The emergence of SARS-CoV-2 created a havoc worldwide, causing high morbidity, serious complications and mortality. The ORF1ab of SARS-CoV-2 has 16 non-structural proteins which are required for genome replication and transcription. All of these are druggable targets, of which NSP12 (RNA-dependent RNA polymerase), was selected as a potential target for drug molecules. Remdesivir is a recommended drug for SARS-CoV-2 and it targets the RdRp protein. Although Remdesivir was given to COVID-19 patients based on their clinical manifestations, yet the transmission and spread of the virus continued and to add to its pandemicity, new variants emerged from time to time. This necessitates the need for molecular modification of existing antiviral drugs so that more precise targets for halting viral replication can be selected. For this, the approach used was repurposing of the existing drugs. In the present study, ten FDA-approved drugs were chosen on the basis of their properties of inhibiting the RdRp protein. These drugs were subjected for checking the docking score with the target protein. Of these, Remdesivir, Ribavirin, Favipiravir and Baloxavir were taken for further analysis on the basis of their best scores. These drugs were then modified to check the efficiency to inhibit the RdRp and to stop the replication rate of the virus. We docked the modified drugs with the macrodomain of RdRp by using the CB-Dock web server and checked the binding affinity and amino acid contact residues. The modified drugs were also checked for bioactivity in the Molinspiration cheminformatics online tool. Our results showed increased affinity for RdRp of SARS-CoV-2 when compared to the original compound. We also checked the synthetic accessibility of the drugs using the SwissADME tool. The study showed promising results when modified. The findings reported need further confirmation through wet lab studies.
Citation: Bhawna Sharma, Bennet Angel, Vankadoth Umakanth Naik, Annette Angel, Vinod Joshi, BM Shareef, Neha Singh, Ambreen Shafaat Khan, Poorna Khaneja, Shilpa Barthwal, Ramesh Joshi, Nuzhat Maqbool Peer, Kiran Yadav, Komal Tomar, Satendra Pal Singh. Repurposed drug molecules targeting NSP12 protein of SARS-CoV-2: An in-silico study[J]. AIMS Molecular Science, 2023, 10(4): 322-342. doi: 10.3934/molsci.2023019
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The emergence of SARS-CoV-2 created a havoc worldwide, causing high morbidity, serious complications and mortality. The ORF1ab of SARS-CoV-2 has 16 non-structural proteins which are required for genome replication and transcription. All of these are druggable targets, of which NSP12 (RNA-dependent RNA polymerase), was selected as a potential target for drug molecules. Remdesivir is a recommended drug for SARS-CoV-2 and it targets the RdRp protein. Although Remdesivir was given to COVID-19 patients based on their clinical manifestations, yet the transmission and spread of the virus continued and to add to its pandemicity, new variants emerged from time to time. This necessitates the need for molecular modification of existing antiviral drugs so that more precise targets for halting viral replication can be selected. For this, the approach used was repurposing of the existing drugs. In the present study, ten FDA-approved drugs were chosen on the basis of their properties of inhibiting the RdRp protein. These drugs were subjected for checking the docking score with the target protein. Of these, Remdesivir, Ribavirin, Favipiravir and Baloxavir were taken for further analysis on the basis of their best scores. These drugs were then modified to check the efficiency to inhibit the RdRp and to stop the replication rate of the virus. We docked the modified drugs with the macrodomain of RdRp by using the CB-Dock web server and checked the binding affinity and amino acid contact residues. The modified drugs were also checked for bioactivity in the Molinspiration cheminformatics online tool. Our results showed increased affinity for RdRp of SARS-CoV-2 when compared to the original compound. We also checked the synthetic accessibility of the drugs using the SwissADME tool. The study showed promising results when modified. The findings reported need further confirmation through wet lab studies.
This paper is a contribution to the recently very active area of quantitative stochastic homogenization of second order uniformly elliptic operators, the main goal of which is to quantify how close is the large scale behavior of the heterogeneous operator
As originally realized in the seminal papers by Papanicolaou and Varadhan [23] and, independently, by Kozlov [20], the central object in the homogenization of elliptic operators with random coefficients is the corrector
−∇x⋅(A(x)∇x(x⋅ξ+ϕξ(A,x)))=0 |
in the whole space
Ahomei:=⟨A(ei+∇ϕei)⟩. |
Since the problem is linear, it clearly suffices to study the
Both mentioned works [20,23] were purely qualitative in the sense that they showed the sublinearity of the corrector in the limit of large scales without any rate. Assuming that the correlation of the coefficient fields decays with a specific rate (either encoded by some functional inequality like the Spectral Gap estimate or the Logarithmic Sobolev Inequality, or by some mixing conditions or even assuming finite range of dependence), one goal of quantitative theory is to quantify the sublinearity (smallness) of the corrector and consequences thereof.
