Strong Influence of a Small Fiber on Shear Stress in Fiber-Reinforced Composites
aa r X i v : . [ m a t h . A P ] O c t Strong Influence of a Small Fiber on Shear Stress inFiber-Reinforced Composites
Mikyoung Lim ∗ KiHyun Yun † November 2, 2018
Abstract
In stiff fiber-reinforced material, the high shear stress concentration occurs in thenarrow region between fibers. With the addition of a small geometric change in cross-section, such as a thin fiber or a overhanging part of fiber, the concentration is sig-nificantly increased. This paper presents mathematical analysis to explain the rapidlyincreased growth of the stress by a small particle in cross-section. To do so, we considertwo crucial cases where a thin fiber exists between a pair of fibers, and where one oftwo fibers has a protruding small lump in cross-section. For each case, the optimallower and upper bounds on the stress associated with the geometrical factors of fibersis established to explain the strongly increased growth of the stress by a small particle.
MSC-class: 35J25, 73C40
In this paper, we concern ourselves with the high stress concentration occurring in thestiff fiber-reinforced composites when fibers are located closely. The primary investigationfocuses on the case when a smaller fiber is located in-between area of two fibers, see Figure1 and Figure 2. This paper reveals that, with the addition of a smaller fiber, the growthof stress is significantly increased: if the diameter d of the fiber in the middle is sufficientlysmall and the distance between adjoining fibers is ǫ , then the stress blows up at the rate of √ dǫ in the narrow region, even though the blow-up rate has been known as √ ǫ as in thecase of a pair of fibers. This means that the defect of fiber as a protrusion causes muchlower strengths in composites than had been thought. To derive it, we estimate the optimallower and upper bounds of the stress concentration in terms of the diameters of fibers andthe distances between them. These bounds explain the dramatic change of the growth ofstress when the diameter of the fiber placed in the middle is relatively smaller than othertwo fibers.In the anti-plane shear model, the stress tensor represents the electric field in the twodimensional space, where the out-of-plane elastic displacement satisfies a conductivity equa-tion, and the cross-section of stiff fibers corresponds to the embedded conductors. In this ∗ Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA([email protected]) † Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA([email protected]) ǫ − / blow-up rate, where ǫ is thedistance between two conductors. Yun [14, 15] has extended this blow-up result for the caseof two adjacent perfect conductors of a sufficiently general shape in two dimensions. In Bao,Li and Yin’s paper [7], it has been also investigated as the blow-up phenomena in higherdimensional spaces, also see [2, 13]. They has also done a natural follow-up in [5, 8] thatthe blow-up rate known only for a pair of fibers is still valid for the multiple inclusions inany dimensions.In contrast, our paper witnesses an unexpected fact on multiple inclusions that thegrowth of stress can be significantly increased by a little geometric change of an inclusion,even though the blow-up rate is still ǫ − / .Figure 1: Case (A) and case (B)Figure 2: Case (C) and case (D)For l = 1 , . . . , L , let D l be conducting inclusions in R , that is cross-sections of stifffibers. Then, under the action of the applied field H , the electric potential u satisfies thefollowing conductivity equation: ∆ u = 0 , in R \∪ Ll =1 D l ,u ( x ) − H ( x ) = O ( | x | − ) , as | x | → ∞ ,u | ∂D l = C l (constant) , for l = 1 , . . . , L, R ∂D l ∂ ν u dS = 0 , for l = 1 , . . . , L, (1)2here H is an entire harmonic function H in R and x = ( x , x ). In this paper, we onlyconsider the case of L = 2 , ǫ − / for ∇ u , and this result isextended by Yun [14, 15] to general shaped fibers. Building on the prior results, we extendthese into the the following interesting direction: we first consider the circular inclusions inCase (A) and Case (B), and second extend the result into general shaped ones in Case (C)and Case (D). In Case (A) and Case (B), we add a small circular inclusion between twoothers so that three disk centers are lined up in one straight line. The additional disk canbe embedded disjointly from other disks, or it partially overlap one of two disks, and weformulate these two cases as follows.