rn−1 , where the function ω : [0, ∞) → [0, 1] is a non-decreasing concave function with limρ↘0 ω(ρ) = ω(0) = 0 satisfying ∫ 1 ω(ρ) dρ < ∞. ρ 0 We remark that there is no regularity assumption on x1 -variable. The main purpose of this paper is to derive the point-wise gradient estimates of solutions to (1.1) by using the following Riesz potential of the right-hand side µ: ∫ r |µ|(Qρ (x)) dρ (x ∈ Rn , r > 0, 0 < α < n). Iα|µ| (x, r) = ρn−α ρ 0 Let x0 = (x1 , x′ ) be any point in QR and 0 < r ≤ R. Also for the continuity condition on the boundary data ϕ ∈ C 1 (Q2R (0) ∩ Rn0 ), we measure the oscillation of the gradient Dx′ ϕ on Qρ (0, x′0 ) ∩ Rn0 , say ∫ r dρ osc Dx′ ϕ ′ )∩Rn ρ Q (0,x ρ 0 0 0 instead of the usual local Dini-continuity condition on Dx′ ϕ. The point-wise potential type gradient estimates for elliptic and parabolic problems have been studied very actively in recent years. When the nonlinearity is Dini-continuous on x-variable, in [28] the author obtained the point-wise gradient estimates for the interior case via Riesz potential of the right-hand side data. The results in [28] have been extended to p-Laplace type problems. We refer to [2,12,17,22,26] for the elliptic problems and [19,20,24,25] for the parabolic problems. See [2,3,8,9] for nonstandard growth problems which have attracted increasing amount of attention recently. Also with [19,22,24,26], we refer to [11,21,23] for the gradient continuity type results by potentials. If the nonlinearity is allowed to be measurable in one spatial variable, the point-wise potential type gradient estimates for parabolic equations are obtained for the interior case in [6]. In this paper, we consider elliptic equations with measurable nonlinearities and extend [6] to the boundary case. We remark that [10] considered linear elliptic equations with measurable coefficients in C 1,Dini -domains instead of the flat boundary. In our opinion, [10] used a proof based on the linear structure of the equation, and we need to use a complicated barrier argument for extending [10] to nonlinear equations. A standard approach for obtaining a boundary regularity is odd or even extension of the given equations. One can extend a linear elliptic equation defined on the half cube Q2r (0)∩Rn+ to the cube Q2r (0) by using odd or even extensions, in which case the discontinuity of the coefficients might occur on Rn0 ∩ Q2r (0). For H¨older continuity results, the coefficients are allowed to be merely measurable for all directions, and so the results for elliptic equations up to the boundary can be obtained by using odd or even extensions and the results for the interior case. On the other hand, the gradients might not be bounded if the coefficients are merely measurable, see for instance [16]. So because of the discontinuity of the coefficients on Rn0 for the extended elliptic equations, the extension approach for obtaining the boundary H¨older continuity is not feasible to the boundary gradient boundedness if the coefficients are assumed to be Dini-continuous in all directions. However, if one can obtain the gradient estimates under the assumption that the coefficients are allowed to be merely measurable in one direction, then one can handle the discontinuity of the coefficients on Rn0 . Actually, in [15,18], Calder´ on–Zygmund type estimates for elliptic equations with measurable nonlinearities with zero boundary condition and Lipschitz regularity for the limiting equations are obtained by using an odd extension. In this paper, we use this extension approach for elliptic equations with measurable nonlinearities Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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to obtain the boundary point-wise gradient estimate. The excess decay estimates of the gradient played a key role in the interior point-wise gradient estimates, and so we need to obtain a corresponding estimate for the boundary case. But when the nonlinearity is allowed to be merely measurable, the gradient might be discontinuous because of the measurable nonlinearities. So instead, we derive an excess decay estimate of the even extension of a1 (Dw, x0 ) and the odd extension of Dx′ w. Then with the comparison estimates in the previous literatures, one can obtain the desired point-wise potential type gradient estimates. 2. Definitions and main results Let y be a typical point and r > 0 be a typical length of cube. To state the main result, we now introduce the following notations: 1. 2. 3. 4. 5.

y = (y 1 , . . . , y n ) = (y 1 , y ′ ). Rn+ = {(y 1 , y ′ ) ∈ Rn : y 1 > 0}. Rn0 = {(y 1 , y ′ ) ∈ Rn : y 1 = 0}. Rn− = {(y 1 , y ′ ) ∈ Rn : y 1 < 0}. n Qr (y) = (y 1 − r, y 1 + r) × · · · × (y n − r, y n + r). Qr = Qr (0). Q+ r = Qr ∩ R+ . ′ ′ n Tr (y ) = Qr (0, y ) ∩ R0 . ∫ ∫ For f ∈ L1 (U ) with |U | = ̸ 0, (f )U = −U f dx = |U1 | U f dx.

Let the right-hand side data µ be a Radon measure defined on Q+ 2R with finite total mass. Without loss n of generality, we may assume that µ is defined on R by letting µ(O) = 0 for every O ⊂ Rn \ Q+ 2R . In Theorem 2.1, we assume that 2n µ ∈ L n+2 (Q+ (2.1) 2R ). This assumption will be removed in Corollary 2.3 where a general solution called SOLA(Solutions Obtained by Limit of Approximations) will be considered. SOLA is a class of very weak solutions which does not necessarily belong to the energy space W 1,2 and was introduced to obtain existence and uniqueness for several cases such as µ does not belong to the L1 spaces, see [4,5]. Also for the boundary data, let ϕ : Q2R ∩ Rn0 → R be a function such that ϕ ∈ C 1 (Q2R ∩ Rn0 ). (2.2) We obtain the following point-wise potential type gradient estimate. Theorem 2.1. Under the assumptions (1.2), (1.3), (1.4), (2.1) and (2.2), there exists c1 = c1 (n, λ, Λ) ≥ 1 such that if r ∈ (0, R] satisfies ∫ 2r ω(ρ)dρ c1 ≤ 1, (2.3) ρ 0 then for a weak solution u ∈ W 1,2 (Q+ 2R ) of { div a(Du, x1 , x′ ) = µ u = ϕ we have the following point-wise estimate [∫ ∫ |Du(x0 )| ≤ c−

Qr (x0 )∩Rn +

|Du| dx + c 0

r

(

in Q+ 2R , on Q2R ∩ Rn0 ,

|µ|(Qρ (x0 ) ∩ Rn+ ) + osc′ Dx′ ϕ ρn−1 Tρ (x0 )

(2.4)

)

] dρ ′ + |Dx′ ϕ(x0 )| , ρ

for any Lebesgue point x0 = (x10 , x′0 ) ∈ Q+ R of Du. When dealing with measure data problems, we need to remove the assumption (2.1). So we introduce a general solution called SOLA. Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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Definition 2.2. A SOLA of (2.4) is a distributional solution u ∈ W 1,1 (Q+ 2R ) to (2.4) such that u is the limit of solutions uh ∈ W 1,2 (Q+ ) to 2R { div a(Duh , x1 , x′ ) = µh in Q+ 2R , uh = ϕ on Q2R ∩ Rn0 , in the sense that uh → u in W 1,1 (Q+ 2R ) and µh ⇀ µ in the sense of measures satisfying lim sup |µh |(Qρ (y) ∩ Rn+ ) ≤ |µ|(Qρ (y) ∩ Rn+ ) for any Qρ (y) ⊂ Q2R , h→∞

where µh ∈

L∞ (Q+ 2R ).

We finally state our main result for SOLA. Without (2.1), Theorem 2.1 continues to hold for SOLA of (2.4), with the estimate ] ) [∫ ( ∫ r |µ|(Qρ (x0 ) ∩ Rn+ ) dρ ′ + osc′ Dx′ ϕ + |Dx′ ϕ(x0 )| , |Du(x0 )| ≤ c− |Du| dx + c ρn−1 ρ Tρ (x0 ) Qr (x0 )∩Rn 0

Corollary 2.3.

+

for any Lebesgue point x0 = (x10 , x′0 ) ∈ Q+ R of Du. Remark 2.4. For the sake of convenience and simplicity, we employ the letter c > 0 and α ∈ (0, 1] throughout this paper to denote any constants which can be explicitly computed in terms of known quantities such as n, λ, Λ. Thus the exact value denoted by c and α may change from line to line in a given computation. 3. Preliminaries The following lemma is the key result to handle measurable nonlinearities, which was proved in [6, Lemma 4.1]. Lemma 3.1.

Under the assumptions (1.2) and (1.3), we have [ ] |ξ| ≤ c |ξ ′ | + (2Λ)−1 |a1 (ξ, x)| ≤ c|ξ|,

for any x ∈ Rn and for any ξ = (ξ 1 , ξ ′ ) ∈ Rn . We use a version of reverse H¨ older’s inequality from [14, Remark 6.12]. Lemma 3.2.

Let G : A → Rn be a measurable map, n ≥ 1 and χ0 > 1, c0 , κ ≥ 0 such that ) χ1

(∫ −

χ0

|G|

0

dx

∫ ≤ c∗ −

Qr (y)

(|G| + s) dx.

