Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. It was known that the optimal blow up rate in dimension $n = 2$ is of order $\varepsilon^{-1/2}$. It has recently been proved that in dimensions $n \ge 3$, an upper bound of the gradient is of order $\varepsilon^{-1/2 + \beta}$ for some $\beta>0$. On the other hand, optimal values of $\beta$ have not been identified. In this paper, we prove that when the inclusions are balls, the optimal value of $\beta$ is $[-(n-1)+\sqrt{(n-1)^2+4(n-2)}~]/4 \in (0,1/2)$ in dimensions $n \ge 3$.


Introduction and main results
First we describe the nature of the domain.Let Ω ⊂ R n be a bounded domain with C 2 boundary containing two C 2,γ (0 < γ < 1) relatively strictly convex open sets D 1 and D 2 with dist(D 1 ∪ D 2 , ∂Ω) > c > 0. Let and Ω := Ω\(D 1 ∪ D 2 ).The conductivity problem can be modeled by the following elliptic equation: div a k (x)∇u k = 0 in Ω, where ϕ ∈ C 2 (∂Ω) is given, and In the context of electric conduction, the elliptic coefficients a k refer to conductivities, and the solutions u k represent voltage potential.From an engineering point of view, it is significant to estimate the magnitude of the electric fields in the narrow region between the inclusions, which is given by |∇u k |.This problem is analogous to the Lamé system studied by Babuška, Andersson, Smith, and Levin [5], where they analyzed numerically that, when the ellipticity constants are bounded away from 0 and infinity, the gradient of solutions remain bounded independent of ε, the distance between inclusions.Later, Bonnetier and Vogelius [12] proved that when ε = 0, |∇u k | is bounded for a fixed k and circular inclusions D 1 and D 2 in dimension n = 2.This result was extended by Li and Vogelius [28] to general second order elliptic equations of divergence form with piecewise Hölder coefficients and general shape of inclusions D 1 and D 2 in any dimension.Furthermore, they established a stronger piecewise C 1,α control of u k , which is independent of ε.Li and Nirenberg [27] further extended this global Lipschitz and piecewise C 1,α result to general second order elliptic systems of divergence form, including the linear system of elasticity.Some higher order derivative estimates in dimension n = 2 were obtained in [15,16,18].When k equals to ∞ (inclusions are perfect conductors) or 0 (insulators), it was shown in [13,22,31] that the gradient of solutions generally becomes unbounded as ε → 0. Ammari et al. in [3] and [4] considered the perfect and insulated conductivity problems with circular inclusions in R 2 , and established optimal blowup rates ε −1/2 in both cases.Yun extended in [33] and [34] the results above allowing D 1 and D 2 to be any bounded strictly convex smooth domains.
The above gradient estimates in dimension n = 2 were localized and extended to higher dimensions by Bao, Li, and Yin in [6] and [7].For the perfect conductor case, they proved in [6] that when n ≥ 4.
These bounds were shown to be optimal in the paper and they are independent of the shape of inclusions, as long as the inclusions are relatively strictly convex.Moreover, many works have been done in characterizing the singular behavior of ∇u, which are significant in practical applications.For further works on the perfect conductivity problem and closely related ones, see e.g.[1, 2, 8-11, 14-17, 20, 21, 23-26, 30] and the references therein.
For the insulated conductivity problem, it was proved in [7] that The upper bound is optimal for n = 2 as mentioned above.
Yun [35] studied the following free space insulated conductivity problem in He proved that for some positive constant C independent of ε, He also showed that this upper bound of |∇u| on the ε-segment connecting D 1 and D 2 is optimal for H(x) ≡ x 1 .Although this result does not provide an upper bound of |∇u| in the complement of the ε-segment, it has added support to a long time suspicion that the upper bound ε −1/2 obtained for dimension n = 3 in [7] is not optimal.The upper bound (1.1) was recently improved by Li and Yang [29] to for some β > 0. When insulators are unit balls, a more explicit constant β(n) was given by Weinkove in [32] for n ≥ 4 by a different method.The constant β(n) obtained in [32] presumably improves that in [29].In particular, it was proved in [32] that β(n) approaches 1/2 from below as n → ∞.However, the optimal blow up rate in dimensions n ≥ 3 remained unknown.We draw reader's attention to a recent survey paper [19] by Kang, where in the conclusions section, the three-dimensional case is described as an outstanding problem.