(A) One disk and a pair of partially overlapping disks, Figure 1: there is a portion of diskprotruding from one of circular inclusions, i.e., L = 2, and D and D are ǫ -distanceddomains defined as D = B r ( c ) and D = B r ( c ) ∪ B r ( c ) , (2)where B r l ( c l ) is the disk with the radius r i and centered at c l , and c = ( − r − ǫ , , c = ( r + ǫ , , and c = ( r + a + ǫ , . (3)Here, B r ( c ) is a small disk protruding from B r ( c ), and we assume B r ( c ) ∩ B r ( c ) = ∅ , i.e., 0 < a < r , dist( B r ( c ) , B r ( c )) ≃ r and 0 < ǫ ≪ r ≪ r ≃ r . (B) Three disjoint disks, Figure 1: a small disk is disjointly embedded into the in-betweenarea of two disks, i.e., L = 3, and D l = B r l ( c l ) , l = 1 , , , (4)where c = ( − r − ǫ , c = ( r + ǫ ,
0) and c = ( r + r + ǫ + ǫ , D and D is ǫ , and the distance between D and D is ǫ . Here, D is regarded as the cross-section of the thin fiber between a pair of fibers with thecross-section D and D . Thus, we assume that0 < ǫ i ≪ r ≪ r ≃ r for i = 1 , . In both cases, the blow-up rate is remarkably increased due to the existence of B r ( c )as follows: Theorem 1.1 (Case A: Protruding small disk)
Let D and D be defined as (2) . Thenthere is a positive constant C independent of ǫ , r , r and r such that u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ C r r r + r √ r √ ǫ, here u is the solution to (1) with H ( x , x ) = x . As a result, by the Mean Value Theorem,there is a point x in the narrow region between D and D such that |∇ u ( x ) | ≥ C r r r + r √ r √ ǫ . For any entire harmonic function H , let u be the solution to (1) with H . Then, there isa positive constant C independent of ǫ , r , r and r such that |∇ u ( x ) | ≤ C r r r + r √ r √ ǫ in the narrow region between D and D . Theorem 1.2 (Case B: Disjointly embedded small disk)
Let D i , i = 1 , , , be ballsdefined as (4) . Then there is a positive constant C independent of ǫ , ǫ , r , r and r suchthat u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ C r r r + r √ r √ ǫ , and u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ C r r r + r √ r √ ǫ , where u is the solution to (1) with H ( x , x ) = x . As a result, by the Mean Value Theorem,there exists points; x in the narrow region between D and D ; x in the narrow regionbetween D and D , which satisfy that |∇ u ( x i ) | ≥ C r r r + r √ r √ ǫ i for i = 1 , . For any entire harmonic function H , let u be the solution to (1) with H . Then, there isa positive constant C independent of ǫ , ǫ , r , r and r such that |∇ u ( x ) | ≤ C r r r + r √ r √ ǫ i for i = 1 , . in the narrow regions between D and D , and between D and D , respectively. In this paper, we first estimate the lower-bounds in terms of the radii of inclusions.Based on this estimates we derive the remarkable blow-up rate increasing phenomena whena small conducting inclusion is located in-between region of two inclusions. This paper isorganized as follows: In section 2, we explain the method to calculate the potential differencein the case of two disks. We then derive the lower bound of Case (A) in section 3; Case(B) in section 4. In the case of the upper bounds, the major part of derivation overlaps inCase (A) and Case (B). Thus, the derivation is presented in Subsection 4.5. Based on thesimilar derivation, we can also obtain the analogues of Theorem 1.1 and 1.2 for the inclusionsassociated by a sufficiently general class of shapes.4 nalogues of Theorem 1.1 and 1.2 for a sufficiently general class of shapes
The proofs of Theorem 1.1 and 1.2 are flexible enough even though the results are restrictedto circular inclusions. The estimates presented in Theorem 1.1 and 1.2 can be extended tothe inclusions associated by a sufficiently general class of shapes. To consider a large classof shapes, we make the geometric assumptions more precise. To define D , D and D ,we consider three domains D right, D center and D left in R . In addition, we assume that ϕ right : C \ B (0) → R \ D right, ϕ center : C \ B (0) → R \ D center and ϕ left : C \ B (0) → R \ D left are conformal mappings in C ( C \ B (0)) such that ϕ ′ right( z ) = 0 and ϕ ′ left( z ) = 0for z ∈ ∂B (0). Here, we do not distinguish R from C . The C regularity condition of theseconformal mappings doses not allow non-smooth inclusions such as polygons, but Riemannmapping theorem yields a sufficiently general class of shapes: refer to Ahlfors [1]. Now, weconsider the analogues of Theorem 1.1 and 1.2 for two cases as follows:(C) One domain and a pair of partially overlapping domains, similarly to Figure 2: thereis a small portion of another domain protruding from a inclusion, i.e., L = 2, and D and D are ǫ -distanced domains defined as D = D left and D = ( r D center) ∪ D right , (5)where r D center is the r times diminished domain of D center. We suppose that D is a connected domain, dist( D , D ) = dist( D , r D center),dist( D , D right) ⋍ r ,D ⊂ R − × R and D ⊂ R + × R . In addition, we also assume that r is small enough and0 < ǫ ≪ r , and that the boundaries ∂D , ∂D and ∂D right are strictly convex in the narrowregion between D and D .