Q2r (y)

whenever Q2r (y) ⊂⊂ A, where A ⊂ Rn is an open subset. Then, for every χ ∈ (0, 1] there exists a constant c = c(n, c∗ , χ, χ0 ) such that (∫ − Qr (y)

) χ1 |G|

χ0

0

dx

(∫ ≤c −

) χ1 χ

χ

(|G| + s ) dx

,

Q2r (y)

for every Q2r (y) ⊆ A. Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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From [14, Theorem 7.7], we have an excess decay estimate for linear elliptic equations. Lemma 3.3.

Under the assumption { 2 aij (x)ζi ζj ≥ λ|ζ|

(x ∈ Qr , ζ ∈ Rn ),

∥aij ∥L∞ (Qr ) ≤ Λ, let w be a weak solution of Di [aij (x)Dj w] = 0 in Qr . Then we have that

∫ ( ρ )2α ∫ 2 2 − |w − (w)Qr | dx − |w − (w)Qρ | dx ≤ c r Qr Qρ

for any 0 < ρ ≤ r. 4. Excess decay estimate Assume that

{

a(ξ, x1 ) is measurable in x1 ∈ R for every ξ ∈ Rn , a(ξ, x1 ) is C 1 -regular in ξ ∈ Rn for every x1 ∈ R,

and

⎧ 1 ⎪ ⎨ |a(ξ, x )| ≤ Λ|ξ| |Dξ a(ξ, x1 )| ≤ Λ, ⎪ ⎩ 2 ⟨Dξ a(ξ, x1 )ζ, ζ⟩ ≥ λ|ζ| ,

for every x1 ∈ R, ξ, ζ ∈ Rn and for some constants 0 < λ ≤ Λ. Under the assumption (4.1), (4.2) and 0 ≤ x10 ≤ r,

(4.1)

(4.2)

(4.3)

let g be a weak solution of {

div a(Dg, x1 )

=

0

in Q3r (x0 ) ∩ Rn+ ,

g

=

β ′ · x′

on Q3r (x0 ) ∩ Rn0 ,

(4.4)

where β ′ = (β 2 , . . . , β n ) ∈ Rn−1 and x0 = (x10 , x20 , . . . , xn0 ) = (x10 , x′0 ). Let k ∈ [2, n] be an arbitrary integer. Since a(ξ, x1 ) in (4.4) is independent of xk -variable, one can use the difference quotient method (see for instance [13]) to find that Dk g is weakly differentiable in Q2r (x0 ) ∩ Rn+ and Dk g ∈ W 1,2 (Q2r (x0 ) ∩ Rn+ ). So we differentiate (4.4) with respect to xk -variable to find that { Di [aij (x)Dj (Dk g)] = 0 in Q2r (x0 ) ∩ Rn+ (4.5) Dk g = β k on Q2r (x0 ) ∩ Rn0 , where aij (x) =

∂ai 1 ∂ξj (Dg, x ).

Then one can easily check from (4.2) that {

aij (x)ζi ζj |aij (x)|

2

≥ λ|ζ| ,

(4.6)

≤ Λ,

for any x ∈ Q2r (x0 ) ∩ Rn+ and ζ ∈ Rn We next extend Eq. (4.5) from Q2r (x0 ) ∩ Rn+ to Q2r (x0 ). By (4.5), Dk g − β k = 0 on Q2r (x0 ) ∩ Rn0

(k ∈ [2, n]).

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We let gk be the odd extension of Dk g − β k from Q2r (x0 ) ∩ Rn+ to Q2r (x0 ). Then

{

Di [aij (x)Dj gk ]

=

0

in Q2r (x0 ) ∩ Rn+

gk

=

0

on Q2r (x0 ) ∩ Rn0 .

(4.7)

(4.8)

Let a ˆij (x) be an extension of aij (x) from Q2r (x0 ) ∩ Rn+ to Q2r (x0 ) such that ⎧ a ˆ11 (−x1 , x′ ) = a11 (x1 , x′ ), ⎪ ⎪ ⎪ ⎨a ˆij (−x1 , x′ ) = aij (x1 , x′ ) ⎪ a ˆ1j (−x1 , x′ ) = −a1j (x1 , x′ ) ⎪ ⎪ ⎩ a ˆi1 (−x1 , x′ ) = −ai1 (x1 , x′ )

for 1 < i ≤ n, 1 < j ≤ n,

(4.9)

for 1 < j ≤ n, for 1 < i ≤ n,

for (x1 , x′ ) ∈ Q2r (x0 ) \ Rn+ . Then one can check from (4.6) that { 2 a ˆij (x)ζi ζj ≥ λ|ζ| , |ˆ aij (x)|

≤

(4.10)

Λ,

for any x ∈ Q2r (x0 ) and ζ ∈ Rn . Recall that gk is an odd extension of Dk g − β k from Q2r (x0 ) ∩ Rn+ to Q2r (x0 ). So we find from (4.8) and (4.9) that Di [ˆ aij (x)Dj gk ]

=

0

in Q2r (x0 ).

(4.11)

In view of (4.10) and (4.11), we obtain the following estimates. Lemma 4.1.

Under the assumption (4.3), we have that ∫ ( ρ )2α ∫ 2 2 − |gk − (gk )Qr (x0 ) | dx, − |gk − (gk )Qρ (x0 ) | dx ≤ c r Qr (x0 ) Qρ (x0 )

for any 0 < ρ ≤ r. Also we have the Caccioppoli type inequality, which is ∫ ∫ c 2 2 − |Dgk | dx ≤ 2 − |gk − (gk )Q2ρ (y) | dx, ρ Q2ρ (y) Qρ (y)

(4.12)

(4.13)

for any Q2ρ (y) ⊂ Q2r (x0 ). Proof . With (4.10) and (4.11), we have from Lemma 3.3 that ∫ ( ρ )2α ∫ 2 2 − |gk − (gk )Qρ (x0 ) | dx ≤ c − |gk − (gk )Qr (x0 ) | dx r Qρ (x0 ) Qr (x0 ) for any 0 < ρ ≤ r and for some α = α(n, λ, Λ) ∈ (0, 1]. So (4.12) holds. Next, we choose a cut-off function η such that η ∈ Cc∞ (Q2ρ (y)),

0 ≤ η ≤ 1,

η = 1 on Qρ (y)

and

|Dη| ≤

c . ρ

(4.14)

[ ] Since Q2ρ (y) ⊂ Q2r (x0 ), we test (4.11) by gk − (gk )Q2ρ (y) η 2 ∈ W01,2 (Q2r (x0 )) to find that ∫ − Q2r (y)

a ˆij (x)Dj gk Di

([

] ) gk − (gk )Q2ρ (y) η 2 dx = 0,

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and it follows from (4.10) that ∫ ∫ 2 2 λ− |Dgk | η dx ≤ − a ˆij (x)Dj gk Di gk η 2 dx Q2r (y) Q2r (y) ∫ [ ] = −− a ˆij (x)Dj gk gk − (gk )Q2ρ (y) 2ηDi η dx ∫ Q2r (y) ⏐ ⏐ ≤ c− |Dgk | ⏐gk − (gk )Q2ρ (y) ⏐ |η| |Dη| dx. Q2r (y)

So by Young’s inequality, ∫ −

∫ 2 |Dgk | η 2 dx ≤ c−

Q2r (y)

Q2r (y)

⏐ ⏐ ⏐gk − (gk )Q (y) ⏐2 |Dη|2 dx, 2ρ

and (4.13) follows from the choice of η in (4.14) and the fact that Q2ρ (y) ⊂ Q2r (x0 ). Since a(ξ, x1 ) depends on x1 -variable, we obtain an excess decay estimate by using a1 (Dg, x1 ) instead of D1 g. To do it, let g1 be the even extension of a1 (Dg, x1 ) from Q2r (x0 ) ∩ Rn+ to Q2r (x0 ) satisfying { g1 (x1 , x′ ) = a1 (Dg(x1 , x′ ), x1 ) in Q2r (x0 ) ∩ Rn+ (4.15) g1 (x1 , x′ ) = a1 (Dg(−x1 , x′ ), −x1 ) in Q2r (x0 ) \ Rn+ . Then we obtain the following estimate on Dg1 . Lemma 4.2.

Under the assumption (4.3), we have that ∫ ∫ c ∑ 2 2 − |gk − (gk )Q2ρ (y) | dx − |Dg1 | dx ≤ 2 ρ Q2ρ (y) Qρ (y) 2≤k≤n

for any Q2ρ (y) ⊂ Q2r (x0 ). Proof . Since g is a weak solution of (4.4), we have that ⏐ ⏐ ⎧ ⏐ ⏐ ⎪ ∑ ⏐ ⏐ ⎪ k 1 ⏐ ⎨ |D [a1 (Dg, x1 )]| = ⏐ Dk [a (Dg, x )]⏐ ≤ c|Dx′ Dg| ∈ L2 (Q2r (x0 ) ∩ Rn+ ), 1 ⏐ ⏐2≤k≤n ⏐ ⎪ ⎪ ⎩ |Dx′ [a1 (Dg, x1 )]| ≤ c|Dx′ Dg| ∈ L2 (Q2r (x0 ) ∩ Rn+ ). Since g1 be the even extension of a1 (Dg, x1 ) in (4.15), we find that |Dg1 | = |D[a1 (Dg, x1 )]| ≤ c|Dx′ Dg| ∈ L2 (Q2r (x0 ) ∩ Rn+ ).