Though the present result is purely deterministic in the sense that it translates the fact that the energy of any
A central assumption in our result involves a minimal radius, a notion introduced by Gloria, Neukamm, and Otto [16]: for given fixed
r∗:=inf{r≥1:∀R≥r:1R2−∫BR|(ϕ,σ)−−∫BR(ϕ,σ)|2≤δ}. | (1) |
Here
The introduction of the minimal radius
the sublinearity of the corrector, as encoded in the definition of the random variable
−∫Br|∇u|2≤C−∫BR|∇u|2. |
The idea that the (large-scale) regularity theory of
Assuming that the ensemble on the coefficient fields satisfies a coarsened version of the Logarithmic Sobolev Inequality, Gloria, Neukamm, and Otto [16] showed that the minimal radius
⟨exp(1Crd(1−β)∗)⟩≤C, |
where
Recently, reviving the parabolic approach used in the discrete setting [17], which has the benefit of conveniently disintegrating contributions to the corrector from different scales, Gloria and Otto [18] obtained a similar results assuming the coefficient fields have finite range of dependence. As a by-product, assuming finite range of dependence Gloria and Otto got the estimates for the minimal radius
⟨exp(1Crd(1−ϵ)∗)⟩<∞,∀ϵ>0. |
As already said, using completely different methods, such almost Gaussian bounds for a related quantity
Finally, on the other side of the spectrum, Fischer and Otto [14] combined Meyer's estimate together with sensitivity analysis to show that for strongly correlated coefficient fields (more precisely, they consider coefficient fields which are
⟨exp(1Crβ∗)⟩≤C. |
In the present paper we will obtain deterministic estimates for the Green's function based on the minimal radii
An obvious advantage of the present approach is that it clearly separates the random effects, described by
Our only goal in this paper is to obtain bounds, and not to show existence (or other properties) of the Green's function. In fact, a well known counterexample of De Giorgi [11] shows that there are uniformly elliptic coefficient fields for which the Green's function does not exist. Nevertheless, as recently shown in [10] by Conlon, Otto, and the second author, this is not a generic behavior. More precisely, in [10] they show that for any uniformly elliptic coefficient field
There are several works studying estimates on the Green's function in the context of uniformly elliptic equations with random coefficients. Using De Giorgi-Nash-Moser approach for a parabolic equation (which is naturally restricted to the scalar case), Delmotte and Deuschel [12] obtained annealed estimates on the first and second gradient of the Green's function, in
Both works [15,21] used De Giorgi-Nash-Moser-type argument, and as such were restricted to a single equation. In contrast, our result is not restricted to the scalar case, a reason why we had to develop different techniques to obtain the estimates.
Before we state the main result, let us mention other works relating the smallness of the corrector and the properties of solutions to the heterogeneous equation. Together with Otto [8], we compare the finite energy solution
−∇⋅A∇u=∇⋅g, |
with
−∇⋅Ahom∇uhom=∇⋅˜g. |
Here by twice corrected we mean that first the right-hand side
1R2−∫BR|(ϕ,σ)−−∫BR(ϕ,σ)|2≤CR−2β,∀R≥r∗,β. | (2) |
Compared to the condition (1) which we use in the present paper, the above condition (2) is obviously stronger. Indeed, while for example
Hence, in comparison with the present work, in [8] we get a stronger statement (since we estimate the difference between the heterogeneous Green's function and corrected constant-coefficient Green's function while in the present paper we only control the heterogeneous Green's function alone), at the expense of stronger assumptions on the smallness of the corrector and a more involved proof. More precisely, here we show that the second mixed derivative of the Green's function
Since we are dealing with linear equations, we make use of a duality argument, first introduced by Avellaneda and Lin in [5]. This allows us to obtain estimates on the
Last, let us mention the work of Otto and the authors [7], where we push farther the results of [8] using higher order correctors. The second and higher order correctors were introduced into the stochastic homogenization setup by Fischer and Otto [13], in order to extend the
The paper is organized as follows: In the next section we will state our assumptions together with the main result, Theorem 1, and its corollaries, Corollary 1, Corollary 2, and Corollary 3. In Section 3 we prove Theorem 1 and in Section 4 we give the argument for Corollary 1, which is the only corollary which does not immediately follow from the theorem.