(D) Three disjoint domains D , D and D , Figure 2: a small inclusion D is disjointlyembedded into the in-between area of two other domains, i.e., L = 3, and D = D left , D = r D center and D = D right (6)where r D center is the r times diminished domain of D center. We assume that D and D are ǫ apart, D and D are ǫ apart, and D distances enough from D that r is sufficiently small and0 < ǫ i ≪ r for i = 1 , , since D is regarded as the cross-section of the thin fiber, that the boundaries ∂D , ∂D and ∂D right are strictly convex in the narrow region between D and D , D ⊂ R − × R and D ∪ D ⊂ R + × R . heorem 1.3 (Case C: Protruding small lump) Let D and D be defined as (5) . Thenthere is a positive constant C independent of ǫ and r such that u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ C √ r √ ǫ, where u is the solution to (1) with H ( x , x ) = x . As a result, by the Mean Value Theorem,there is a point x in the narrow region between D and D such that |∇ u ( x ) | ≥ C √ r √ ǫ . For any entire harmonic function H , let u be the solution to (1) with H . Then, there isa positive constant C independent of r and ǫ such that |∇ u ( x ) | ≤ C √ r √ ǫ in the narrow region between D and D . Theorem 1.4 (Case D: Disjointly embedded small inclusion)
Let D i , i = 1 , , , beballs defined as (6) . Then there is a positive constant C independent of r , ǫ and ǫ suchthat u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ C √ r √ ǫ , and u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ C √ r √ ǫ , where u is the solution to (1) with H ( x , x ) = x . As a result, by the Mean Value Theorem,there exists points; x in the narrow region between D and D ; x in the narrow regionbetween D and D , which satisfy that |∇ u ( x i ) | ≥ C √ r √ ǫ i for i = 1 , . For any entire harmonic function H , let u be the solution to (1) with H . Then, there isa positive constant C independent of r , ǫ and ǫ such that |∇ u ( x i ) | ≤ C √ r √ ǫ i for i = 1 , . in the narrow regions between D and D , and D and D , respectively. We derive the lower bound of Case (C) in section 3; Case (D) in section 4. In the case ofthe upper bounds, the major part of derivation overlaps in Case (A), Case (B), Case (C)and Case (D). Thus, the main idea is presented in Subsection 4.5.6
Preliminary
We explain the main idea to calculate the difference of potential between two adjacent,possibly disconnected, conductors.In this section, differently from (1), D i , i = 1 ,
2, could be also the union of two disjointdomains. Define u as the solution to (1), where it is assigned one constant value throughout D i even when D i is disconnected. Now, define h as the solution to ∆ h = 0 , in R \ ( D ∪ D ) ,h = O ( | x | − ) , as | x | → ∞ ,h | ∂D i = k i (constant) , for i = 1 , , Z ∂D i ∂ ν h dS = ( − i , for i = 1 , , (7)where ν is the outward unit normal vector of R n \ ( D ∪ D ), i.e., directed inward of D i . Toindicate the dependence of u and h on D and D , we denote them as u = Φ[ D , D ] , (8) h = Ψ[ D , D ] . (9)The potential difference of u in D and D is represented in terms of h as follows. Lemma 2.1 ([14]) u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D = Z ∂D H∂ ν h dS + Z ∂D H∂ ν h dS. (10)The lemma above can be derived by the Divergence Theorem, see [14]. R Using Lemma 2.1, we can easily calculate the potential difference u | D − u | D of the solution u to (1) when D = B r ( c ) and D = B r ( c ) , (11)where c = ( − r − ǫ ,
0) and c = ( r + ǫ , R i be the reflection with respect to D i , in other words, R i ( x ) = r i ( x − c i ) | x − c i | + c i , i = 1 , , and p ∈ D be the fixed point of R ◦ R , then R ( p )(=: p ) is the fixed point of R ◦ R ,and p = (cid:16) − √ r r r r + r √ ǫ + O ( ǫ ) , (cid:17) and p = (cid:16) √ r r r r + r √ ǫ + O ( ǫ ) , (cid:17) . Moreover, we can easily show thatΨ[ D , D ] = 12 π (log | x − p | − log | x − p | ) . (12)7y an elementary calculation, it can be shown that the middle point p + p exists betweentwo approaching points (cid:0) − ǫ , (cid:1) and (cid:0) ǫ , (cid:1) . Applying the middle point property to estimatefor Ψ[ D , D ] (cid:0) ± ǫ , (cid:1) , we can get the following lemma. Lemma 2.2