In this paper, we focus on the following insulated conductivity problem in dimensions n ≥ 3, and give an optimal gradient estimate for a certain class of inclusions including two balls of any size: where ϕ ∈ C 2 (∂Ω) is given, and ν = (ν 1 , . . ., ν n ) denotes the inner normal vector on ∂D 1 ∪ ∂D 2 .We use the notation x = (x ′ , x n ), where x ′ ∈ R n−1 .After choosing a coordinate system properly, we can assume that near the origin, the part of ∂D 1 and ∂D 2 , denoted by Γ + and Γ − , are respectively the graphs of two C 2,γ (0 < γ < 1) functions in terms of x ′ .That is, for some R 0 > 0, where f and g are C 2,γ functions satisfying with a > 0.Here and throughout the paper, we use the notation O(A) to denote a quantity that can be bounded by CA, where C is some positive constant independent of ε.For 0 < r ≤ R 0 , we denote By standard elliptic estimates, the solution u ∈ H 1 ( Ω) of (1.3) satisfies We will focus on the following problem near the origin: (1.8) It was proved in [7] that for u ∈ H 1 (Ω R0 ) satisfying (1.8), where C is a positive constant depending only on n, R 0 , a, f C 2 , and g C 2 , and is in particular independent of ε.The above mentioned improvement on (1.1) in [29,32] also apply to (1.9).
Our main results of this paper are as follows.
Theorem 1.1.For n ≥ 3, ε ∈ (0, 1/4), let u ∈ H 1 (Ω R0 ) be a solution of (1.8) with f, g satisfying (1.4) and (1.5).Then there exists a positive constant C depending only on n, R 0 , γ, a positive lower bound of a, and an upper bound of f C 2,γ and g C 2,γ , such that where Note that α(n) is monotonically increasing in n, and For n = 3, the exponent α−1 is the same as the exponent in (1.2).For n ≥ 4, the exponent α−1 2 is strictly greater than the one obtained in [32].A consequence of Theorem 1.1 is, in view of (1.7), as follows.
Estimate (1.12) is optimal as shown in the following theorem.
be the solution of (1.3).Then there exists positive constant C depending only on n, such that where α is given by (1.11).
Remark 1.4.Estimate (1.13) holds for all C 2 domains Ω and C 4 relatively strictly convex open sets D 1 , D 2 that are axially symmetric with respect to x n -axis.A modification of the proof of the theorem yields the result.
Let us give a brief description of the proof of Theorem 1.3.Consider In the polar coordinates, ū(x ′ ) = ū(r, ξ), where x ′ = (r, ξ), 0 < r < 1, and ξ ∈ S n−2 .Since the boundary value ϕ depends only on x 1 and is odd in x 1 , the projection of ū(r, •) to the span of the spherical harmonics is We analyze the equations satisfied by U 1,1 (r) and û(r, x n ) and establish a lower bound and, consequently, there exists Estimate (1.13) follows since u(0) = 0 by the oddness of u in x 1 .Theorems 1.1 and 1.3 will be proved in Sections 2 and 3, respectively.

Proof of Theorem 1.1
In this section, we prove Theorem 1.1.Without loss of generality, we may assume a = 1.Namely, we consider We perform a change of variables by setting This change of variables maps the domain Ω R0 to a cylinder of height ε, denoted by Q R0,ε , where ) be a solution of (1.8) and let v(y) = u(x).Then v satisfies with v L ∞ (QR 0 ,ε) ≤ 1, where the coefficient matrix (a ij (y)) is given by and for i = 1, . . ., n − 1.By (1.4), we know that for i = 1, . . ., n − 1, (2.4) Note that e 1 , . . ., e n−1 depend only on y ′ and are independent of y n .We define (2.5) with v L ∞ (BR 0 ) ≤ 1, where a in ∂ n v is the average of a in ∂ n v with respect to y n in (−ε, ε).Since ∂u ∂ν = 0 on Γ + and Γ − , we have, by (1.4) and (1.9) that By the harmonicity of ∂ n u, the estimate (1.7), and the maximum principle, and consequently, Therefore, the equation (2.6) can be rewritten as where F i := −a in ∂ n v − e i ∂ i v satisfies, using (1.9) and (2.4), For γ, s ∈ R, we introduce the following norm Then for any R ∈ (0, R 0 /2), we have (2.11) where α := min{α, 1 + γ − 2s}, α is given in (1.11), and C is some positive constant depending only on n, γ, s, and R 0 , and is independent of ε.
For the proof, we use an iteration argument based on the following two lemmas.Lemma 2.2.For n ≥ 3, ε > 0, and (2.12) Then for any 0 < ρ < R ≤ R 0 , we have , where α is given in (1.11).