Notation. Throughout the article, we denote by
We fix a coefficient field
∫Rd∇φ⋅A(x)∇φdx≥λ∫Rd|∇φ|2,∀φ∈C∞c(Rd),|A(x)ξ|≤|ξ|,∀a.e. x∈Rd,∀ξ∈Rd, | (3) |
where
Theorem 1. Let
−∫Br(x)|∇u|2≤C(d,λ)−∫BR(x)|∇u|2. | (4) |
Let
−∇x⋅A∇xG(A;⋅,y)=δ(⋅−y), |
assuming it exists for a.e.
∫B1(x0)∫B1(y0)|∇x∇yG(A;x,y)|2dxdy≤C(d,λ)(r∗(x0)r′∗(y0)|x0−y0|2)d, | (5) |
∫B1(x0)∫B1(y0)|∇yG(A;x,y)|2dxdy≤C(d,λ)|x0−y0|2(r∗(x0)r′∗(y0)|x0−y0|2)d, | (6) |
∫B1(x0)∫B1(y0)|∇xG(A;x,y)|2dxdy≤C(d,λ)|x0−y0|2(r′∗(x0)r∗(y0)|x0−y0|2)d, | (7) |
∫B1(x0)∫B1(y0)|G(A;x,y)|2dxdy≤C(d,λ)|x0−y0|4(r∗(x0)r′∗(y0)+r′∗(x0)r∗(y0)|x0−y0|2)d. | (8) |
where
Though the Green's function does not have to exist in
Corollary 1. Let
ˉA(x,x3):=(A(x)0 01 ) | (9) |
there exists two points
Then for a.e.
−∇x⋅A∇G(A;⋅,y)=δ(⋅−y). |
Moreover, given
∫B1(x0)∫B1(y0)|∇y∇G(A;x,y)|2dxdy≤C(λ)(r∗(A,x0)r∗(At,y0))2|x0−y0|4, | (10) |
∫B1(x0)∫B1(y0)|∇G(A;x,y)|2dxdy≤C(λ)(r∗(ˉAt,(x0,0))r∗(ˉA,(y0,0)))2|x0−y0|2. | (11) |
Remark 1. Assuming that the coefficient field
Remark 2. It is clear from the proof of Theorem 1 that all the above estimates, i.e. (5)-(11), are true also if the domains of integration
Remark 3. The appearance of different minimal radii in (10) and (11) (in (10) the minimal radii are related to the equation in
For notational convenience we state the result for a single equation. Since in the proof of Theorem 1 we do not use any scalar methods (like for example De Giorgi-Nash-Moser iteration), the result holds also in the case of elliptic systems -for that one just considers that
Using the Gaussian bounds on
Corollary 2. Suppose
⟨exp((C|x0−y0|2d∫B1(x0)∫B1(y0)|∇x∇yG(A;x,y)|2dxdy)d(1−ϵ))⟩<∞,⟨exp((C|x0−y0|2d−2∫B1(x0)∫B1(y0)|(∇x,∇y)G(A;x,y)|2dxdy)d(1−ϵ))⟩<∞, |
and in
⟨exp((C|x0−y0|2d−4∫B1(x0)∫B1(y0)|G(A;x,y)|2dxdy)d(1−ϵ))⟩<∞. |
In the case of coefficient fields with stronger correlations we can use the result from [16]:
Corollary 3. Suppose
diam(D)≤(dist(D)+1)β≤C(d)diam(D). |
Moreover, let us assume that there is
⟨F2logF2⟩−⟨F2⟩log⟨F2⟩≤1ρ⟨‖∂F∂A‖2⟩, |
where the carré-du-champ of the Malliavin derivative is defined as
‖∂F∂A‖2:=∑D(∫D|∂F∂A|2). |
Then there exists a constant
⟨exp((C|x0−y0|2d∫B1(x0)∫B1(y0)|∇x∇yG(A;x,y)|2dxdy)d(1−β))⟩<∞,⟨exp((C|x0−y0|2d−2∫B1(x0)∫B1(y0)|(∇x,∇y)G(A;x,y)|2dxdy)d(1−β))⟩<∞, |
and in
⟨exp((C|x0−y0|2d−4∫B1(x0)∫B1(y0)|G(A;x,y)|2dxdy)d(1−β))⟩<∞. |