There is a constant
C > independent of ǫ , r and r such that C r r + r r r √ ǫ ≤ Ψ[ D , D ] (cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12) ∂D ≤ C r r + r r r √ ǫ for small ǫ > . From Lemma 2.1, we calculate the potential difference of u . Lemma 2.3
Let H ( x , x ) be an entire harmonic function. The solution u to (1) where L = 2 and D l , l = 1 , , are given as (11) satisfies u | ∂D − u | ∂D = H ( p ) − H ( − p )= 2 √ ∂ x H (0 , q r r r + r √ ǫ + O ( ǫ ) . (13) Remark 2.4
Referring to the mean value theorem, there exists a point x between ∂D and ∂D such that |∇ u ( x ) | ≥ √ | ∂ x H (0 , | r r r r + r √ ǫ . (14) for any sufficiently small ǫ > . Moreover, as a result in [4], there is a constant C indepen-dent of ǫ , r and r such that k∇ u k L ∞ (Ω \ ( D ∪ D )) ≤ C k∇ H k L ∞ (Ω) r r r r + r √ ǫ where Ω = B r + r ) (0 , . In this section, we consider two ǫ -distanced domains D and D , see Case (A) at Figure 1,where D = B r ( c ) and D = B r ( c ) ∪ B r ( c ) , where c = ( − r − ǫ , , c = ( r + ǫ , , and c = ( r + a + ǫ , . Here, B r ( c ) is a small lump of B r ( c ), and we assume B r ( c ) ∩ B r ( c ) = ∅ , i.e., 0 < a < r , and and 0 < ǫ ≪ r ≪ min( r , r ) . Define h = Ψ[ D , D ] , and u = Φ[ D , D ] , (15)and h j = Ψ[ D , B r j ( c j )] , and u j = Φ[ D , B r j ( c j )] , j = 2 , , (16)where Ψ and Φ defined in section 2.1. 8 .1 Properties of h and h j , j = 2 , Lemma 3.1
Let h = Ψ[ D , D ] , then we have ∂ ν h ( x ) = O ( √ ǫ ) , x ∈ ∂D \ B r ( c ) . (17) Proof . Set B i = B r i ( c i ) , for i = 1 , , . We choose a smooth domain e Ω as follows: e Ω ⊂ ( B ∪ B ) , B ⊂ e Ω ∂ e Ω \ B = ∂B \ B ( ∂ e Ω ∩ ∂B ) \ ∂B = ( 12 ǫ, e h = Ψ[ B , e Ω]. Then, we consider V defined in R \ ( B ∪ B ∪ B ) as follows: V = h − h | ∂B − h | ∂ ( B ∪ B ) e h | ∂B − e h | ∂ e Ω e h. Then, it follows that V | ∂B = V | ∂ e Ω \ B = a constant . Since h | ∂ ( B ∪ B ) > h | ∂B and e h | ∂ e Ω > e h | ∂B , the minimum of V attains on ∂ e Ω \ B and ∂B .Thus, we have ∂ ν h − h | ∂B − h | ∂ ( B ∪ B ) e h | ∂B − e h | ∂ e Ω ∂ ν e h ≤ ∂B ∪ ( ∂ e Ω \ B ) . (18)By the integration on ∂B , we have0 < h | ∂B − h | ∂ ( B ∪ B ) e h | ∂B − e h | ∂ e Ω ≤ . Using the bound (18) once more, we have ∂ ν h ≤ ∂ ν e h | ∂ e Ω on ∂ e Ω \ B = ∂B \ B . The domain e Ω is smooth so that we can use the method presented by Yun [14, 15]. Then,up to a conformal mapping to a circle, ∂ ν h is bounded by constant times the Poisson Kernelwith respect to a interior point √ ǫ distanced from the boundary (refer to the inequality (9)in [15]). Note that ∂B \ B distances enough from ( ǫ, ∂ ν h ≤ ∂ ν e h | ∂ e Ω ≤ C √ ǫ on ∂B \ B . Therefore, we have completed the proof of the lemma. (cid:3) emma 3.2 ∂ ν h ( x ) ≤ M ∂ ν h ( x ) , x ∈ ∂D , (19) where M = h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D h (cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12) ∂D . (20) Proof . Define W = h − M h , in R \ ( D ∪ D ) . Since h is constant on ∂D , M >
0, and h takes it’s maximum on ∂B r ( c ), W (cid:12)(cid:12)(cid:12) ∂D \ B r ( c ) − W (cid:12)(cid:12)(cid:12) ∂D \ B r ( c ) = − M ( h (cid:12)(cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12)(cid:12) ∂D \ B r ( c ) ) < , and W (cid:12)(cid:12)(cid:12) ∂D \ B r ( c ) − W (cid:12)(cid:12)(cid:12) ∂D = h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D − M ( h (cid:12)(cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12)(cid:12) ∂D ) = 0 . Therefore, W takes its minimum on ∂D , and ∂ ν W ≤ , on ∂D . (cid:3) Lemma 3.3 h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D = h (cid:12)(cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12)(cid:12) ∂D + O ( ǫ ) . (21) Proof . Note that Z ∂D ∂ ν ( h − h ) dS = 0 , and Z ∂D ∂ ν ( h − h ) dS = Z ∂D ∂ ν h dS − Z ∂B r ( c ) ∂ ν h dS − Z ∂ ( D \ B r ( c )) ∂ ν h dS = 1 − − . With the fact that h | ∂D and h | ∂D are constants and the (exterior) Divergence Theorem,we have that 0 = Z ∂D ∂ ν ( h − h ) h dS + Z ∂D ∂ ν ( h − h ) h dS = Z ∂D ( h − h ) ∂ ν h dS + Z ∂D ( h − h ) ∂ ν h dS. Hence, h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D = Z ∂D h∂ ν h dS + Z ∂D h∂ ν h dS = Z ∂D h ∂ ν h dS + Z ∂D h ∂ ν h dS = h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂B r ( c ) + Z ∂D ( h − h (cid:12)(cid:12)(cid:12) ∂B r ( c ) ) ∂ ν h dS C dependent of a , see (3), such that (cid:12)(cid:12)(cid:12) ( h − h (cid:12)(cid:12)(cid:12) ∂B r ( c ) )( x ) (cid:12)(cid:12)(cid:12) ≤ C √ ǫ, for all x ∈ ∂D \ B r ( c ) . Therefore, with (17) as well, we obtain (21). (cid:3)