(4.16)

Also by using the fact that g1 is the even extension of a1 (Dg, x1 ) from Q2r (x0 ) ∩ Rn+ to Q2r (x0 ), we obtain that |Dg1 (x1 , x′ )| = |Dg1 (−x1 , x′ )| ≤ c|Dx′ Dg(−x1 , x′ )| ∈ L2 (Q2r (x0 ) \ Rn+ ). We prove the lemma by considering the cases (1) y 1 ≥ 0 and (2) y 1 < 0. If (1) y 1 ≥ 0, then by the fact that g1 is the even extension of a1 (Dg, x1 ), ∫ ∫ 2 2 − |Dg1 | dx ≤ 2− |Dg1 | dx. Qρ (y)

Qρ (y)∩Rn +

(4.17)

(4.18)

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Recall that gk is the odd extension of Dk g − β k . In view of (4.16) and (4.13) in Lemma 4.1, we obtain that ∫ ∫ 2 2 − |Dg1 | dx ≤ c− |Dx′ Dg| dx Qρ (y)∩Rn +

Qρ (y)∩Rn +

∑ ∫ −

≤c

2

n 2≤k≤n Qρ (y)∩R+

∑ ∫ ≤c −

|Dgk | dx (4.19) 2

|Dgk | dx

2≤k≤n Qρ (y)

∫ c ∑ 2 − |gk − (gk )Q2ρ (y) | dx. ρ2 Q2ρ (y)

≤

2≤k≤n

So from (4.18) and (4.19), the lemma holds for y 1 ≥ 0. Next, suppose that (2) y 1 < 0. The proof is similar to that of the case that y 1 ≥ 0. By the fact that g1 is the even extension of a1 (Dg, x1 ), we find that ∫ ∫ 2 2 − |Dg1 | dx ≤ 2− |Dg1 | dx. (4.20) n Qρ (y)

Qρ (y)∩R−

In view of (4.17) and (4.13) in Lemma 4.1, we obtain that ∫ ∫ 2 2 − |Dg1 | dx ≤ c− |Dx′ Dg(−x1 , x′ )| dx Qρ (y)∩Rn −

Qρ (y)∩Rn −

∑ ∫ −

≤c

2

Qρ (y)∩Rn −

2≤k≤n

∑ ∫ −

≤c

|Dgk (−x1 , x′ )| dx (4.21) 2

|Dgk | dx

2≤k≤n Qρ (y)

∫ c ∑ 2 − |gk − (gk )Q2ρ (y) | dx. ≤ 2 ρ Q2ρ (y) 2≤k≤n

So from (4.20) and (4.21), the lemma holds for y 1 < 0. With (4.7) and (4.15), set G : Q2r → Rn as G = (g1 , g2 , . . . , gn ).

(4.22)

For the Sobolev conjugate 2∗ , the following reverse H¨older type inequality holds. Lemma 4.3.

Under the assumption (4.3), let G as in (4.22). Then we have that (∫ −

2∗

|G − ζ|

) 1∗ ∫ 2 dx ≤ c−

Qr (x0 )

|G − ζ| dx

Q2r (x0 )

for any ζ ∈ Rn . Proof . Fix any Q2ρ (y) ⊂⊂ Q2r (x0 ). Then by the Sobolev type embedding, (∫ − Qρ (y)

|G − ζ|

2∗

) dx

2 2∗

∫ ≤ c−

2

2

ρ2 |DG| + |G − ζ| dx,

(4.23)

Qρ (y)

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where 2∗ > 2 is the Sobolev conjugate. By (4.13) in Lemmas 4.1 and 4.2, ∫ ∫ 2 2 2 − ρ2 |DG| dx = ρ2 − |Dg1 | + · · · + |Dgn | dx Qρ (y) Qρ (y) ∑ ∫ 2 ≤c − |gk − (gk )Q2ρ (y) | dx 2≤k≤n Q2ρ (y)

∑ ∫ ≤c −

(4.24)

2

|gk − ζ k | dx

2≤k≤n Q2ρ (y)

∫ ≤ c−

2

|G − ζ| dx.

Q2ρ (y)

In light of (4.23) and (4.24), we obtain (∫ −

|G − ζ|

2∗

) 2∗ ∫ 2 ≤ c− dx

2

|G − ζ| dx.

Q2ρ (y)

Qρ (y)

Since Q2ρ (y) ⊂⊂ Q2r (x0 ) was arbitrary chosen, the lemma holds from Lemma 3.2. So we obtain the following excess decay estimate on G. Lemma 4.4.

Under the assumption (4.3), let G as in (4.22). Then we have that ∫ ( ρ )α ∫ − |G − (G)Qρ (x0 ) | dx ≤ c − |G − (G)Q2r (x0 ) | dx r Qρ (x0 ) Q2r (x0 )

for any 0 < 2ρ ≤ r. Proof . From the Sobolev–Poincar´e’s inequality, Lemma 4.2 and (4.12) in Lemma 4.1, we find that ∫ ∫ 2 2 − |g1 − (g1 )Qρ (x0 ) | dx ≤ cρ2 − |Dg1 | dx Qρ (x0 ) Qρ (x0 ) ∑ ∫ 2 ≤c − |gk − (gk )Q2ρ (x0 ) | dx ≤c ≤c

2≤k≤n Q2ρ (x0 ) ( ρ )2α ∑ ∫

r

−

2≤k≤n Qr (x0 )

2

|gk − (gk )Qr (x0 ) | dx

( ρ )2α ∫ 2 − |G − (G)Qr (x0 ) | dx. r Qr (x0 )

From Lemma 4.1, we obtain that ( ρ )2α ∫ ∑ ∫ 2 2 − |gk − (gk )Qρ (x0 ) | dx ≤ c − |G − (G)Qr (x0 ) | dx. r Qρ (x0 ) Qr (x0 ) 2≤k≤n

By combining the above two estimates, we obtain from Lemma 4.3 that ∫ ( ρ )2α ∫ 2 2 − |G − (G)Qρ (x0 ) | dx ≤ c − |G − (G)Qr (x0 ) | dx r Qρ (x0 ) Qr (x0 ) ( )2 ( ρ )2α ∫ ≤c − |G − (G)Q2r (x0 ) | dx . r Q2r (x0 ) by taking ζ = (G)Q2r (x0 ) . So the lemma follows from H¨older’s inequality. Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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5. Comparison estimates We obtain the comparison estimates by separating the interior and the boundary cases. For the interior comparison estimates, we obtain an excess decay estimate on Qr (x0 ) with the assumption x10 > r. Also for the boundary comparison estimates, we obtain an excess decay estimate on Q4r (x0 ) with the assumption 0 ≤ x10 ≤ r. 5.1. Interior comparison estimates Under the assumption (1.2)–(1.4), let u be a weak solution of div a(Du, x1 , x′ ) = µ in Qr . In [12,17], the desired excess decay estimate is obtained for the case that a(ξ, x1 , x′ ) is Dini-continuous on (x1 , x′ ). In our problem, we assume that a(ξ, x1 , x′ ) is measurable on x1 -variable, but the proof for the excess decay estimate in our problem similar to that of [2,12,17,22,26]. So for the interior case, we only state results in this subsection. We also remark that the desired interior excess decay estimate for parabolic problems was obtained in [7]. Let v be the weak solution of { div a(Dv, x1 , x′ ) = 0 in Qr (x0 ), v

= u

on ∂Qr (x0 ),

div a(Dg, x1 , x′0 )

=

0

in Q r2 (x0 ),

g

=

v

on ∂Q r2 (x0 ).

and g be the weak solution of {

The comparison estimate for Du and Dv was obtained in several papers. By repeating the proof such as in [12, Lemma 3.3], [22, Lemma 2.5], and [26, Lemma 5.2], one can prove that ∫ c|µ|(Qr (x0 )) − |Du − Dv| dx ≤ . rn−1 Qr (x0 ) Also the comparison estimate for Dv and Dg was obtained in several papers. By repeating the proof such as in [12, Lemma 3.4, 3.5] and [17, Lemma 3.4], one can prove that ∫ ∫ − |Dv − Dg| dx ≤ c ω(r)− |Dv| dx. Q r (x0 )

Qr (x0 )

2

So we obtain that [

] ∫ |µ|(Qr (x0 )) |Du − Dg| dx ≤ c + ω(r)− |Du| dx . rn−1 Q r (x0 ) Qr (x0 )

∫ −

(5.1)

2

We set

{

U = (a1 (Du, x1 , x′0 ), D2 u, . . . , Dn u), G = (a1 (Dg, x1 , x′0 ), D2 g, . . . , Dn g).