The proof is inspired by a duality argument of Avellaneda and Lin [5,Theorem 13], which they used to obtain Green's function estimates in the periodic homogenization. After stating and proving two auxiliary lemmas, we first prove the estimate on the second mixed derivative (5). Then, (6) will follow from (5) using Poincaré inequality and one additional estimate. Next we observe that (7) can be obtained from (6) by replacing the role of
We thus start with the following two auxiliary lemmas. The first one is very standard:
Lemma 1 (Caccioppoli inequality). Let
∫Bρ|∇u|2≤C(d)λρ2δ2∫B(1+δ)ρ|u−c|2 | (12) |
for any
Proof. By considering
∫Rd|∇(ηu)|2≤C(d)λ∫|∇η|2u2. |
Since
Lemma 2. Let
−∫Br∗(0)|u|2≤C(d,λ)−∫BR0|u|2. | (13) |
Proof. Throughout the proof we write
−∫Br∗|u−ur∗|2Poincaré≲r2∗−∫Br∗|∇u|2(4)≲r2∗−∫BR0/2|∇u|2(12)≲(r∗R0)2−∫BR0|u|2≤−∫BR0|u|2. |
Hence, to prove (4) it is enough to show
|ur∗|2=|−∫Br∗u|2≲−∫BR0|u|2. | (14) |
To prove it, we use the following estimate
|ur−u2r|≲r(−∫B2r|∇u|2)12, | (15) |
which in fact holds for any function
We first argue how to obtain (14) thanks to estimate (15): Let
|ur∗−u2nr∗|≤n−1∑k=0|u2kr∗−u2k+1r∗|(15)≲n−1∑k=02kr∗(−∫B2k+1r∗|∇u|2)12(4)≲(−∫BR0/2|∇u|2)12n−1∑k=02kr∗(12)≲R0(1R20−∫BR0|u|2)12=(−∫BR0|u|2)12. |
Using Jensen's inequality and the fact that
|u2nr∗|=|−∫B2nr∗u|≤(−∫B2nr∗|u|2)12≲(−∫BR0|u|2)12. |
Combination of the two previous estimates then gives (14).
It remains to prove (15). Using Jensen's and Poincaré's inequalities we get
|ur−u2r|=|−∫Br(u−ur)−(u−u2r)|≲−∫Br|u−ur|+−∫B2r|u−u2r|≲(−∫Br|u−ur|2)12+(−∫B2r|u−u2r|2)12≲r(−∫B2r|∇u|2)12. |
We denote
∫B1(y0)|Fρ(∇x∇yG(⋅,y))|2dy≲(r∗(x0)r′∗(y0)R20)d | (16) |
for any
|Fρ(∇v)|2≤∫Bρ(x0)|∇v|2, | (17) |
for any
Step 1. Proof of (16) (inspired by the duality argument of Avellaneda and Lin [5]).
Let
−∇⋅A∇u=−∇⋅f |
in
∫Rd|∇u|2≲∫Rd|f|2. | (18) |
Then on the one hand, the Green's function representation formula yields
∇u(x)=∫BR0(y0)∇x∇yG(x,y)f(y)dy. | (19) |
If
|Fρ(∇u)|2≤∫Bρ(x0)|∇u|2dx≤∫Br∗(x0)(x0)|∇u|2≲(r∗(x0)R0)d∫BR0(x0)|∇u|2(18)≲(r∗(x0)R0)d∫Rd|f|2. |
If
|Fρ(∇u)|2≤∫Bρ(x0)|∇u|2dx(18)≲∫Rd|f|2dx≤(r∗(x0)R0)d∫Rd|f|2. |
Since
Fρ(∇u)=∫BR0(y0)Fρ(∇x∇yG(⋅,y))f(y)dy, |
where the dot means that
|∫BR0(y0)Fρ(∇x∇yG(⋅,y))f(y)dy|2≲(r∗(x0)R0)d∫BR0(x0)|f|2. |
Using definition of the norm
∫BR0(y0)|Fρ(∇x∇yG(⋅,y))|2dy≲(r∗(x0)R0)d. | (20) |
Let