Let H ( x , x ) = x and ν be the unit normal vector of R \ ( D ∪ D ), i.e., directed inwardto D i , i = 1 ,
2. Remind that we defined h = Ψ[ D , D ] , and u = Φ[ D , D ] , (22)and h j = Ψ[ D , B r j ( c j )] , and u j = Φ[ D , B r j ( c j )] , j = 2 , , (23)where Ψ and Φ defined in section 2.1.Note that ∂ ν h (cid:12)(cid:12)(cid:12) ∂D < H < ∂D and H > ∂D , and, as a result, from Lemma2.1, we have u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D = Z ∂D ( ∂ ν h ) H dS + Z ∂D ( ∂ ν h ) H dS ≥ Z ∂D H∂ ν h dS. (24)Applying the lemma 3.2, Lemma 3.3, (24) becomes u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D h (cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12) ∂D Z ∂D H∂ ν h dS ≥ h (cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12) ∂D + O ( ǫ ) h (cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12) ∂D √ r r r r + r r . It follows from Lemma 2.2 that h (cid:12)(cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12)(cid:12) ∂D ≥ C r r + r r r √ ǫ + O ( ǫ )and h (cid:12)(cid:12)(cid:12) ∂B r ( c ) − h (cid:12)(cid:12)(cid:12) ∂D ≤ C r r + r r r √ r + O ( r ) . Therefore, u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ C r r r + r √ r √ ǫ. This proves Theorem 1.1. (cid:3) .3 Proof Theorem 1.3 We consider the general shaped domain in Theorem 1.3. But, we take an advantage ofthe properties of circular inclusions. To make a connection between circular domains andgeneral shaped domains, we need to establish the monotonic property of Ψ as follows:
Lemma 3.4 [Monotonic property of Ψ ] Let D A , D B , e D A and e D B be domains. Assumethat D A ⊆ e D A and D B ⊆ e D B . Then, we have ≤ Ψ[ e D A , e D B ] (cid:12)(cid:12) ∂ e D B − Ψ[ e D A , e D B ] (cid:12)(cid:12) ∂ e D A ≤ Ψ[ D A , D B ] (cid:12)(cid:12) ∂D B − Ψ[ D A , D B ] (cid:12)(cid:12) ∂D A . Proof . Without any loss of generality, we consider only the case of D A = e D A . Let G = Ψ[ D A , e D B ] − M Ψ[ D A , D B ]where M = Ψ[ D A , e D B ] (cid:12)(cid:12) ∂D B − Ψ[ D A , e D B ] (cid:12)(cid:12) ∂D A Ψ[ D A , D B ] (cid:12)(cid:12) ∂D B − Ψ[ D A , D B ] (cid:12)(cid:12) ∂D A . The minimum of G attains on ∂D A . By the Hopf’s Lemma, we have ∂ ν G ≤ ∂D A . Integrating ∂ ν G on ∂D A , we have − M ≤
0. Therefore, we have0 ≤ Ψ[ D A , e D B ] (cid:12)(cid:12) ∂ e D B − Ψ[ D A , e D B ] (cid:12)(cid:12) ∂D A ≤ Ψ[ D A , D B ] (cid:12)(cid:12) ∂D B − Ψ[ D A , D B ] (cid:12)(cid:12) ∂D A . Repeating the same argument again, we can obtain the disable inequality. (cid:3)
Applying D left, r D center and D right instead of B r i ( c i ), i = 1 , ,
3, to the argumentpresented in the proof of Theorem 1.1, we can obtain u (cid:12)(cid:12)(cid:12) ∂r D center − u (cid:12)(cid:12)(cid:12) ∂D left ≥ C h (cid:12)(cid:12) ∂ ( r D center) − h (cid:12)(cid:12) ∂D left + O ( ǫ ) h (cid:12)(cid:12) ∂D right − h (cid:12)(cid:12) ∂D left √ r when H ( x , x ) = x . Here, h = Ψ[ D left , r D center] and h = Ψ[ D left , D right].It follows that from Yun [14, 15] that h (cid:12)(cid:12)(cid:12) ∂D right − h (cid:12)(cid:12)(cid:12) ∂D left ≃ √ r . To estimate h (cid:12)(cid:12)(cid:12) ∂r D center − h (cid:12)(cid:12)(cid:12) ∂D left , we choose two disks B left and B center containing D left and D center such that the distance between B left and r B center is ǫ . Using Lemma3.4 and 2.2, we have h (cid:12)(cid:12)(cid:12) ∂ ( r D center) − h (cid:12)(cid:12)(cid:12) ∂D left & r ǫr . D = D left, D = r D center and D = D right in this theorem. Therefore, u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D ≥ C √ r √ ǫ. This proves Theorem 1.3. (cid:3)
We consider three disjoint inclusion case, see Figure 1 and 2, a small one is disjointlyembedded into the in-between area of two others, and prove Theorem 1.2 and 1.4. Weassume that D and D are closely spaced with the distance ǫ , and D and D are closelyspace with ǫ , but D and D are not close, and that D , D and D have the boundaryregularity given in Theorem 1.4. u Let H c be a harmonic function outside of ∪ i =1 D i and have the same constant value in ∪ i =1 D i satisfying that ∆ H c = 0 , in R \∪ i =1 D i ,H c ( x ) − H ( x ) = O ( | x | − ) , as | x | → ∞ ,H c | ∪ i =1 ∂D i = C H (constant) . (25)Since H c − H is harmonic at infinity, H c − H attains maximum only at the boundary pointsof D i , i = 1 , ,