Then by [6, Lemma 4.3], we have the following excess decay estimate for G. ∫ ( ρ )α ∫ − |G − (G)Qρ (x0 ) | dx ≤ c − |G − (G)Q r (x0 ) | dx, r 2 Qρ (x0 ) Q r (x0 )

(5.2)

(5.3)

2

for any 0 < 2ρ ≤ r. Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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From Lemma 3.1, we have that |Du| ≤ c|U |. Since |U − G| ≤ c|Du − Dg|, we obtain from (5.1) that [ ] ∫ ∫ |µ|(Qr (x0 )) |U − G| dx ≤ c + ω(r)− − |Du| dx rn−1 Qr (x0 ) Q r (x0 ) 2 [ ] ∫ |µ|(Qr (x0 )) + ω(r)− ≤c |U | dx , rn−1 Qr (x0 ) and we find from (5.3) that ∫ ( ρ )α ∫ − − |U − (U )Qρ (x0 ) | dx ≤ c |U − (U )Q r (x0 ) | dx r 2 Q r (x0 ) Qρ (x0 ) 2 ] [ ( )n ∫ r |µ|(Qr (x0 )) |U | dx . + ω(r)− +c ρ rn−1 Qr (x0 ) One can easily check that ∫ ∫ − |U − (U )Q r (x0 ) | dx ≤ − Q r (x0 )

2

Q r (x0 )

2

∫ |U − (U )Qr (x0 ) | dx + −

Q r (x0 )

2

|(U )Qr (x0 ) − (U )Q r (x0 ) | dx 2

2

∫ n+1 ≤2 − Qr (x0 )

|U − (U )Qr (x0 ) | dx,

and we obtain that ∫ − |U − (U )Qρ (x0 ) | dx Qρ (x0 )

] ( )n [ ∫ ( ρ )α ∫ r |µ|(Qr (x0 )) − |U − (U )Qr (x0 ) | dx + c + ω(r)− |U | dx ≤c r ρ rn−1 Qr (x0 ) Qr (x0 )

(5.4)

for any 0 < 2ρ ≤ r. 5.2. Comparison estimates near the boundary We assume that 0 ≤ x10 ≤ r.

(5.5)

Under the assumptions (1.2), (1.3), (1.4), (2.1) and (2.2), let u be a weak solution of { div a(Du, x1 , x′ ) = µ in Q4r (x0 ) ∩ Rn+ , u = ϕ

on Q4r (x0 ) ∩ Rn0 ,

where ϕ ∈ C 1 (Q4r (x0 ) ∩ Rn0 ). With (5.5), let v be the weak solution of { div a(Dv, x1 , x′ ) = 0 in Q4r (x0 ) ∩ Rn+ , [ ] v = u on ∂ Q4r (x0 ) ∩ Rn+ , let w be the weak solution of { div a(Dw, x1 , x′ )

=

0

w

=

v − ϕ + Dx′ ϕ(x′0 ) · x′

and g be the weak solution of { div a(Dg, x1 , x′0 ) g

=

0

= w

in Q4r (x0 ) ∩ Rn+ , [ ] on ∂ Q4r (x0 ) ∩ Rn+ ,

in Q3r (x0 ) ∩ Rn+ , [ ] on ∂ Q3r (x0 ) ∩ Rn+ .

(5.6)

(5.7)

(5.8)

(5.9)

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n In (5.8), we regarded the boundary data ϕ as a function on Q2R (0) ∩ R+ by defining ϕ(x1 , x′ ) = ϕ(0, x′ ) for 1 every 0 < x < 2R. In view of (5.5), we find that

w = v − ϕ + Dx′ ϕ(x′0 ) · x′ on Q4r (x0 ) ∩ Rn0 = T4r (x′0 ).

(5.10)

The comparison estimate between Du and Dv is similar to that of the interior case as in the following lemma. Lemma 5.1.

Under the assumption (5.5), we have that ∫ c|µ|(Q4r (x0 ) ∩ Rn+ ) . − |Du − Dv| dx ≤ rn−1 Q4r (x0 )∩Rn +

∫ Proof . Suppose that Q (x )∩Rn |Du − Dv| dx ̸= 0, other-wise the lemma holds trivially. For fixed h > 0 4r 0 + and θ > 1, we claim that ∫ 2 |Du − Dv| dx ch1−θ ≤ · |µ|(Q4r (x0 ) ∩ Rn+ ). (5.11) θ n (h + |u − v|) θ − 1 Q4r (x0 )∩R +

The claim (5.11) was obtained for the interior case in [23, Lemma 10.1], and the approach for [23, Lemma 10.1] can be also applied to the boundary case. For sake of the convenience of the reader, we give a proof here. For φ ∈ W01,2 (Q4r (x0 ) ∩ Rn+ ) ∩ L∞ (Q4r (x0 ) ∩ Rn+ ), (5.6) and (5.7) imply that ∫ ∫ ⟨a(Du, x) − a(Dv, x), Dφ⟩ dx = φ dµ. (5.12) Q4r (x0 )∩Rn +

Q4r (x0 )∩Rn +

Take the test function φ± = h1−θ − (h + |u − v|± )1−θ ∈ W01,2 (Q4r (x0 ) ∩ Rn+ ) ∩ L∞ (Q4r (x0 ) ∩ Rn+ ) for (5.12) ⏐ to find that ⏐ ∫ ⏐ ⟨a(Du, x) − a(Dv, x), D(u − v)± ⟩ dx ⏐⏐ ⏐ |I± | = ⏐(θ − 1)− ⏐ ⏐ ⏐ (h + (u − v)± )θ Q4r (x0 )∩Rn + ⏐∫ ⏐ ⏐ ⏐ ⏐ ⏐ =⏐ ⟨a(Du, x) − a(Dv, x), Dφ± ⟩ dx⏐ ⏐ Q4r (x0 )∩Rn ⏐ + ⏐∫ ⏐ ⏐ ⏐ ⏐ ⏐ =⏐ φ± dµ⏐ ⏐ Q4r (x0 )∩Rn ⏐ +

≤ h1−θ |µ|(Q4r (x0 ) ∩ Rn+ ), where the fact that 0 ≤ φ± ≤ h1−θ was used in the above estimate. So the claim (5.11) follows from the ellipticity condition (1.3). ∫ We choose n θ= and h=r |Du − Dv| dx. (5.13) n−1 Q4r (x0 )∩Rn +

By H¨ older’s inequality and (5.11), ∫ ∫ |Du − Dv| dx = Q4r (x0 )∩Rn +

|Du − Dv|

Q4r (x0 )∩Rn +

(h + |u − v|)

[∫ ≤ Q4r (x0 )∩Rn +

2

θ

θ 2

· (h + |u − v|) 2 dx

|Du − Dv| dx (h + |u − v|)θ

] 21 [∫

] 12 θ

(h + |u − v|) dx Q4r (x0 )∩Rn +

(5.14)

] 21 ] 12 [∫ h1−θ θ ≤c · |µ|(Q4r (x0 ) ∩ Rn+ ) hθ + |u − v| dx . θ−1 Q4r (x0 )∩Rn [

+

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By the Sobolev embedding and (5.13), we have that (

∫

θ

|u − v| dx ≤

)θ

∫ |Du − Dv| dx

cr

Q4r (x0 )∩Rn +

≤ chθ .

(5.15)

Q4r (x0 )∩Rn +

So from (5.13)–(5.15), we find that ∫ Q4r (x0 )∩Rn +

|Du − Dv| dx ≤ cr|µ|(Q4r (x0 ) ∩ Rn+ )

and the lemma follows from (5.5). Lemma 5.2.

Under the assumption (5.5), we have that ∫ − |Dv − Dw| dx ≤ c osc′ Dx′ ϕ. Q4r (x0 )∩Rn +

T4r (x0 )

Proof . We test (5.7) and (5.8) by v − w + ϕ − Dx′ ϕ(x′0 ) · x′ . Then we find that ∫ − ⟨a(Dv, x1 , x′ ) − a(Dw, x1 , x′ ), Dv − Dw + Dx ϕ − Dx ϕ(x′0 )⟩ dx = 0, Q4r (x0 )∩Rn +

and it follows from (1.3) that ∫ −

∫ 2 |Dv − Dw| dx ≤ c−

Q4r (x0 )∩Rn +

2

Q4r (x0 )∩Rn +

∫ = c−

|Dx ϕ − Dx ϕ(x′0 )| dx 2

Q4r (x0 )∩Rn +

|Dx′ ϕ − Dx′ ϕ(x′0 )| dx )2

( ≤c

osc Dx′ ϕ

T4r (x′0 )

.

So the lemma follows by H¨ older’s inequality. To compare Dw and Dg, we need the following reverse H¨older type inequality. Lemma 5.3.

Under the assumption (5.5), we have that ) 21

(∫ −

2

|Dw| dx

(∫ ≤c −

) |Dw| +

Q4r (x0 )∩Rn +

Q3r (x0 )∩Rn +

|Dx′ ϕ(x′0 )| dx

.

Proof . For the sake of the simplicity, we denote β ′ = Dx′ ϕ(x′0 )

and

β = (0, β ′ ) = (0, Dx′ ϕ(x′0 )).

In view of (5.5)–(5.7), we have that v=u=ϕ

on T4r (x′0 ).

(5.16)

It follows from (5.10) and (5.16) that w − β ′ · x′ = v − ϕ = 0

on

T4r (x′0 ).

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Let w ˆ be the zero extension of w − β ′ · x′ from Q4r (x0 ) ∩ Rn+ to Q4r (x0 ). Then { w − β ′ · x′ in Q4r (x0 ) ∩ Rn+ , w ˆ= 0 on Q4r (x0 ) \ Rn+ .

