Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

Salvatore Leonardi; Francesco Leonetti; Eugenio Rocha; Vasile Staicu

doi:10.1515/anona-2021-0205

Open Access Published by De Gruyter November 29, 2021

Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

Salvatore Leonardi , Francesco Leonetti , Eugenio Rocha and Vasile Staicu

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0205

Abstract

We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

Keywords: Quasilinear; elliptic; system; weak; solution; regularity

MSC 2010: Primary: 35J47; Secondary: 35B65; 49N60

1 Introduction

This paper deals with quasilinear elliptic systems in divergence form

−div(a(x,u(x))Du(x))=0,x∈Ω, (1.1)

where u: Ω ⊂ ℝⁿ → ℝ^N and a:Ω × ℝ^N → ℝ^N²n² is matrix valued with components ai,jα,β (x, y) where i, j ∈ {1, ..., n} and α, β ∈ {1, ..., N}.

On the coefficients ai,jα,β (x, y) we set the usual conditions, that is they are measurable with respect to x, continuous with respect to y, bounded and elliptic. When N = 1, that is in the case of one single equation, the celebrated De Giorgi-Nash-Moser theorem ensures that weak solutions u ∈ W^1,2(Ω) are locally bounded and even Hölder continuous, see section 2.1 in [27].

But in the vectorial case N ≥ 2, the aforementioned result is no longer true due to the De Giorgi’s counterexample, see [6], section 3 in [27] and the recent paper [29]; see also [32] and [20].

So it arises the question of finding additional structural restrictions on the coefficients ai,jα,β that keep away De Giorgi’s counterexample and allow for local boundedness of weak solutions u, see Section 3.9 in [28].

In the present work we assume a condition on the support of off-diagonal coefficients: there exists L₀ ∈ [0, +∞) such that ∀ L ≥ L₀, when α ≠ β,

(ai,jα,βx,y≠0andyα>L)⇒yβ>L, (1.2)

(see Figure 1 and note that the support has the shape of a butterfly in the plane y^β − y^α).

$Figure 1 Assumption (𝓐3): off-diagonal coefficients ai,jα,β $\begin{array}{} \displaystyle a^{\alpha,\beta}_{i,j} \end{array}$ vanish on the white part of the picture; they might be non zero only on the grey part.$

Figure 1

Assumption (𝓐₃): off-diagonal coefficients ai,jα,β vanish on the white part of the picture; they might be non zero only on the grey part.

Under such a restriction we are able to prove local boundedness of weak solutions. All the necessary assumptions and the result will be listed in section 2 while proofs will be performed in section 3.

It is worth to stress out that systems with special structure have been studied in [33], [26] and off-diagonal coefficients with a particular support have been successfully used when proving maximum principles in [21], L^∞-regularity in [22], when obtaining existence for measure data problems in [23], [24] and, for the degenerate case, in [7].

Higher integrability has been studied as well in [10] when off-diagonal coefficients are small and have staircase support and in [11] when off-diagonal coefficients are proportional to diagonal ones.

Let us mention as well that when the ratio between the largest and the smallest eigenvalues of ai,jα,β is close to 1, then regularity of u is studied at page 183 of [12]; see also [31], [18], [17], [19].

Let us also say that proving boundedness for weak solutions could be an important tool for getting fractional differentiability, see the estimate after (4.15) in [18]. In the present paper we deal with local boundedness of solutions. If the reader is interested in regularity up to a rough boundary it could be worth looking at [25].

2 Assumptions and Result

Assume Ω is an open bounded subset of ℝⁿ, with n ≥ 3. Consider the system of N ≥ 2 equations

−∑i=1n∂∂xi∑α,β=1N∑j=1nai,jα,βx,u∂∂xjuβ=0 in Ω, for α=1,...,N. (2.1)

Note that u^β is the β component of u = (u¹, u², ..., u^N). We list our structural conditions.