∫B1(y0)|Fρ(∇x∇yG(⋅,y))|2dy≤∫Br′∗(y0)(y0)|Fρ(∇x∇yG(⋅,y))|2dy≲(r′∗(y0)R0)d∫BR0(y0)|Fρ(∇x∇yG(⋅,y))|2dy(20)≲(r∗(x0)r′∗(y0)R20)d. | (21) |
If
∫B1(y0)|Fρ(∇x∇yG(⋅,y))|2dy≤∫BR0(y0)|Fρ(∇x∇yG(⋅,y))|2dy(20)≲(r∗(x0)R0)d≤(r∗(x0)r′∗(y0)R20)d. |
Step 2. Let
∫B(1+δ)ρ|v|2=∞∑k=1|Fk(v)|2 and ∫B(1+δ)ρ|∇v|2=∞∑k=1λk|Fk(v)|2≥λN∞∑k=N|Fk(v)|2, |
where the last inequality follows from the monotonicity of
∫B(1+δ)ρ|v|2=N−1∑k=1|Fk(v)|2+∞∑k=N|Fk(v)|2≤N−1∑k=1|Fk(∇v)|2+1λN∫B(1+δ)ρ|∇v|2, | (22) |
where we used that
Step 3. Combination of Step 1 and Step 2 (applied to
∫B1(y0)∫Bρ(x0)|∇x∇yG(x,y)|2dxdy(12)≲1δ2∫B1(y0)∫B(1+δ)ρ(x0)|∇yG(x,y)−Gavg(y)|2dxdy(22)≲1δ2(N−1∑k=0∫B1(y0)|Fk(∇x∇yG(⋅,y))|2+1λN∫B1(y0)∫B(1+δ)ρ(x0)|∇x∇yG(x,y)|2dxdy)(21)≲1δ2(N(r∗(x0)r′∗(y0)R20)d+1λN∫B1(y0)∫B(1+δ)ρ(x0)|∇x∇yG(x,y)|2dxdy), | (23) |
where we defined
Step 4. For a given sequence
Mk:=(r∗(x0)r′∗(y0)R20)−d∫B1(y0)∫Bρk(x0)|∇x∇yG(x,y))|2dxdy. |
For any
Mk≤Cδ2kNk+Cδ2k1λNMk+1, | (24) |
where the values of
Cδ2k1λN≤C′k4(αk2d2d)−2d=C′α2d14≤14. |
For this choice (24) turns into
Mk≤Cαk4k2d2d+14Mk+1. |
Iterating this we get
M1≤CαK∑k=14−kk4k2d2d+(14)KMK+1. |
Assuming we have
M1≤Cα2d∞∑k=14−kk4+2d. |
Since the sum on the right-hand side is summable, we get that
It remains to justify the assumption
(r∗(x0)r′∗(y0)R20)−d∫B1(y0)(∫B1(x0)|∇x∇yG(x,y))|2dx)χΛ(y)dy≤C, |
where the right-hand side does not depend on
M1=(r∗(x0)r′∗(y0)R20)−d∫B1(y0)∫B1(x0)|∇x∇yG(x,y))|2dxdy≤C |
by the Monotone Convergence Theorem. This completes the proof of (5).
We first observe that using Poincaré's inequality we can control the difference between
Step 1. By Poincaré inequality in the
∫B1(y0)(∫B1(x0)|∇yG(x,y)−(−∫B1(x0)∇yG(x′,y)dx′)|2dx)dy≲∫B1(y0)∫B1(x0)|∇x∇yG(x,y)|2dxdy(5)≲(r∗(x0)r′∗(y0)R20)d. | (25) |
By the triangle inequality we have
∫B1(y0)∫B1(x0)|∇yG(x,y)|2dxdy≲∫B1(y0)(∫B1(x0)|∇yG(x,y)−(−∫B1(x0)∇yG(x′,y)dx′)|2dx)dy+|B1|∫B1(y0)(−∫B1(x0)∇yG(x,y)dx)2dy, |
and so (6) follows from (25) provided we show
∫B1(y0)(−∫B1(x0)∇yG(x,y)dx)2dy≲(r∗(x0)r′∗(y0))dR2d−20. | (26) |
Step 2. Proof of (26). Similarly as for (5), consider arbitrary
−∇⋅A∇u=−∇⋅f |
in
∫Rd|∇u|2≲∫Rd|f|2. | (27) |
Let
|F(u)|2≤∫B1(x0)|u|2≤∫Br∗(x0)(x0)|u|2Lemma 2≲rd∗(x0)−∫BR0(x0)|u|2Jensen≤rd∗(x0)(−∫BR0(x0)|u|2dd−2)d−2d≲rd∗(x0)Rd−20(∫Rd|u|2dd−2)d−2dSobolev≲rd∗(x0)Rd−20∫Rd|∇u|2(27)≲rd∗(x0)Rd−20∫Rd|f|2. |
If otherwise
|F(u)|2≤∫B1(x0)|u|2≤∫BR0(x0)|u|2≲rd∗(x0)−∫BR0(x0)|u|2 |