(5.17)

2n Let 2∗ = n+2 . Then by considering two cases (1) Q 3ρ (y) ⊂ Rn+ and (2) Q 3ρ (y) ̸⊂ Rn+ , we prove the following 2 2 assertion that (∫ ) 22∗ ∫ 2 2 2 − (5.18) |Dw| ˆ dx ≤ c− |Dw| ˆ ∗ + |β| ∗ dx Qρ (y)

Q2ρ (y)

for any Q2ρ (y) ⊂⊂ Q4r (x0 ). Choose any Q2ρ (y) ⊂⊂ Q4r (x0 ). First, suppose that Q 3ρ (y) ⊂ Rn+ . Then Q 3ρ (y) ⊂ Q4r (x0 ) ∩ Rn+ . Choose 2 2 a cut-off function η ∈ Cc∞ (Q 3ρ (y)) such that 2

η ∈ Cc∞ (Q 3ρ (y)),

0 ≤ η ≤ 1,

η = 1 on Qρ (y)

2

[ Take a test function w − (w)Q 3ρ

]

2 (y)

∫ −

⟨

|Dη| ≤

and

c . ρ

(5.19)

η 2 ∈ W01,2 (Q 3ρ (y)) for (5.8) to find that 2

([ a(Dw, x), D w − (w)Q 3ρ

] )⟩ η2 dx = 0,

2 (y)

Q 3ρ (y) 2

and so the ellipticity condition (1.3) gives that ∫ ∫ 2 |Dw| η 2 dx ≤ − λ−

⟨a(Dw, x), η 2 Dw⟩ dx

Q 3ρ (y)

Q 3ρ (y) 2

∫2 = −−

⟨ [ a(Dw, x), w − (w)Q 3ρ

] ⟩ 2ηDη dx

2 (y)

Q 3ρ (y) 2

∫ ≤ c− Q 3ρ (y)

⏐ ⏐ |Dw| ⏐w − (w)Q 3ρ

2 (y)

⏐ ⏐ ⏐ |η| |Dη| dx.

2

By Young’s inequality, ∫ −

∫ 2 |Dw| η 2 dx ≤ c−

Q 3ρ (y)

Q 3ρ (y)

2

⏐ ⏐ ⏐w − (w)Q 3ρ

2 (y)

⏐2 ⏐ 2 ⏐ |Dη| dx,

2

and the choice of the cut-off function η in (5.19) yields that ∫ ∫ 2 2 − |Dw| dx ≤ − |Dw| η 2 dx Qρ (y)

Q 3ρ (y)

∫ 2 ≤ c− Q 3ρ (y)

⏐ ⏐ ⏐w − (w)Q 3ρ

2 (y)

⏐2 ⏐ 2 ⏐ |Dη| dx

(5.20)

2

≤

∫ ⏐2 ⏐ c ⏐ ⏐ − − (w) ⏐w ⏐ dx. Q 3ρ ρ2 Q 3ρ (y) 2 (y) 2

From Sobolev–Poincar´e’s inequality, we have ⎛ ∫ ⎝− Q 3ρ (y) 2

⏐ ⏐ ⏐w − (w)Q 3ρ

2 (y)

⎞ 21 ⎛ ∫ ⏐2 ⏐ ⎠ ⎝ ≤ cρ − ⏐ dx

⎞ 21

∗

2∗

|Dw|

dx⎠

.

(5.21)

Q 3ρ (y) 2

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So it follows from (5.20) and (5.21) that ⎞ 22

⎛ ∫ 2 |Dw| dx ≤ c ⎝−

∫ − Qρ (y)

∗

|Dw|

2∗

dx⎠

.

Q 3ρ (y) 2

Since Q 3ρ (y) ⊂ Rn+ , recall from (5.17) that w ˆ = w − β ′ · x′ in Q 3ρ (y). 2

2

) 22

(∫ 2 |Dw ˆ + β| dx ≤ c −

∫ − Qρ (y)

∗

|Dw ˆ + β|

2∗

,

dx

Q2ρ (y)

and the assertion (5.18) holds for the case Q 3ρ (y) ⊂ Rn+ . 2 Now, suppose that Q 3ρ (y) ̸⊂ Rn+ . Then by the fact that Q2ρ (y) ⊂ Q4r (x0 ), 2

|Q2ρ (y) ∩ [Q4r (x0 ) \ Rn+ ]| ≥ cρn ≥ c|Q2ρ (y)|.

(5.22)

Since w ˆ = 0 in Q4r (x0 ) \ Rn+ , we have from (5.22) and the Sobolev–Poincar´e type inequality in [14, Theorem 3.16] that ) 22 (∫ ∫ ∗

2

|w| ˆ dx ≤ cρ2

− Q2ρ (y)

2∗

−

|Dw| ˆ

dx

.

(5.23)

Q2ρ (y)

Choose a cut-off function η such that η ∈ Cc∞ (Q2ρ (y)),

0 ≤ η ≤ 1,

η = 1 on Qρ (y)

and

|Dη| ≤

c . ρ

(5.24)

Since Q2ρ (y) ⊂ Q4r (x0 ), we have from (5.24) that η ∈ Cc∞ (Q4r (x0 )). By (5.17), we also have that w ˆ = 0 in n Q4r (x0 ) \ R+ . Thus we find that ( ) wη ˆ 2 ∈ W01,2 Q4r (x0 ) ∩ Rn+ . ( ) Take the test function wη ˆ 2 ∈ W01,2 Q4r (x0 ) ∩ Rn+ for (5.8) to find that ∫

⟨a(Dw, x), D[wη ˆ 2 ]⟩ dx = 0.

Q4r (x0 )∩Rn +

Since Q2ρ (y) ⊂ Q4r (x0 ), we obtain from (5.24) that ∫

⟨a(Dw, x), D[wη ˆ 2 ]⟩ dx = 0,

Q2ρ (y)∩Rn +

and by a direct calculation, ∫

⟨a(Dw, x) − a(β, x), η 2 Dw⟩ ˆ dx

Q2ρ (y)∩Rn +

∫ =−

∫ ⟨a(Dw, x) − a(β, x), w2ηDη⟩ ˆ dx −

Q2ρ (y)∩Rn +

(5.25) 2

⟨a(β, x), η Dw ˆ + w2ηDη⟩ ˆ dx. Q2ρ (y)∩Rn +

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From (5.17), we have that Dw ˆ = Dw − β in Rn+ . So (1.3) and (5.25) imply that ∫ ∫ 2 |Dw| ˆ η 2 dx ≤ c |η||Dw∥ ˆ w||Dη| ˆ + |β|(η 2 |Dw| ˆ + |w||η∥Dη|) ˆ dx. Q2ρ (y)∩Rn +

Q2ρ (y)∩Rn +

By Young’s inequality, ∫

∫

2 2

2

2

2

|w| ˆ |Dη| + |β| η 2 dx.

|Dw| ˆ η dx ≤ c Q2ρ (y)∩Rn +

(5.26)

Q2ρ (y)∩Rn +

By (5.17), w ˆ = 0 in Q4r (x0 ) \ Rn+ . Since Q2ρ (y) ⊂ Q4r (x0 ), we have that w ˆ = 0 in Q2ρ (y) \ Rn+ , and it follows from (5.26) that ∫ ∫ 2 |Dw| ˆ η 2 dx ≤ c Q2ρ (y)

2

∫

2

2

|β| η 2 dx.

|w| ˆ |Dη| dx + c

Q2ρ (y)

(5.27)

Q2ρ (y)

From (5.23), (5.27) and the choice of cut-off function η in (5.24), we have that ∫ ∫ 2 2 − |Dw| ˆ dx ≤ − |Dw| ˆ η 2 dx Qρ (y) Q2ρ (y) ∫ ∫ 2 2 2 ≤ c− |w| ˆ |Dη| dx + c− |β| η 2 dx Q2ρ (y) Q2ρ (y) ∫ ∫ c 2 2 |w| ˆ dx + c− |β| dx ≤ 2− ρ Q2ρ (y) Q2ρ (y) ) 22 (∫ ∗

2∗

≤c −

|Dw| ˆ

dx

2

+ c|β| .

Q2ρ (y)

So (5.18) holds for the case Q 3ρ (y) ̸⊂ 2

Rn+ .

By considering two cases (1) Q 3ρ (y) ⊂ Rn+ and (2) Q 3ρ (y) ̸⊂ Rn+ , we have the assertion (5.18). Since 2

2

Q2ρ (y) ⊂⊂ Q4r (x0 ) in (5.18) was arbitrary chosen, by Lemma 3.2 for s = |β| and χ0 = (∫ −

) 21 2

|Dw| ˆ dx

Q3r (x0 )

∫ ≤ c−

2 2∗

> 1, we have that

|Dw| ˆ + |β| dx.

Q4r (x0 )

So the lemma follows from (5.5), (5.17) and the fact that β = (0, Dx′ ϕ(x′0 )). With the reverse H¨ older type inequality in Lemma 5.3, we obtain the following comparison estimate for Dw and Dg. Lemma 5.4.

Under the assumption (5.5), we have that ∫ ∫ − |Dw − Dg| dx ≤ cω(3r)− Q3r (x0 )∩Rn +

Q4r (x0 )∩Rn +

|Dw| + |Dx′ ϕ(x′0 )| dx.