For all i, j ∈ {1, ..., n} and all α, β ∈ {1, ..., N}, we require that ai,jα,β : Ω × ℝ^N → ℝ satisfies the following conditions:
x ↦ ai,jα,β (x, y) is measurable and y ↦ ai,jα,β (x, y) is continuous;
(boundedness of all the coefficients) for some constant c > 0, we have

|ai,jα,βx,y|≤c

for almost all x ∈ Ω and for all y ∈ ℝ^N;
(ellipticity of all the coefficients)} for some constant ν > 0, we have

∑α,β=1N∑i,j=1nai,jα,βx,yξiαξjβ≥ν|ξ|2

for almost all x ∈ Ω, for all y ∈ ℝ^N and for all ξ ∈ ℝ^N×n;
(“butterfly” support of off-diagonal coefficients) there exists L₀ ∈ [0, +∞) such that ∀ L ≥ L₀, when α ≠ β,

(ai,jα,βx,y≠0 and |yα|>L)⇒|yβ|>L,

(see Figure 1).

Remark 2.1

Assumption (𝓐₃) guarantees equality (3.2): such an equality is a basic tool for proving boundedness of solutions.

Example 2.2

An example of coefficients which readily satisfy the aforementioned assumptions are defined as follows:

ai,jα,β(x,y)=ai,jα,β(y)=δijmax(|yβ|−|yα|,0)1+2|y|if α≠βδijif α=β

where α, β = 1, 2 and i, j = 1, …, n with n ≥ 3 and N = 2. In this case we have c = 1, ν = 1/2 and we can pick for instance L₀ = 0.

We say that a function u:Ω → ℝ^N is a weak solution of the system (2.1), if u ∈ W^1,2(Ω, ℝ^N) and

∫Ω∑α,β=1N∑i,j=1nai,jα,β(x,u(x))Djuβ(x)Diφα(x)dx=0, (2.2)

for all φ ∈ W01,2 (Ω, ℝ^N).

Theorem 2.3

Let u ∈ W^1,2(Ω, ℝ^N) be a weak solution of system (2.1) under the set (𝓐) of assumptions. Then u ∈ Lloc∞ (Ω, ℝ^N) and we have the following estimate

supB(x0,r)|uα|≤2maxL0;2(n−1)(n−2)n4+16c2n4N4ν2n/224n+2+nn/2(R−r)n∑β=1N∫B(x0,R)|uβ|21/2 (2.3)

for every α = 1, ..., N and for every r, R with 0 < r < R and B(x₀, R) ⊂ Ω, where c is the constant involved in assumption (𝓐₁), ν is given in (𝓐₂) and L₀ appears in (𝓐₃).

Remark 2.4

The present local L^∞–regularity result improves on [22] since assumption (𝓐₃) allows off diagonal coefficients to have a larger support than in [22].

Remark 2.5

“Butterfly” support (𝓐₃) has been used in [7] when proving the existence of at least one globally bounded solution to a (possibly) degenerate problem with zero boundary value problem. In the present work we prove local boundedness of every solution to a non degenerate system regardless of boundary values.

3 Proof of the result

The proof of Theorem 2.3 will be performed in several steps

STEP 1. Caccioppoli inequality

Lemma 3.1

(Caccioppoli inequality on superlevel sets) Let u ∈ W^1,2(Ω, ℝ^N) be a weak solution of system (2.1) under assumptions (𝓐₀), (𝓐₁), (𝓐₂), (𝓐₃). For 0 < s < t, let B(x₀, s) and B(x₀, t) be concentric open balls centered at x₀ with radii s and t respectively. Assume that B(x₀, t) ⊂ Ω and L ≥ L₀. Then

∑α=1N∫uα>L∩Bx0,sDuα2dx≤16c2n4N4ν2∑α=1N∫{uα>L}∩B(x0,t)uα−Lt−s2dx, (3.1)

where c is the constant involved in assumption (𝓐₁), ν is given in (𝓐₂) and L₀ appears in (𝓐₃).

Proof of Lemma 3.1

Let u ∈ W^1,2(Ω, ℝ^N) be a weak solution of system (2.1). Let η:ℝⁿ → ℝ be the standard cut-off function such that 0 ≤ η ≤ 1, η ∈ C01 (B(x₀, t)), with B(x₀, t) ⊂ Ω and η = 1 in B(x₀, s). Moreover, |D η| ≤ 2/(t − s) in ℝⁿ. For every level L ≥ L₀, consider

TLs=−Lifs<−Lsif−L≤s≤LLifs>L

and

GLs=s−TLs.