and proceed as in the previous inequality. As before, we use linearity of
|F(u)|=|∫BR0(y0)F(∇yG(⋅,y))f(y)dy|. |
Since
∫BR0(y0)|F(∇yG(⋅,y))|2dy≲rd∗(x0)Rd−20. | (28) |
As before, it remains to argue that by going from
∫B1(y0)|F(∇yG(⋅,y))|2dy=∫B1(y0)|∇v|2dy≤∫Br′∗(y0)(y0)|∇v|2dy≲(r′∗(y0)R0)d∫BR0(y0)|∇v|2≲(r∗(x0)r′∗(y0))dR2d−20. | (29) |
For the choice
Similarly to the proof of (6), we use Poincaré's inequality (Step 1) to show that (8) follows from (7) provided we control averages of
Step 1. By Poincaré's inequality in the
∫B1(y0)(∫B1(x0)|G(x,y)−(−∫B1(x0)G(x′,y)dx′)|2dx)dy≲∫B1(y0)∫B1(x0)|∇xG(x,y)|2dxdy(7)≲R20(r′∗(x0)r∗(y0)R20)d. |
Then by the triangle inequality we have
∫B1(y0)∫B1(x0)|G(x,y)|2dxdy≲∫B1(y0)(∫B1(x0)|G(x,y)−(−∫B1(x0)G(x′,y)dx′)|2dx)6dy+|B1|∫B1(y0)(−∫B1(x0)G(x,y)dx)2dy, |
and so (8) follows provided we show
∫B1(y0)(−∫B1(x0)G(x,y)dx)2dy≲(r∗(x0)r′∗(y0))dR2d−40. | (30) |
Step 2. Proof of (30). Similarly as for (6), consider arbitrary
−∇⋅A∇u=f |
in
λ∫Rd|∇u|2≤∫BR0(y0)fu≤Rd20(∫BR0(y0)|f|2)12(−∫BR0(y0)|u|2)12Jensen,d≥3≤Rd20(∫BR0(y0)|f|2)12(−∫BR0(y0)|u|2dd−2)d−22d=R0(∫BR0(y0)|f|2)12(∫BR0(y0)|u|2dd−2)d−22dSobolev≲R0(∫BR0(y0)|f|2)12(∫Rd|∇u|2)12, |
and so
∫Rd|∇u|2≲R20∫BR0(y0)|f|2. | (31) |
We point out that compared to the proof of (5) or (6), we got additional
Let
|F(u)|2≤∫B1(x0)|u|2≤∫Br∗(x0)(x0)|u|2Lemma 2≲rd∗(x0)−∫BR0(x0)|u|2Jensen,d≥3≤rd∗(x0)(−∫BR0(x0)|u|2dd−2)d−2d≲rd∗(x0)Rd−20(∫Rd|u|2dd−2)d−2dSobolev≲rd∗(x0)Rd−20∫Rd|∇u|2(31)≲rd∗(x0)Rd−40∫Rd|f|2. |
If otherwise
|F(u)|2≤∫B1(x0)|u|2≤∫BR0(x0)|u|2≲rd∗(x0)−∫BR0(x0)|u|2 |
and proceed analogously to the other case. Using the Green's function representation formula we have
|F(u)|=|∫BR0(y0)F(G(⋅,y))f(y)dy|. |
Since
∫BR0(y0)|F(G(⋅,y))|2dy≲rd∗(x0)Rd−40. | (32) |
As before, it remains to argue that by going from
Now we use Lemma 2 with
∫B1(x0)|v|2≤∫Br′∗(y0)|v|2Lemma 2≲(r′∗(y0)R0)d∫BR0(y0)|v|2(32)≲(r∗(x0)r′∗(y0))dR2d−40. | (33) |
For the choice
We provide a generalization of (6)-(7) in the two-dimensional case. When
ˉA(x,x3):=(A(x)0 01 ), |
and the three-dimensional Green's function
−∇ˉx⋅ˉA∇ˉxˉG(ˉA;⋅,ˉy)=δ(⋅−ˉy). |
It will become clear below that the argument for the representation formula for
Step 1. We argue that for almost every
∇G(A;⋅,y):=∫R∇xˉG(ˉA;(⋅,x3),(y,y3))dx3, | (34) |
satisfies for every
∫∇xζ(x)⋅A(x)∇G(A;x,y)dx=ζ(y), | (35) |