Proof . We test (5.8) and (5.9) by w − g to find that ∫ − ⟨a(Dw, x1 , x′0 ) − a(Dg, x1 , x′0 ), Dw − Dg⟩ dx Q3r (x0 )∩Rn +

∫ =− Q3r (x0 )∩Rn +

⟨a(Dw, x1 , x′0 ) − a(Dw, x1 , x), Dw − Dg⟩ dx.

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It follows from (1.3) and (1.4) that ∫ ∫ 2 − |Dw − Dg| dx ≤ cω(3r)− Q3r (x0 )∩Rn +

17

|Dw||Dw − Dg| dx.

Q3r (x0 )∩Rn +

By Young’s inequality and Lemma 5.3 we have that ∫ ∫ 2 2 − |Dw − Dg| dx ≤ c[ω(3r)] − Q3r (x0 )∩Rn +

2

|Dw| dx

Q3r (x0 )∩Rn +

)2

(∫ ≤ c[ω(3r)]2 − Q4r (x0 )∩Rn +

|Dw| + |Dx′ ϕ(x′0 )| dx

,

and the lemma follows by H¨ older’s inequality. By combining Lemmas 5.1, 5.2 and 5.4, we obtain the following comparison estimates for Du and Dg. Lemma 5.5.

Under the assumption (5.5), we have that ] [ ∫ |µ|(Q4r (x0 ) ∩ Rn+ ) + osc Dx′ ϕ − |Du − Dg| dx ≤ c rn−1 T4r (x′0 ) Q3r (x0 )∩Rn + ∫ + cω(4r)− |Du| + |Dx′ ϕ(x′0 )| dx. Q4r (x0 )∩Rn +

Proof . By Lemma 5.4, ∫ ∫ − |Dw − Dg| dx ≤ cω(3r)− Q3r (x0 )∩Rn +

Q4r (x0 )∩Rn +

∫ ≤ cω(3r)− Q4r (x0 )∩Rn +

|Dw| + |Dx′ ϕ(x′0 )| dx |Du| + |Du − Dw| + |Dx′ ϕ(x′0 )| dx.

In view of Lemmas 5.1 and 5.2, ∫ ∫ − |Du − Dw| dx ≤ − Q4r (x0 )∩Rn +

|Du − Dv| + |Dv − Dw| dx

Q4r (x0 )∩Rn +

] |µ|(Q4r (x0 ) ∩ Rn+ ) + osc Dx′ ϕ . ≤c rn−1 T4r (x′0 ) [

and the lemma follows by combining the above two estimates. For G in (4.22), which is an extension of (a1 (Dg, x1 , x′0 ), D2 g, . . . , Dn g), we obtained an excess decay estimate of G in Lemma 4.4. So we derive an excess decay estimate for an extension U : Q4r (x0 ) → Rn of (a1 (Du, x1 , x′0 ), D2 u − D2 ϕ, . . . , Dn u − Dn ϕ) which is defined as U = (u1 , . . . , un ),

(5.28)

where {

u1 is the even extension of a1 (Du, x1 , x′0 ) from Q4r (x0 ) ∩ Rn+ to Q4r (x0 ), uk is the odd extension of Dk u − Dk ϕ from Q4r (x0 ) ∩ Rn+ to Q4r (x0 )

(k ∈ [2, n]).

(5.29)

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Lemma 5.6.

Under the assumption (5.5), we have that ∫ ( ρ )α ∫ − |U − (U )Q4r (x0 ) | dx − |U − (U )Qρ (x0 ) | dx ≤ c r Q4r (x0 ) Qρ (x0 ) ] ( )n [ |µ|(Q4r (x0 ) ∩ Rn+ ) r + osc′ Dx′ ϕ +c ρ rn−1 T4r (x0 ) ] ( )n [ ∫ r +c ω(4r)− |U | + |Dx′ ϕ(x′0 )| dx , ρ Q4r (x0 )

for any 0 < 2ρ ≤ r. Proof . We set G : Q3r (x0 ) → Rn as G = (g1 , . . . , gn ), where {

(5.30)

g1 is the even extension of a1 (Dg, x1 ) from Q3r (x0 ) ∩ Rn+ to Q3r (x0 ), gk is the odd extension of Dk g − Dk ϕ(x′0 ) from Q3r (x0 ) ∩ Rn+ to Q3r (x0 )

(k ∈ [2, n]).

In view of (5.5), we obtain from Lemma 4.4 that ∫ ( ρ )α ∫ − |G − (G)Q2r (x0 ) | dx − |G − (G)Qρ (x0 ) | dx ≤ c r Q2r (x0 ) Qρ (x0 )

(5.31)

(5.32)

for any 0 < 2ρ ≤ r. In view of (5.29) and (5.31), we find that for any k ∈ [2, n], |uk − gk | ≤ |Dk u − Dk g| + |Dk ϕ − Dk ϕ(x′0 )| ≤ |Du − Dg| + osc Dx′ ϕ in Q3r (x0 ) ∩ Rn+ . T3r (x′0 )

Since 0 ≤ x10 ≤ r, we have from (5.29) and (5.31) that for any k ∈ [2, n], ∫ ∫ − |uk − gk | dx ≤ 2− |uk − gk | dx Q3r (x0 )∩Rn +

Q3r (x0 )

∫ ≤ 2− Q3r (x0 )∩Rn +

(5.33) |Du − Dg| dx + 2 osc′ Dx′ ϕ. T3r (x0 )

In view of (5.29) and (5.31), we have that |u1 − g1 | ≤ |a1 (Du, x1 , x′0 ) − a1 (Dg, x1 , x′0 )| ≤ c|Du − Dg| in Q3r (x0 ) ∩ Rn+ , and that

∫ − Q3r (x0 )

∫ |u1 − g1 | dx ≤ 2−

Q3r (x0 )∩Rn +

∫ |u1 − g1 | dx ≤ 2−

|Du − Dg| dx.

(5.34)

Q3r (x0 )∩Rn +

Here, we have used the fact that 0 ≤ x10 ≤ r in (5.34). So by the definition of U and G in (5.28) and (5.30), it follows from (5.33) and (5.34) that ∫ ∫ − |U − G| dx = − |(u1 , . . . , un ) − (g1 , . . . , gn )| dx Q3r (x0 ) Q3r (x0 ) [∫ ] (5.35) ≤c − |Du − Dg| dx + osc′ Dx′ ϕ . Q3r (x0 )∩Rn +

T3r (x0 )

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In the case of (5.5), we find from Lemma 5.5 that ∫ − |Du − Dg| dx Q3r (x0 )∩Rn +

] ∫ |µ|(Q4r (x0 ) ∩ Rn+ ) |Du| + |Dx′ ϕ(x′0 )| dx. + osc′ Dx′ ϕ + cω(4r)− ≤c rn−1 T4r (x0 ) Q4r (x0 )∩Rn [

(5.36)

+

We find from Lemma 3.1 that |Du| ≤ c|U |. So it follows from (5.35) and (5.36) that ∫ − |U − G| dx Q3r (x0 ) ] [ ∫ |µ|(Q4r (x0 ) ∩ Rn+ ) ≤c |U | + |Dx′ ϕ(x′0 )| dx. + osc′ Dx′ ϕ + cω(4r)− rn−1 T4r (x0 ) Q4r (x0 )

(5.37)

From (5.32) and (5.37), we have that ∫ ( ρ )α ∫ − |U − (U )Qρ (x0 ) | dx ≤ c − |U − (U )Q2r (x0 ) | dx r Qρ (x0 ) Q2r (x0 ) ] ( )n [ |µ|(Q4r (x0 ) ∩ Rn+ ) r +c + osc′ Dx′ ϕ ρ rn−1 T4r (x0 ) ] ( )n [ ∫ r ′ +c ω(4r)− |U | + |Dx′ ϕ(x0 )| dx , ρ Q4r (x0 ) for any 0 < 2ρ ≤ r. One can easily check that ∫ ∫ ∫ − |U − (U )Q2r (x0 ) | dx ≤ − |U − (U )Q4r (x0 ) | dx + − |(U )Q4r (x0 ) − (U )Q2r (x0 ) | dx Q2r (x0 ) Q2r (x0 ) Q2r (x0 ) ∫ ≤ 2n+1 − |U − (U )Q4r (x0 ) | dx, Q4r (x0 )

and the lemma follows. 6. Point-wise Riesz potential estimates To prove the main theorem, we first obtain the desired excess decay estimate by combining the interior case and the boundary case. For a weak solution u of (2.4), we set U : Q2R → Rn as U = (u1 , . . . , un ), where

{

u1 is the even extension of a1 (Du, x1 , x′0 ) from Q+ 2R to Q2R , uk is the odd extension of Dk u − Dk ϕ from Q+ 2R to Q2R

Lemma 6.1.

(6.1)

(k ∈ [2, n]).