We define φ:ℝⁿ → ℝ^N with φ = (φ¹, ..., φ^N), where

φα:=η2GLuα, for all α∈{1,...,N}.

Then

Diφα=η21uα>LDiuα+2η(Diη)1uα>LGLuα for all i∈{1,...,n} and α∈{1,...,N}.

Using this test function in the weak formulation (2.2) of system (2.1), we have

0=∫Ω∑α,β=1N∑i,j=1nai,jα,βDjuβDiφαdx=∫Ω∑α,β=1N∑i,j=1nai,jα,βDjuβη21uα>LDiuαdx+∫Ω∑α,β=1N∑i,j=1nai,jα,βDjuβ2η(Diη)1uα>LGLuαdx.

Now, assumption (𝓐₃) guarantees that

ai,jα,βx,ux1uα>Lx=ai,jα,βx,ux1uβ>Lx1uα>Lx (3.2)

when β ≠ α and L ≥ L₀. It is worthwhile to note that (3.2) holds true when α = β as well; then

∫Ω∑α,β=1N∑i,j=1nai,jα,β1uβ>LDjuβη21uα>LDiuαdx=−∫Ω∑α,β=1N∑i,j=1nai,jα,β1uβ>LDjuβ2η(Diη)1uα>LGLuαdx. (3.3)

Now we can use ellipticity assumption (𝓐₂) with ξiα=1uα>LDiuα and we get

ν∫Ωη2∑α=1N1uα>LDuα2dx≤∫Ω∑α,β=1N∑i,j=1nai,jα,β1uβ>LDjuβη21uα>LDiuαdx. (3.4)

Moreover

GLuα=uα−L where uα>L (3.5)

and

−∫Ω∑α,β=1N∑i,j=1nai,jα,β1uβ>LDjuβ2η(Diη)1uα>LGLuαdx≤∫Ωc∑β=1N∑j=1n1uβ>L|Djuβ|∑α=1N∑i=1n2η|Diη|1uα>LGLuαdx≤∫Ωc∑β=1Nn1uβ>L|Duβ|∑α=1Nn2η|Dη|1uα>LGLuαdx≤∫Ωcn2ϵη2∑β=1N1uβ>L|Duβ|2+∫Ωcn2ϵ|Dη|2∑α=1N1uα>LGLuα2dx≤∫Ωcn2N2ϵη2∑β=1N1uβ>L|Duβ|2+∫Ωcn2N2ϵ|Dη|2∑α=1N1uα>LGLuα2dx, (3.6)

where we used the inequality 2ab ≤ ϵ a² + b²/ϵ, provided ϵ > 0. Merging (3.5), (3.4) and (3.6) into (3.3) we get

ν∫Ωη2∑α=1N1uα>L|Duα|2dx≤∫Ωcn2N2ϵη2∑β=1N1uβ>L|Duβ|2+∫Ωcn2N2ϵ|Dη|2∑α=1N1uα>L(uα−L)2dx.

We choose ϵ = ν/(2cn²N²) and we have

ν2∫Ωη2∑α=1N1uα>L|Duα|2dx≤∫Ω2c2n4N4ν|Dη|2∑α=1N1uα>L(uα−L)2dx.

Using the properties of the cut off function η we deduce

∑α=1N∫{uα>L}∩B(x0,s)|Duα|2dx≤16c2n4N4ν2∑α=1N∫{uα>L}∩B(x0,t)uα−Lt−s2dx. (3.7)

Note that

Diuα=Diuα;

this ends the proof of Lemma 3.1. □

STEP 2. Sup estimate for general vectorial functions

In the next Lemma we state and prove a general result that holds true for some general vectorial function v ∈ W^1,p(Ω, ℝ^N). Eventually, we will use such a result with v = (|u¹|, ..., |u^N|) and p = 2.