Proof.By the elliptic theory, v 1 ∈ C ∞ (B R0 ).Without loss of generality, we assume that v 1 (0) = 0.By scaling, it suffices to prove the lemma for R = 1.Denote y ′ = (r, ξ) ∈ (0, 1) × S n−2 .We can rewrite (2.12) as Take the decomposition where Y k,i is a k-th degree spherical harmonics, that is, and {Y k,i } k,i forms an orthonormal basis of L 2 (S n−2 ).Here we used the fact that and satisfies for each k ∈ N, i = 1, 2, . . ., N (k).For any k ∈ N, let For any c ∈ R, we have, by a direct computation, Thus for c > 0 sufficiently small, we have Therefore, for any γ > 0, ), we know that V k,i (r) is bounded in (0, 1), so we have By the maximum principle, Sending γ → 0, we have It follows from (2.13) and (2.14), (2.15) where C > 0 depends only on n, γ, and s, and is in particular independent of ε.

16)
We use Moser's iteration argument.By the definitions, The second term on the left-hand side is equal to Therefore, by Hölder's inequality and using 1 + γ − 2s > 0, (p − 1) , where δ > 0 is chosen sufficiently small so that . (2.17) Taking p = 2 in the above, we have, by Hölder's inequality, .
Applying the Sobolev-Poincaré inequality on the left-hand side, we have From (2.17), by Hölder's inequality, , which implies that .
Then by the Sobolev inequality and Young's inequality, we have For k ≥ 0, let and Iterating the relations above, we have, by (2.18), where C is a positive constant depending on n, γ, and s, and is in particular independent of k.The lemma is concluded by taking k → ∞ in (2.19).
Proof of Proposition 2.1.Without loss of generality, we assume that v(0) = 0 and Consider and where F (y ′ ) := R −1 F (Ry ′ ) and G(y we apply Lemma 2.3 to ṽ2 with ε replaced with R −2 ε to obtain For a positive integer k, we take ρ = 2 −i−1 R 0 and R = 2 −i R 0 in (2.22) and iterate from i = 0 to k − 1.We have, using 1 For any ρ ∈ (0, R 0 /2), let k be the integer such that 2 Therefore, (2.11) is proved.

Now we define
for any integer l, and Note that Q 2,ε 1/2 = S 0 .We take the even extension of ṽ with respect to y n = ε 1/2 and then take the periodic extension (so that the period is equal to 4ε 1/2 ).More precisely, we define, for any l ∈ Z, a new function v by setting v(y) := ṽ y ′ , (−1) l y n − 2lε 1/2 , ∀y ∈ S l .
Then v and âij are defined in the infinite cylinder Q 2,∞ .In particular, by using the conormal boundary conditions, it is easily seen that v satisfies the equation By [27, Proposition 4.1] and [29, Lemma 2.1], which implies, after reversing the changes of variables, For any R ∈ (ε 1/2 , R 0 /4), by Proposition 2.1 and (2.23), we have This implies We make a change of variables by setting This change of variables maps the domain where It is straightforward to verify that for any µ > 0. We can apply the "flipping argument" as above to get for any R ∈ (ε 1/2 , R 0 /4).Therefore, we have improved the upper bound |∇u(x)| ≤ C(ε 2 , we take s 1 = s 0 − γ 2 and repeat the argument above.We may decrease γ if necessary so that α−1 2 = −s 0 +k γ 2 for any k = 1, 2, . ... After repeating the argument finite times, we obtain the estimate (1.10).
Proof of Theorem 1.3.
Step 1.By the elliptic theory, the fact that Ω is symmetric in x 1 , and the fact that ϕ is odd in x 1 , we know that u is smooth and u is odd in , f and g can be written as respectively.In Ω 1 , where Ω r is defined as in (1.6), we define where ), and by (1.10), Again, we denote Y k,i to be a k-th degree normalized spherical harmonics so that {Y k,i } k,i forms an orthonormal basis of L 2 (S n−2 ), Y 1,1 to be the one after normalizing x 1 | S n−2 , and x ′ = (r, ξ).Since ū is odd with respect to x 1 = 0, and in particular ū(0) = 0, we have the following decomposition where where and Step 2. We will prove, for some constant C 1 (ε), that where and ν is the unit inner normal of ∂ D i .Clearly v(r) = r satisfies the first line of (3.8), and ∂v ∂ν < 0 on ∂ D i , i = 1, 2. Thus, we know that r is a subsolution of (3.8), and therefore û ≥ r.Then where r 0 is a small constant independent of ε, which implies that Step 4. Completion of the proof of Theorem 1.3.