(6.2)

For any x0 ∈ Q+ R , we have that ∫ ( ρ )α ∫ − |U − (U )Q4r (x0 ) | dx − |U − (U )Qρ (x0 ) | dx ≤ c r Q4r (x0 ) Qρ (x0 ) ] ( )n [ |µ|(Q4r (x0 ) ∩ Rn+ ) r +c + osc Dx′ ϕ ρ rn−1 T4r (x′0 ) [ ] ( )n ∫ r ′ +c ω(4r)− |U | + |Dx′ ϕ(x0 )| dx , ρ Q4r (x0 )

for any 0 < ρ ≤ 4r ≤ R. Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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Proof . We prove the lemma by considering the cases (1) 2r < ρ ≤ 4r and (2) 0 < ρ ≤ 2r . If (1) 2r < ρ ≤ 4r, then from the fact that ∫ ∫ − |U − (U )Qρ (x0 ) | dx ≤ − |U − (U )Q4r (x0 ) | + |(U )Q4r (x0 ) − (U )Qρ (x0 ) | dx Qρ (x0 )

Qρ (x0 )

)n ∫ 4r − |U − (U )Q4r (x0 ) | dx ρ Q4r (x0 ) ( ρ )α ∫ − ≤c |U − (U )Q4r (x0 ) | dx, r Q4r (x0 ) (

≤2

the lemma holds. Next, suppose that (2) 0 < ρ ≤ 2r holds. If x10 > r, then we have that Qr (x0 ) ⊂ Rn+ . So by comparing (5.2) and (6.1), we obtain from (5.4) that ∫ − |U − (U )Qρ (x0 ) | dx Qρ (x0 ) ] ( )n [ ∫ ( ρ )α ∫ r |µ|(Qr (x0 )) + ω(r)− |U | dx ≤c − |U − (U )Qr (x0 ) | dx + c r ρ rn−1 Qr (x0 ) Qr (x0 ) ] ( )n [ ∫ ( ρ )α ∫ |µ|(Q4r (x0 ) ∩ Rn+ ) r + ω(4r)− |U | dx , ≤c − |U − (U )Q4r (x0 ) | dx + c r ρ rn−1 Q4r (x0 ) Q4r (x0 ) and the lemma holds for x10 > r. If 0 ≤ x10 ≤ r, then by comparing (5.28) and (6.1), Lemma 5.6 implies that ∫ ( ρ )α ∫ − |U − (U )Qρ (x0 ) | dx ≤ c − |U − (U )Q4r (x0 ) | dx r Qρ (x0 ) Q4r (x0 ) ] ( )n [ |µ|(Q4r (x0 ) ∩ Rn+ ) r +c + osc Dx′ ϕ ρ rn−1 T4r (x′0 ) [ ] ( )n ∫ r ′ +c ω(4r)− |U | + |Dx′ ϕ(x0 )| dx . ρ Q4r (x0 ) So the lemma holds for 0 ≤ x10 ≤ r. This completes the proof. We are now ready to prove Theorem 2.1. Proof . Fix a Lebesgue point x0 = (x10 , x′0 ) ∈ Q+ R of Du. Then from (6.1) and Lemma 3.1, we find that |Du| ≤ c(|U | + |Dx′ ϕ|) ≤ c(|Du| + |Dx′ ϕ|)

in Q+ 2R .

It follows from Lemma 6.1 that ( )α ∫ ∫ ρ1 − |U − (U )Qρ1 (x0 ) | dx ≤ c2 − |U − (U )Qρ2 (x0 ) | dx ρ2 Qρ1 (x0 ) Qρ2 (x0 ) ] ( )n [ |µ|(Qρ2 (x0 ) ∩ Rn+ ) ρ2 + osc Dx′ ϕ +c ρ1 Tρ2 (x′0 ) ρn−1 2 ] ( )n [ ∫ ρ2 +c ω(ρ2 )− |U | + |Dx′ ϕ(x′0 )| dx , ρ1 Qρ (x0 )

(6.3)

(6.4)

2

for any 0 < ρ1 ≤ ρ2 ≤ R. So there exists a small universal constant δ = δ(n, λ, Λ) ∈ (0, 1/2] such that c2 δ α ≤

1 . 16

(6.5)

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It follows from (6.4) and (6.5) that ∫ ∫ 1 − |U − (U )Qδρ (x0 ) | dx ≤ − |U − (U )Qρ (x0 ) | dx 16 Qρ (x0 ) Qδρ (x0 ) ] [ |µ|(Qρ (x0 ) ∩ Rn+ ) + osc′ Dx′ ϕ + c3 ρn−1 Tρ (x0 ) ] [ ∫ ′ + c3 ω(ρ)− |U | + |Dx′ ϕ(x0 )| dx ,

(6.6)

Qρ (x0 )

for any 0 < ρ ≤ R. Set c1 = c1 (n, λ, Λ) ≥ 1 in (2.3) as { c1 = max

} 16δ −2n c3 ,1 . log 2

Then by the assumption (2.3) in Theorem 2.1, δ −2n c3 log 2

∫ 0

2r

ω(ρ) dρ 1 ≤ ρ 16

and

r ∈ (0, R].

(6.7)

For i ≥ 0, set ri = δ i r,

Qi = Qri (x0 ),

∫ Ei = − |U − (U )Qi | dx,

⏐ ⏐∫ ⏐ ⏐ Fi = ⏐⏐− U dx⏐⏐

(6.8)

Qi

Qi

and |µ|(Qi ∩ Rn+ ) + osc′ Dx′ ϕ + ω(ri )|Dx′ ϕ(x′0 )|. Tri (x0 ) rin−1

νi =

(6.9)

Then by taking ρ = ri in (6.6), we find that Ei+1

[ ] ∫ Ei ≤ + c3 ω(ri )− |U | dx + νi 16 Qi

(i ≥ 0).

So from the fact that ∫ ∫ − |U | dx ≤ − |U − (U )Qi | dx + |(U )Qi | ≤ Ei + Fi , Qi

Qi

we find that ] 1 + c3 ω(ri ) Ei + c3 [ω(ri )Fi + νi ] ≤ 16 [

Ei+1

(i ≥ 0).

(6.10)

With the fact that δ = δ(n, λ, Λ) ∈ (0, 1/2] and ⎡

⎤ ∫ 2r ∫ rj ∞ ∑ 1 ω(ρ) dρ 1 ω(ρ) dρ ⎦ δ −2n c3 ω(rj ) ≤ δ −2n c3 ⎣ + log 2 ρ log(1/δ) ρ r r j+1 j=0 j=0 ∫ 2r −2n δ c3 ω(ρ) dρ ≤ , log 2 0 ρ ∞ ∑

we obtain from (6.7) that δ −2n c3

∞ ∑ j=0

ω(rj ) ≤

1 16

and

c3 ω(ri ) ≤

1 16

(i ≥ 0).

(6.11)

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For the sake of the simplicity, by using (6.9) and (6.11), we discover ν := 4δ −n c3

∞ ∑

νj

j=0 [ ∞ ∑

] |µ|(Qj ∩ Rn+ ) ′ + osc Dx′ ϕ + ω(rj )|Dx′ ϕ(x0 )| = 4δ c3 Trj (x′0 ) rjn−1 j=0 [∫ ( ) ] 2r |µ|(Qρ (x0 ) ∩ Rn+ ) dρ ′ ≤c + osc′ Dx′ ϕ + |Dx′ ϕ(x0 )| . ρn−1 ρ Tρ (x0 ) 0 −n

(6.12)

It follows from (6.10) and (6.11) that Ej + c3 [ω(rj )Fj + νj ] (j ≥ 0). 8 We now consider the above estimate for j ∈ {0, 1, . . . , i − 1} and sum up over j, thereby gaining Ej+1 ≤

i ∑

Ei ≤

j=1

i−1 ∑ E0 +2 c3 [ω(rj )Fj + νj ] 4 j=0

(i ≥ 1).

(6.13)

By using the induction, we prove the claim that ∫ Fi ≤ 4δ −n − |U | dx + ν.

(6.14)

Q0

⏐ ⏐∫ ⏐ ⏐ For i = 0, the claim holds from the fact that δ ∈ (0, 1/2] and F0 = ⏐−Q U dx⏐. Next, suppose that (6.14) 0 holds for 0, 1, . . . , i, that is, ∫ F0 , F1 , . . . , Fi ≤ 4δ −n − |U | dx + ν.

(6.15)

Q0

Then we prove that (6.14) holds by showing that ∫ Fi+1 ≤ 4δ −n − |U | dx + ν.

(6.16)

Q0

By the triangle inequality, for any i ≥ 0, ⏐∫ ⏐ ⏐ ⏐ ⏐ ⏐ Fi+1 = ⏐− U dx⏐ ⏐ Qi+1 ⏐ ⏐ ⏐ ⏐∫ ⏐ ∑ ∫ i ⏐∫ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ≤ ⏐− U dx⏐ + U dx − − U dx⏐ ⏐− ⏐ Qj+1 ⏐ Q0 Qj j=0

≤ F0 +

i ∑ j=0

∫ −n δ − |U − (U )Qj | dx, Qj

and so (6.13) gives that Fi+1 ≤ F0 + δ −n

i ∑ j=0

⎡ Ej ≤ F0 + δ −n ⎣

⎤

i ∑

5E0 +2 c3 [ω(rj )Fj + νj ]⎦ 4 j=0

(i ≥ 0).