Lemma 3.2

Assume that Ω is a bounded open subset of ℝⁿ and v = (v¹, ..., v^N) ∈ W^1,p(Ω, ℝ^N) with 1 < p < n. We require the existence of constants c₁ > 0 and L₀ ≥ 0 such that

∑α=1N∫vα>L∩Bx0,sDvαpdx≤c1∑α=1N∫{vα>L}∩B(x0,t)vα−Lt−spdx, (3.8)

for every s, t, L, where 0 < s < t, B(x₀, t) ⊂ Ω and L ≥ L₀. Then,

supB(x0,r)vα≤2maxL0;(n−1)p(n−p)n[2p+c1]n/p24n+p+nn/p(R−r)n∑β=1N∫B(x0,R)(max{vβ;0})p1/p (3.9)

for every α = 1, ..., N and for every r, R with 0 < r < R and B(x₀, R) ⊂ Ω.

Proof of Lemma 3.2

Let us consider balls B(x₀, r₁) and B(x₀, r₂) with 0 < r₁ < r₂ and B(x₀, r₂) ⊂ Ω. Let η : ℝⁿ → ℝ be the standard cut-off function such that 0 ≤ η ≤ 1, η ∈ C01 (B(x₀, (r₁ + r₂)/2)), with η = 1 in B(x₀, r₁). Moreover, |Dη| ≤ 4/(r₂ – r₁) in ℝⁿ. Let us set

AL,rα=:{x∈B(x0,r):vα>L}.

Then, using Hölder inequality, Sobolev embedding and the properties of the cut-off function,

∫AL,r1α(vα−L)p≤∫AL,r1α(vα−L)p∗p/p∗|AL,r1α|1−(p/p∗)=∫AL,r1α[η(vα−L)]p∗p/p∗|AL,r1α|1−(p/p∗)=∫B(x0,r1)[η(max{vα−L;0})]p∗p/p∗|AL,r1α|1−(p/p∗)≤∫B(x0,(r1+r2)/2)[η(max{vα−L;0})]p∗p/p∗|AL,r1α|1−(p/p∗)≤c2∫B(x0,(r1+r2)/2)|D[η(max{vα−L;0})]|p|AL,r1α|1−(p/p∗)=c2∫B(x0,(r1+r2)/2)|(Dη)(max{vα−L;0})+ηD(max{vα−L;0})|p|AL,r1α|1−(p/p∗)=c2∫AL,(r1+r2)/2α|(Dη)(vα−L)+ηDvα|p|AL,r1α|1−(p/p∗)≤c22p∫AL,(r1+r2)/2α|(Dη)(vα−L)|p+∫AL,(r1+r2)/2α|ηDvα|p|AL,r1α|1−(p/p∗)≤c22p4p∫AL,(r1+r2)/2αvα−Lr2−r1p+∫AL,(r1+r2)/2α|Dvα|p|AL,r1α|1−(p/p∗) (3.10)

where c₂ = [(n – 1)p/(n – p)]^p. Now we sum upon α from 1 to N obtaining

∑α=1N∫AL,r1α(vα−L)p≤c22p∑α=1N4p∫AL,(r1+r2)/2αvα−Lr2−r1p+∫AL,(r1+r2)/2α|Dvα|p|AL,r1α|1−(p/p∗)≤c22p∑α=1N4p∫AL,(r1+r2)/2αvα−Lr2−r1p+∫AL,(r1+r2)/2α|Dvα|p∑β=1N|AL,r1β|1−(p/p∗). (3.11)

In order to control ∑ ∫ |Dv^α|^p we use our assumption (3.8) with s = (r₁ + r₂)/2 and t = r₂: we get

∑α=1N∫AL,r1α(vα−L)p≤c22p4p∑α=1N∫AL,(r1+r2)/2αvα−Lr2−r1p+c12p∑α=1N∫AL,r2αvα−Lr2−r1p∑β=1N|AL,r1β|1−(p/p∗)≤c22p[4p+c12p]∑α=1N∫AL,r2αvα−Lr2−r1p∑β=1N|AL,r1β|1−(p/p∗). (3.12)