i.e., in a weak sense it solves
By definition of
∫∇ˉxˉζ(ˉx)⋅ˉA∇ˉxˉG(ˉA;ˉx,ˉy)dˉx=ˉζ(ˉy). |
Thus, for any
∫ˉρ(ˉy)∫∇ˉxˉζ(ˉx)⋅ˉA∇ˉxˉG(A;ˉx,ˉy)dˉxdˉy=∫ˉρ(y)ˉζ(ˉy)dˉy. |
We now choose a sequence
∫ˉρ(ˉy)∫ζ(x)η′n(x3)∂x3ˉG(ˉA;ˉx,ˉy)dˉxdˉy+∫ˉρ(ˉy)∫ηn(x3)∇ζ(x)⋅A∇ˉG(ˉA;ˉx,ˉy)dˉxdˉy=∫ˉρ(y)ζ(y)dˉy. |
We now want to send
∫supp(ˉρ)∫supp(ζ)×R|∇ˉxˉG(ˉA;ˉx,ˉy)|dˉxdˉy<+∞, | (36) |
then by the Dominated Convergence Theorem we may conclude that
∫ˉρ(ˉy)∫∇ζ(x)⋅A(∫R∇ˉG(ˉA;ˉx,ˉy)dx3)dxdˉy=∫ˉρ(ˉy)ζ(y)dˉy, |
and thus (35) by the arbitrariness of the test function
To argue inequality (36) we proceed as follows: We define a finite radius
M≥max(r∗(ˉAt,ˉX),r∗(A,ˉY))andsupp(ˉρ)⊂ˉBM(ˉY), supp(ζ)⊂BM/2(X), |
and observe that inequality (36) is implied by
∫ˉBM(ˉY)∫BM/2(X)×R|∇ˉxˉG|dˉxdˉy<+∞. | (37) |
Since
∫ˉBM(ˉY)∫ˉBM((X,X3))|ˉ∇xˉG(ˉA;ˉx,ˉy)|2dˉxdˉy≲M6|Y−(X,X3)|4≤M6|Y3−X3|4 | (38) |
provided
We now cover the cylinder
∫ˉBM(ˉY)∫BM/2(X)×R|∇ˉxˉG|dˉxdˉy≤+∞∑n=0∫ˉBM(ˉY)∫ˉBM(X,±Mn)|∇ˉxˉG|dˉxdˉy≲∫ˉBM(ˉY)∫ˉB4M((X,0))|∇ˉxˉG|dˉxdˉy+∑n>4∫ˉBM(ˉY)∫ˉBM(X,±Mn)|∇ˉxˉG|dˉxdˉy. | (39) |
We claim that
Here we only sketch the idea why
(∫ˉBr(ˉx)∫ˉBr(ˉy)|ˉ∇ˉxˉG|2)12≲|ˉBr|r2, |
where
∫ˉBr(ˉx)∫ˉBr(ˉy)|ˉ∇ˉxˉG|≲|ˉBr|2r2. |
Using a simple covering argument, the above estimate holds also in the case when the balls are replaced by cubes. Since
∫ˉBR(0)∫ˉBR(0)|ˉ∇ˉxˉG|≲∫ˉB2R(0)∫ˉB2R(0)|ˉx−ˉy|−2dˉxdˉy<∞, |
where we used that for
Going back to the second term on the right-hand side of (39), an application of Hölder's inequality in both variables
∑n>4∫ˉBM(ˉY)∫ˉBM(X,±Mn)|∇ˉxˉG|≲M3∑n>4(∫ˉBM(ˉY)∫ˉBM(X,±Mn)|ˉ∇ˉxˉG|2)12. |
We now may apply to the r.h.s. the bound (38) and thus obtain
∑n>4∫ˉBM(ˉY)∫ˉBM(X,±Mn)|∇ˉxˉG|≲M6∑n>4(Mn)−2≲M4<∞. |
We have established (36).
Before concluding Step 1, we show that the representation formula (34) does not depend on the choice of the coordinate
∫R∇xˉG(ˉA;(x0,x3),(y0,y0,3))dx3=∫R∇xˉG(ˉA;(x0,x3),(y0,y1,3))dx3. |
Without loss of generality we assume
ˉG(ˉA;ˉx+ˉz,ˉy+ˉz)=ˉG(ˉA(⋅+ˉz);ˉx,ˉy), |
by choosing
ˉG(ˉA;ˉx+ˉz,ˉy+ˉz)=ˉG(ˉA;ˉx,ˉy). | (40) |
Let
−∫Bδ(x0)−∫ˉBδ((y0,y1,3))∫R∇xˉG(ˉA;ˉx,ˉy)dˉxdˉy=−∫Bδ(x0)−∫ˉBδ((y0,y1,3))∫R∇xˉG(ˉA;(x,x3−y1,3+y1,3),(y,y3−y1,3+y1,3))dˉxdˉy, |
and use (40) with
−∫Bδ(x0)−∫ˉBδ((y0,y1,3))∫R∇xˉG(ˉA;ˉx,ˉy)dˉxdˉy=−∫Bδ(x0)−∫ˉBδ((y0,0))∫R∇xˉG(ˉA;ˉx,ˉy)dˉxdˉy. |
We now appeal to Lebesgue's theorem and conclude (38).