3. To make H c − H attains zero at infinity, C H should satisfy − (cid:13)(cid:13) H (cid:13)(cid:13) L ∞ ( ∪ i =1 D i ) ≤ C H ≤ (cid:13)(cid:13) H (cid:13)(cid:13) L ∞ ( ∪ i =1 D i ) . (26)Moreover, H c satisfies P i =1 R ∂D i ∂ ν H c dS = 0 . The solution u to (1) is represented as u ( x ) = H c ( x ) + c h ( x ) + c h ( x ) , (27)where h = Ψ (cid:2) D , ( D ∪ D ) (cid:3) , h = Ψ (cid:2) ( D ∪ D ) , D (cid:3) , and (cid:18) c c (cid:19) = − − Z ∂D ∂ ν h dS Z ∂D ∂ ν h dS Z ∂D ∂ ν h dS − Z ∂D ∂ ν H c dS Z ∂D ∂ ν H c dS , (28)where Ψ is defined as (7) and (9). The equality (28) is from the integration of ∂ ν u on ∂D and ∂D .Applying the upper bound on the gradient of solution without the potential differenceamong the boundaries to conductivity equation derived in Bao et al. [7], we can show that13 H c does not blow-up (also refer to [14]). Using Lemma 4.3 in the following section, wehave R ∂D ∂ ν h dS = 1 + O ( √ ǫ ). This implies that (cid:18) c c (cid:19) ≈ − (cid:18) − − (cid:19) − Z ∂D ∂ ν H c dS Z ∂D ∂ ν H c dS . Thus, the coefficient c i , i = 1 ,
2, is bounded independently of ǫ and ǫ . Therefore, theblow-up rate of ∇ u essentially relies on ∇ h i . In this respect, we consider the properties of h i in the following section. h and h We build the optimal bounds of u based on (27); it is essential to drive properties of h and h in the narrow regions between inclusions. Let h and h be as follows: h = Ψ (cid:2) D , ( D ∪ D ) (cid:3) and h = Ψ (cid:2) ( D ∪ D ) , D (cid:3) . Proposition 4.1
There are the following estimates for h and h :(i) In the narrow region between D and D , we have ∇ h = O (cid:16) √ ǫ (cid:17) and ∇ h = O ( √ ǫ ) . (ii) In the narrow region between D and D , we have ∇ h = O ( √ ǫ ) and ∇ h = O (cid:16) √ ǫ (cid:17) . (iii) h | ∂D ∪ ∂D − h | ∂D ≃ √ ǫ and h | ∂D − h | ∂D ∪ ∂D ≃ √ ǫ . Proof . We consider ∇ h . By Lemma 4.2 and 4.4, we have0 > ∂ ν h ≥ C∂ ν Ψ[ D , D ] on ∂D and 0 < ∂ ν h ≤ ∂ ν Ψ[ D , D ] on ∂D , and by Lemma 4.3, 0 < | ∂ ν h | ≤ C √ ǫ on ∂D , where D is defined in Lemma 4.4. Without any loss of generality, we assume that (cid:16) − ǫ , (cid:17) ∈ ∂D , (cid:16) ǫ , (cid:17) ∈ ∂D and dist( D , D ) = ǫ . p ( x ) = log | x − ( √ ǫ , | − log | x + ( √ ǫ , | . Referring to the inequality (9) in [15], there is a constant C such that0 < |∇ h | ≤ C |∇ p | on ∂ ( D ∪ D ∪ D ) . Regarding ( x , x ) as a complex number z = x + x i , we consider ρ ( z ) = ∂ h ( z ) − ∂ h ( z ) iC ( ∂ p ( z ) − ∂ p ( z ) i ) . Then, ρ ( z ) can be extended to ∞ as an analytic function. From definition, | ρ ( z ) | < ∂D ∪ ∂D ∪ ∂D . By the maximum principle, | ρ ( z ) | < C \ ( D ∪ D ∪ D ) . Thus, we have |∇ h | ≤ C |∇ p | in R \ ( D ∪ D ∪ D ) . Therefore, ∇ h = O (cid:16) √ ǫ (cid:17) in the narrow region between D and D , and ∇ h = O ( √ ǫ )in the narrow region between D and D . Similarly, we have ∇ h = O (cid:16) √ ǫ (cid:17) in the narrowregion between D and D , and ∇ h = O ( √ ǫ ) in the narrow region between D and D .We have proven (i) and (ii).The estimate (iii) is presented by Lemma 4.4. (cid:3) Lemma 4.2
We have the following properties:(i) < h | ∂D − h | ∂D ≤ Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D . (ii) < ∂ ν h ≤ ∂ ν Ψ[ D , D ] on ∂D . Proof . Let M = Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D ∪ ∂D Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D , and G ( x ) = Ψ[ D , ( D ∪ D )]( x ) − Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D ∪ ∂D − M (cid:18) Ψ[ D , D ]( x ) − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D (cid:19) . Then, G = 0 on ∂D ∪ ∂D , and G > ∂D . By Hopf’s lemma, ∂ ν G < ∂D . ∂ ν h ≤ M ∂ ν Ψ[ D , D ] on ∂D . (29)Note that h = Ψ[ D , ( D ∪ D )]. By integrating G on ∂D , we have the inequality (i).On the other hand, by Hopf’s lemma, ∂ ν G < ∂D . This means that ∂ ν h ≤ M ∂ ν Ψ[ D , D ] on ∂D . From the inequality (i),
M <
1. Therefore, we have (ii). (cid:3)
Lemma 4.3
There is a constant C such that ≤ ∂ ν h ≤ C √ ǫ on ∂D . Proof . We use the method similar to Lemma 4.2. Let M = Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D ∪ ∂D Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D , and G ( x ) = Ψ[ D , ( D ∪ D )]( x ) − Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D ∪ ∂D − M (cid:18) Ψ[ D , D ]( x ) − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D (cid:19) . Then, G = 0 on ∂D ∪ ∂D , and G > ∂D . By Hopf’s lemma, ∂ ν G < ∂D . Since h = Ψ[ D , ( D ∪ D )], this inequality means that0 ≤ ∂ ν h ≤ M ∂ ν Ψ[ D , D ] on ∂D . Now, we estimate the gradient of M Ψ[ D , D ]. To do so, we consider the potentialdifference between ∂D and ∂D as follows: M Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − M Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D = h | ∂D − h | ∂D = h | ∂D − h | ∂D ≤ Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D ≤ C √ ǫ The last inequality above was proven by Yun in his paper [14, 15], since Ψ[ D , D ] is onlyfor two domains. Note that D is not close to D . Owing to the method in Bao et al. [7],we have k ∂ ν M Ψ[ D , D ] k L ∞ ( ∂D ) ≤ C √ ǫ . Therefore, we can obtain the result. (cid:3) emma 4.4 Let D is a disk containing D and D withdist ( D , D ) = dist ( D , D ) . (i) There is a positive constant C such that > ∂ ν h ≥ C∂ ν Ψ[ D , D ] on ∂D . (ii) h | ∂D − h | ∂D ≥ Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D . (iii) h | ∂D ∪ ∂D − h | ∂D ≃ √ ǫ . Proof . To prove (i) and (ii), we use the same derivation to Lemma 4.2. So, we set M = Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D ∪ ∂D Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D , and G ( x ) = Ψ[ D , ( D ∪ D )]( x ) − Ψ[ D , ( D ∪ D )] (cid:12)(cid:12)(cid:12) ∂D ∪ ∂D − M (cid:18) Ψ[ D , D ]( x ) − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D (cid:19) . Then G (cid:12)(cid:12)(cid:12) ∂D = 0 and G ≤ ∂D . By Hopf’s lemma, we have ∂ ν G > ∂D . By the integration on ∂D , we have (ii) and M <
1. Therefore, the inequality ∂ ν G > h | ∂D − h | ∂D ≥ Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D and h | ∂D − h | ∂D ≤ Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D . The potential Ψ[ D , D i ] ( i = 1 ,
4) is only for two domains and thus, its difference between D and D i ( i = 1 ,
2) was already estimated in Yun [14, 15] as follows: for i = 1 , D , D i ] (cid:12)(cid:12)(cid:12) ∂D i − Ψ[ D , D i ] (cid:12)(cid:12)(cid:12) ∂D ≃ √ ǫ . Therefore, we have (iii). (cid:3) emma 4.5 We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∪ i =1 ∂D i H∂ ν h dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ ǫ . Proof . Without loss of generality, we assume that (cid:16) − ǫ , (cid:17) ∈ ∂D , (cid:16) ǫ , (cid:17) ∈ ∂D , dist( D , D ) = ǫ and ( − , ∈ D . We consider e H as follows: e H = H − ∂ H (0 , x | x − (1 , | . It follows from the Divergence Theorem that Z ∂D ∪ ∂D ∪ ∂D x | x − (1 , | ∂ ν hdS = Z ∂D ∪ ∂D ∪ ∂D ∂ ν (cid:18) x | x − (1 , | (cid:19) hds = 0 , since x | x − (1 , | = O ( | x | − ) as | x | → ∞ . Hence, we have Z ∪ i =1 ∂D i H∂ ν h dS = Z ∂D ∪ ∂D e H∂ ν h dS + Z ∂D e H∂ ν h dS. We first consider R ∂D ∪ ∂D e H∂ ν h dS . By Lemma 4.2 and 4.4, we have0 > ∂ ν h ≥ C∂ ν Ψ[ D , D ] on ∂D and 0 < ∂ ν h ≤ ∂ ν Ψ[ D , D ] on ∂D . From definition, ∂ e H = 0. Hence, we can use Lemma 3.2 in [15] so that (cid:12)(cid:12)(cid:12)(cid:12)Z ∂D e H∂ ν h dS (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z ∂D (cid:12)(cid:12)(cid:12) e H Ψ[ D , D ] (cid:12)(cid:12)(cid:12) dS ≤ C √ ǫ and (cid:12)(cid:12)(cid:12)(cid:12)Z ∂D e H∂ ν h dS (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∂D (cid:12)(cid:12)(cid:12) e H Ψ[ D , D ] (cid:12)(cid:12)(cid:12) dS ≤ C √ ǫ . We second consider R ∂D e H∂ ν h dS . By Lemma 4.3, we can have (cid:12)(cid:12)(cid:12)(cid:12)Z ∂D e H∂ ν h dS (cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ ǫ . Therefore, we have done it. (cid:3)
Remark 4.6
We draw attention of readers to the independent work of Bao, Li and Yin in[5] and [8]. Bao et al. have shown that the blow-up rate know only for a pair of inclusion isstill valid to the multiple inclusions cases. As a byproduct of our work, the blow-up rate ofthe gradient for three inclusions is established in Theorem 4.7. heorem 4.7 Let D , D and D be as assumed in the beginning of Section 4. Note that D is not assumed to be smaller than the others. (i) Optimal upper bounds: For any entire harmonic function H ( x , x ) , we have the fol-lowing: in the narrow region between D ∪ D , |∇ u | ≤ C √ ǫ , and, in the narrow region between D ∪ D , |∇ u | ≤ C √ ǫ . (ii) Existence of blow-up: Without loss of generality, we assume that (cid:16) − ǫ , (cid:17) ∈ ∂D , (cid:16) ǫ , (cid:17) ∈ ∂D and dist ( D , D ) = ǫ . For H ( x , x ) = x , there exist x in the narrow region between D and D such that |∇ u ( x ) | ≥ C √ ǫ , and, similarly, there is a linear function H ( x , x ) with y between D and D suchthat |∇ u ( y ) | ≥ C √ ǫ . Proof . From Subsection 4.1, we have a representation (28) for u and the coefficient c i , i = 1 ,