We want to estimate |AL,r1β| by means of ∫(v^β – L)^p. We are able to do that for a lower level L̃. Indeed, for L > L̃ ≥ L₀, we have

|AL,r1β|=1(L−L~)p(L−L~)p|AL,r1β|=1(L−L~)p∫AL,r1β(L−L~)p≤1(L−L~)p∫AL,r1βvβ−L~p≤1(L−L~)p∫AL~,r1βvβ−L~p≤1(L−L~)p∫AL~,r2βvβ−L~p. (3.13)

Note that

1−(p/p∗)=p/n. (3.14)

Inserting (3.13) and (3.13) into (3.12) we deduce

∑α=1N∫AL,r1α(vα−L)p≤c22p[4p+c12p](r2−r1)p(L−L~)pp/n∑α=1N∫AL,r2αvα−Lp∑β=1N∫AL~,r2βvβ−L~pp/n. (3.15)

We want to estimate ∫(v^α – L)^p with ∫(v^α – L̃)^p. Since L > L̃, we have

∫AL,r2αvα−Lp≤∫AL,r2αvα−L~p≤∫AL~,r2αvα−L~p. (3.16)

Inserting (3.16) into (3.15) we get

∑α=1N∫AL,r1α(vα−L)p≤c22p[4p+c12p](r2−r1)p(L−L~)pp/n∑β=1N∫AL~,r2βvβ−L~p1+(p/n). (3.17)

Now we fix 0 < r < R, with B(x₀, R) ⊂ Ω, and we take the following sequence of radii

ρi=r+R−r2i (3.18)

for i = 0, 1, 2, ...; then ρ₀ = R and ρ_i – ρ_i+1 = (R – r)/2ⁱ⁺¹ > 0, so ρ_i strictly decreases and r < ρ_i ≤ R.

Let us fix a level d ≥ L₀ and we take the following sequence of levels

ki=2d1−12i+1 (3.19)

for i = 0, 1, 2, ...; then k₀ = d and k_i+1 – k_i = d/2ⁱ⁺¹ > 0, so k_i strictly increases and L₀ ≤ d ≤ k_i < 2d. We can use (3.17) with levels L = k_i+1 > k_i = L̃ and radii r₁ = ρ_i+1 < ρ_i = r₂:

∑α=1N∫Aki+1,ρi+1α(vα−ki+1)p≤c22p[4p+c12p]((R−r)/2i+1)p(d/2i+1)pp/n∑β=1N∫Aki,ρiβvβ−kip1+(p/n)=c24p[2p+c1]2(i+1)p2(i+1)pp/n(R−r)pdpp/n∑β=1N∫Aki,ρiβvβ−kip1+(p/n). (3.20)

Let us set

Ji=:∑α=1N∫Aki,ρiα(vα−ki)p; (3.21)

then (3.20) can be written as follows

Ji+1≤c24p[2p+c1]2(1+(p/n))p(R−r)pdpp/n2(1+(p/n))piJi1+(p/n). (3.22)

We would like to get

limi→∞Ji=0; (3.23)

this is true provided

J0≤c24p[2p+c1]2(1+(p/n))p(R−r)pdpp/n−n/p2(1+(p/n))p−nn/(pp), (3.24)

as Lemma 7.1 says at page 220 in [13]. Let us try to check (3.24): we first rewrite it as follows

∑α=1N∫Ak0,ρ0α(vα−k0)p≤c24p[2p+c1]2(1+(p/n))p(R−r)pdpp/n−n/p2(1+(p/n))p−nn/(pp); (3.25)

we keep in mind that k₀ = d and ρ₀ = R; so, (3.25) can be written in the following way

c24p[2p+c1]2(1+(p/n))p(R−r)pn/p2(1+(p/n))pnn/(pp)∑α=1N∫Ad,Rα(vα−d)p≤dp. (3.26)

Note that d ≥ L₀ ≥ 0 so, when v^α > d, we have v^α – d ≤ v^α = max{v^α; 0}; then

∫Ad,Rα(vα−d)p≤∫Ad,Rα(max{vα;0})p≤∫B(x0,R)(max{vα;0})p. (3.27)