(6.17)

In view of (6.11), (6.12) and (6.15), we obtain that ⎡ ⎤ ( ) ∑ ∫ i i i ∑ ∑ 2δ −n c3 [ω(rj )Fj + νj ] ≤ 2δ −n c3 ⎣ ω(rj ) 4δ −n − |U | dx + ν + νj ⎦ j=0

j=0

Q0

j=0

(6.18)

[∫ ] 1 ν − |U | dx + ν + . ≤ 2 Q0 2 Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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From (6.8), we have that ∫ F0 ≤ − |U | dx

23

∫ ∫ 5δ −n E0 5δ −n 5δ −n = − |U − (U )Q0 | dx ≤ − |U | dx. 4 4 Q0 2 Q0

and

Q0

So we obtain from (6.17) and (6.18) that ∫ Fi+1 ≤ 4δ −n − |U | dx + ν, Q0

and (6.16) holds. So by an induction, the claim (6.14) follows. Now (6.13) yields that for any i ≥ 0, ∫ ∫ − |U | dx ≤ − |U − (U )Qi | dx + |(U )Qi | Qi

Qi

≤ Ei + Fi ≤

i ∑ E0 +2 c3 [ω(rj )Fj + νj ] + Fi . 4 j=0

It follows from (6.11) and (6.14) that for any i ≥ 0, i ∑

c3 [ω(rj )Fj + νj ] + Fi ≤ c

i ∑

[ ( ) ] ( ) ∫ ∫ −n −n c3 ω(rj ) 4δ − |U | dx + ν + νj + 4δ − |U | dx + ν Q0

j=0

j=0

Q0

[∫ ] ≤ c − |U | dx + ν . Q0

Thus we obtain that

[∫ ] ∫ − |U | dx ≤ c − |U | dx + ν Qi

(i ≥ 0).

Q0

Since Qi = Qδi r (x0 ), we have that [∫ |U | dx ≤ c −

∫ − Qδi r (x0 )

] |U | dx + ν

(i ≥ 0).

(6.19)

Qr (x0 )

Since x0 ∈ Q+ R , we find from (6.3) that for any i ≥ 0, [∫ ∫ −

]

|Du| dx ≤ c −

Qδi r (x0 )∩Rn +

|U | dx +

Qδi r (x0 )∩Rn +

[∫ ≤c −

|U | dx +

Since x0 ∈ Q+ R , we also find from (6.2) and (6.3) that ∫ ∫ ∫ − |U | dx ≤ c− |U | dx ≤ − Qr (x0 )∩Rn +

Since supT

δi r

(x′0 )

(6.20)

]

Qδi r (x0 )

Qr (x0 )

sup |Dx′ ϕ|

Tδi r (x′0 )

Qr (x0 )∩Rn +

sup |Dx′ ϕ| .

Tδi r (x′0 )

|Du| dx + sup |Dx′ ϕ|.

(6.21)

Tr (x′0 )

|Dx′ ϕ| ≤ supTr (x′ ) |Dx′ ϕ| for i ≥ 0, it follows from (6.19)–(6.21) that

∫ − Qδi r (x0 )∩Rn +

0

[∫ |Du| dx ≤ c − Qr (x0 )∩Rn +

] |Du| dx + ν + sup |Dx′ ϕ|

(i ≥ 0).

Tr (x′0 )

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By the triangle inequality and (6.12), we have that sup |Dx′ ϕ| ≤ |Dx′ ϕ(x′0 )| + osc′ Dx′ ϕ ≤ |Dx′ ϕ(x′0 )| + ν,

Tr (x′0 )

and it follows that ∫ −

Tr (x0 )

[∫ |Du| dx ≤ c −

Qδi r (x0 )∩Rn +

] |Du| dx +

Qr (x0 )∩Rn +

|Dx′ ϕ(x′0 )|

Since x0 is a Lebesgue point of Du, by letting i → ∞, we find that [∫ |Du(x0 )| ≤ c −

|Du| dx +

|Dx′ ϕ(x′0 )|

+ν

(i ≥ 0).

(6.22)

] +ν ,

Qr (x0 )

and Theorem 2.1 follows from (6.12). 7. SOLA (Solution obtained by Limit of Approximations) The approximation argument related to SOLA was discussed in many papers such as [2, Section 7], [12, Section 5], [22, Section 5], [23, Section 14.4], [27, Section 4] and [1, Section 6]. So we just give a sketch of the proof for Corollary 2.3 by using an argument given in the papers mentioned before. Let uh ∈ W 1,2 (Q+ 2R ) be a weak solution of (2.4) with ⎧ ⎨ uh → u in W 1,1 (Q+ 2R ), n ⎩ µh ⇀ in the sense of measures with lim sup |µh |(Qρ (y) ∩ R+ ) ≤ |µ|(Qρ (y) ∩ Rn+ ) for any Qρ (y) ⊂ Q2R . h→∞

By replacing u and µ with uh and µh respectively, we return to (6.22) in the proof of Theorem 2.1: ] [∫ ) ∫ ∫ 2r ( |µh |(Qρ (x0 ) ∩ Rn+ ) dρ ′ + |Dx′ ϕ(x0 )| − |Duh | dx ≤ c − |Duh | dx + + osc′ Dx′ ϕ ρn−1 ρ Tρ (x0 ) Q i (x0 ) Qr (x0 ) 0 δ r

for any i ≥ 0. So by letting h → ∞, we find that [∫ ) ] ∫ ∫ 2r ( |µ|(Qρ (x0 ) ∩ Rn+ ) dρ − + |Dx′ ϕ(x′0 )| + osc Dx′ ϕ |Du| dx ≤ c − |Du| dx + ρn−1 ρ Tρ (x′0 ) Q i (x0 ) Qr (x0 ) 0 δ r

for any i ≥ 0. Since x0 is a Lebesgue point of Du, Corollary 2.3 follows by letting i → ∞. Acknowledgments Y. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (No. NRF-2017R1D1A1B03034302). Y. Youn was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (No. NRF-2017R1C1B1010966). References [1] B. Avelin T. Kuusi, G. Mingione, Nonlinear Calder´ on-Zygmund theory in the limiting case, Arch. Ration. Mech. Anal. 227 (2) (2018) 663–714. [2] P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differ. Equ.. 53 (3–4) (2015) 803–846. [3] P. Baroni, J. Habermann, Elliptic interpolation estimates for non-standard growth operators, Ann. Acad. Sci. Fenn. Math. 39 (1) (2014) 119–162. Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.

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[4] L. Boccardo, T. Gallou¨ et, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1) (1989) 149–169. [5] L. Boccardo, T. Gallou¨ et, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17 (1992) 641–655. [6] S.-S. Byun, Y. Kim, Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math. 288 (2016) 152–200. [7] S.-S. Byun, Y. Kim, Riesz potential estimates for parabolic equations with measurable nonlinearities, Int. Math. Res. Not. IMRN (21) (2018) 6737–6779. [8] S. Byun, Y. Youn, Optimal gradient estimates via riesz potentials for p(·)-Laplacian type equations, Q. J. Math. 68 (4) (2017) 1071–1115. [9] S. Byun, Y. Youn, Riesz potential estimates for a class of double phase problems, J. Differential Equations 264 (2) (2018) 1263–1316. [10] H. Dong, J. Xiong, Boundary gradient estimates for parabolic and elliptic systems from linear laminates, Int. Math. Res. Not. IMRN 17 (2015) 7734–7756. [11] F. Duzaar, G. Mingione, Gradient continuity estimates, Calc. Var. Partial Differ. Equ. 39 (2010) 379–418. [12] F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math. 133 (4) (2011) 1093–1149. [13] L.C. Evans, Partial Differential Equations: Second Edition, American Mathematical Society, 2010. [14] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co. Inc., River Edge, NJ, 2003. [15] Y. Jang, Y. Kim, Global gradient estimates for nonlinear equations of p-Laplace type from composite materials, J. Math. Anal. Appl. 471 (1–2) (2019) 1–14. [16] T. Jin, V. Maz’ya, J. Van Schaftingen, Pathological solutions to elliptic problems in divergence form with continuous coefficients, C. R. Math. Acad. Sci. Paris 347 (13–14) (2009) 773–778. [17] Y. Kim, Riesz potential type estimates for nonlinear elliptic equations, J. Differential Equations 263 (10) (2017) 6844–6884. [18] Y. Kim, S. Ryu, Global gradient estimates for parabolic equations with measurable nonlinearities, Nonlinear Anal. 164 (2017) 77–99. [19] T. Kuusi, G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl. (9) 98 (4) (2012) 390–427. [20] T. Kuusi, G. Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems, Rev. Mat. Iberoam. 28 (2) (2012) 535–576. [21] T. Kuusi, G. Mingione, Universal potential estimates, J. Funct. Anal. 262 (2012) 4205–4269. [22] T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal. 207 (2013) 215–246. [23] T. Kuusi, G. Mingione, Guide to nonlinear potential estimates, Bull. Math. Sci. 4 (1) (2014) 1–82. [24] T. Kuusi, G. Mingione, Riesz potentials and nonlinear parabolic equations, Arch. Ration. Mech. Anal. 212 (3) (2014) 727–780. [25] T. Kuusi, G. Mingione, The Wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc. 16 (2014) 835–892. [26] T. Kuusi, G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc. (JEMS) 20 (4) (2018) 929–1004. [27] G. Mingione, The Calder´ on-Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2) (2007) 195–261. [28] G. Mingione, Gradient potential estimates, J. Eur. Math. Soc. 13 (2011) 459–486.

Please cite this article as: Y. Kim and Y. Youn, Boundary Riesz potential estimates for elliptic equations with measurable nonlinearities, Nonlinear Analysis (2019), https://doi.org/10.1016/j.na.2019.02.001.