Step 2. Proof of (11). For this part we denote
∫B∫B1(x0)|∫R∇ˉxˉG(ˉx,ˉy)dx3|2dxdˉy=ry∫B1(y0)∫B1(x0)|∫R∇ˉxˉG(ˉx,(y,0))dx3|2dxdy(34)=ry∫B1(y0)∫B1(x0)|∇G(A;x,y)|2dxdy. |
Since
ry∫B1(x0)∫B1(y0)|∇xG(A;x,y)|2dxdy=∫B∫B1(x0)|∫R∇xˉG(ˉA;ˉx,ˉy)dx3|2dxdˉy≲∫ˉBry((y0,0))∫B1(x0)|∫R∇xˉG(ˉA;ˉx,ˉy)dx3|2dxdˉy≤∫ˉBry((y0,0))∫B1(x0)(∞∑n=−∞∫(n+1)rxnrx|∇xˉG(ˉA;ˉx,ˉy)|dx3)2dxdˉy. |
We define a sequence
an:=(rxry)34(|x0−y0|2+n2(rx)2)12 |
and observe that
(∞∑n=−∞∫(n+1)rxnrx|∇xˉG(ˉA;ˉx,ˉy)|dx3)2=(∞∑n=−∞anrxan−∫(n+1)rxnrx|∇xˉG(ˉA;ˉx,ˉy)|dx3)2Hölder≤(∞∑n=−∞a2n)(∞∑n=−∞(rx)2a2n(−∫(n+1)rxnrx|∇xˉG(ˉA;ˉx,ˉy)|dx3)2)Jensen≤(∞∑n=−∞a2n)(∞∑n=−∞rxa2n∫(n+1)rxnrx|∇xˉG(ˉA;ˉx,ˉy)|2dx3). |
Since
∞∑n=−∞a2n≲(rxry)32|x0−y0|rx, | (41) |
where for simplicity we assumed
ry∫B1(x0)∫B1(y0)|∇xG(A;x,y)|2dxdy≲(rxry)32|x0−y0|rx∑nrxa2n×∫ˉBry((y0,0))∫ˉBrx(x0,(n+1/2)rx)|∇xˉG(ˉA;ˉx,ˉy)|2dˉxdˉy(7),d=3≲(rxry)32|x0−y0|rx)∑nrxa2na4n(41)≲(rxry)3|x0−y0|2rx, |
which is exactly (11).
Concerning (10), there are two possible ways how to proceed. For the first we observe that (35) implies for every test function
∫∇ϕ(x)⋅A(x)(∫∇y∇G(x,y)⋅f(y)dy)dx=∫∇ϕ⋅f=∫∇ϕ⋅A∇u, |
where
∇u(x)=∫∇y∇G(x,y)⋅f(y)dy, |
and the proof of (5) applies verbatim. A different way would be to mimic the argument for (11), i.e., to define
We warmly thank Felix Otto for introducing us into the world of stochastic homogenization and also for valuable discussions of this particular problem. This work was begun while both authors were at the Max Planck Institute for Mathematics in the Sciences in Leipzig.
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1. | Mitia Duerinckx, Antoine Gloria, Felix Otto, The Structure of Fluctuations in Stochastic Homogenization, 2020, 377, 0010-3616, 259, 10.1007/s00220-020-03722-3 | |
2. | Marc Josien, Claudia Raithel, Quantitative Homogenization for the Case of an Interface Between Two Heterogeneous Media, 2021, 53, 0036-1410, 813, 10.1137/20M1311983 | |
3. | Antoine Gloria, Stefan Neukamm, Felix Otto, A Regularity Theory for Random Elliptic Operators, 2020, 88, 1424-9286, 99, 10.1007/s00032-020-00309-4 | |
4. | Mitia Duerinckx, Julian Fischer, Antoine Gloria, Scaling limit of the homogenization commutator for Gaussian coefficient fields, 2022, 32, 1050-5164, 10.1214/21-AAP1705 |