Using (3.27), we get the following sufficient condition when checking (3.26):

c24p[2p+c1]2(1+(p/n))pn/p2(1+(p/n))nn/p(R−r)n∑α=1N∫B(x0,R)(max{vα;0})p≤dp. (3.28)

Then, we fix d verifying (3.28) and L₀ ≤ d; then (3.24) is satisfied and (3.23) holds true. We keep in mind that r < ρ_i and k_i < 2d, so we can use (3.16) with r₂ = r < ρ_i, L = 2d and L̃ = k_i:

∫{vα>2d}∩B(x0,r)(vα−2d)p≤∫{vα>ki}∩B(x0,r)(vα−ki)p≤∫{vα>ki}∩B(x0,ρi)(vα−ki)p, (3.29)

so that

0≤∑α=1N∫{vα>2d}∩B(x0,r)(vα−2d)p≤∑α=1N∫{vα>ki}∩B(x0,ρi)(vα−ki)p=Ji; (3.30)

since (3.23) holds true, we have lim_iJ_i = 0, so

∑α=1N∫{vα>2d}∩B(x0,r)(vα−2d)p=0; (3.31)

this means that |{v^α > 2d} ∩ B(x₀, r)| = 0, so that

vα≤2dalmost everywhere in B(x0,r). (3.32)

Level d can be selected as follows

d=maxL0;c24p[2p+c1]2(1+(p/n))pn/p2(1+(p/n))nn/p(R−r)n∑β=1N∫B(x0,R)(max{vβ;0})p1/p

and claim (3.9) is proved after noting that (4^p 2^(1+(p/n))p)^n/p 2^{(1+(p/n))nn/p} = 2^4n+p+nn/p and c₂ = [(n – 1)p/(n – p)]^p. This ends the proof of Lemma 3.2.□

STEP 3. Proof of Theorem 2.3

Caccioppoli inequality proved in Lemma 3.1 allows us to use Lemma 3.2 with v^α = |u^α|, p = 2 and c₁ = 16c2n4N4ν2 : this gives estimate (2.3) and the proof of Theorem 2.3 ends here.□

Remark 3.3

In the present work we used a test function φ that modifies every component of u; this gives the summation on the index α in Caccioppoli’s inequality (3.1). In [4], [1] and [3] only one component of u is modified and a Caccioppoli’s inequality without the summation on α is proved.

Moreover, the Caccioppoli’s inequality proved in [4] and [1] has an exponent p^* on the right–hand side in contrast with the same p that we have on both sides of (3.8), see also [30], [9], [2], [5], [14], [15], [16].

Remark 3.4

In [22] it is used max{u^α – L; 0} in the test function φ, see Figure 2 (left), while in the present paper we use G_L(u^α) instead, see Figure 2 (right). Such a function G_L(u^α) allows us to deal with support larger than in [22] for off diagonal coefficients.

Figure 2

(left) graph of u^α → max{u^α – L; 0}; (right) graph of u^α → G_L(u^α).

Acknowledgement

Leonardi has been supported by Piano della Ricerca di Ateneo 2020-2022–PIACERI: Project MO.S.A.I.C. “Monitoraggio satellitare, modellazioni matematiche e soluzioni architettoniche e urbane per lo studio, la previsione e la mitigazione delle isole di calore urbano”.

Leonetti acknowledges the support from RIA-UNIVAQ.

Leonardi and Leonetti have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Rocha and Staicu acknowledge the partial support by the Portuguese Foundation for Science and Technology (FCT), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2019(CIDMA).

Conflict of interest

Conflict of interest statement: Authors state no conflict of interest.

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Received: 2021-04-21

Accepted: 2021-08-05

Published Online: 2021-11-29

Published in Print: 2021-11-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

Abstract

1 Introduction

2 Assumptions and Result

Remark 2.1

Example 2.2

Theorem 2.3

Remark 2.4

Remark 2.5

3 Proof of the result

STEP 1. Caccioppoli inequality

Lemma 3.1

Proof of Lemma 3.1

STEP 2. Sup estimate for general vectorial functions

Lemma 3.2

Proof of Lemma 3.2

STEP 3. Proof of Theorem 2.3

Remark 3.3

Remark 3.4

Acknowledgement

References

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