2, is bounded independently of ǫ and ǫ . Proposition 4.1 yields the upper bound ofTheorem 4.7.Now, we consider the existence of the blow-up. Using the result of Subsection 4.1 again,we have a constant C independent of ǫ such that k u k L ∞ ( ∪ i =1 ∂D i ) ≤ C k H k L ∞ ( ∪ i =1 D i ) . Applying the Green’s identity to R ∪ i =1 ∂D i u∂ ν h dS , we have Z ∪ i =1 ∂D i H∂ ν h dS = Z ∪ i =1 ∂D i u∂ ν h dS = − u (cid:12)(cid:12) ∂D + u (cid:12)(cid:12) ∂D (cid:0) Z ∂D ∂ ν h dS (cid:1) + u (cid:12)(cid:12) ∂D (cid:0) Z ∂D ∂ ν h dS (cid:1) = − u (cid:12)(cid:12) ∂D + u (cid:12)(cid:12) ∂D (cid:0) − Z ∂D ∂ ν h dS (cid:1) + u (cid:12)(cid:12) ∂D (cid:0) Z ∂D ∂ ν h dS (cid:1) . (30)By Lemma 4.3, we have u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D ≤ C √ ǫ , (31)19here the constant C above depends on k H k L ∞ ( ∪ i =1 ∂D i ) . Similarly, we have u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D ≤ C √ ǫ . (32)Using (30) again, we have u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D + O ( √ ǫ ǫ ) = Z ∪ i =1 ∂D i H∂ ν h dS ≥ Z ∂D H∂ ν h dS. (33)The last inequality can be derived from the fact that H > ∂D ∪ ∂D .To get the last inequality above, we took an advantage of H = x . By (29), ∂ ν h ≤ h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D ∪ ∂D Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D ∂ ν Ψ[ D , D ] < ∂D . By (iii) in Proposition 4.1, we have h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D ∪ ∂D ⋍ √ ǫ and Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D − Ψ[ D , D ] (cid:12)(cid:12)(cid:12) ∂D ⋍ √ ǫ . The inequality (33) implies u (cid:12)(cid:12)(cid:12) ∂D − u (cid:12)(cid:12)(cid:12) ∂D & √ ǫ. By the Mean Value Theorem, we have the desirable lower bound in the narrow regionbetween D and D . Similarly, we can also obtain the other lower bound. (cid:3) We derive the optimal bounds of the gradient of the solution to (1), when there are adjacentthree disks: D l = B r l ( c l ) , l = 1 , , , (34)where c = ( − r − ǫ , c = ( r + ǫ ,
0) and c = ( r + r + ǫ + ǫ , h = Ψ[ D , ( D ∪ D )]. Let w = Ψ[ D , D ].We begin the proof by showing that w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D ≃ h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D . (35)By the monotonic property of Lemma 3.4, we have h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D ≤ w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D . Considering h − h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D ! w ,
20e can obtain, from the Hopf’s Lemma, Z ∂D ∂ ν h dS ≤ h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D ! Z ∂D ∂ ν w dS. By Lemma 4.3, we have Z ∂D ∂ ν h dS = O ( √ ǫ ) . Since R ∂D ∪ ∂D ∂ ν h dS = 1, we have (cid:16) w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D (cid:17) (1 + O ( √ ǫ )) ≤ h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D . Therefore, we can obtain (35). Owing to the estimate for w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D in Lemma 2.2,we have h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D ≃ r r + r r r √ ǫ . (36)Let w = Ψ[ D , D ]. Considering h − h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D ! w , from the Hopf’s Lemma, we obtain ∂ ν h ≤ h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D ! ∂ ν w ≤ ∂D . Here, we estimate the coefficient in the right hand side. Note that h (cid:12)(cid:12) ∂D = h (cid:12)(cid:12) ∂D . Thus,we have h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D ≃ r r + r r r √ ǫ . Since r ≪ r and r ≪ r , we also have w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D ≃ r r + r r r √ r . This implies that ∂ ν h . q r + r r r √ ǫ q r + r r r √ r ∂ ν w ≤ ∂D . Therefore, we have Z ∂D H∂ ν h dS & r r + r r + r √ r r √ ǫ Z ∂D H∂ ν w dS & r r + r r + r √ r r √ ǫ r r r r + r √ r ≥ r r r + r √ r √ ǫ ≥ . (37)21wing to (30), (31) and (32), we have Z ∪ i =1 ∂D i u∂ ν h dS = (cid:18) − Z ∂D ∂ ν h dS (cid:19) (cid:16) u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D (cid:17) + (cid:18)Z ∂D ∂ ν h dS (cid:19) (cid:16) u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D (cid:17) = (1 − O ( √ ǫ )) (cid:16) u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D (cid:17) + O ( √ ǫ ) ( O ( √ ǫ ) + O ( √ ǫ )) . Therefore, we have u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D ≥ Z ∪ i =1 ∂D i u∂ ν h dS + O ( √ ǫ ) ( O ( √ ǫ ) + O ( √ ǫ ))= 12 Z ∪ i =1 ∂D i H∂ ν h dS + O ( √ ǫ ) ( O ( √ ǫ ) + O ( √ ǫ )) ≥ Z ∂D H∂ ν h dS + O ( √ ǫ ) ( O ( √ ǫ ) + O ( √ ǫ )) ≥ C r r r + r √ r √ ǫ + O ( √ ǫ ) ( O ( √ ǫ ) + O ( √ ǫ )) . Therefore, we have completed the proof. (cid:3)
We pursuit the proof of Theorem 1.2, taking an advantage of the monotonic property ofLemma 3.4. The domains D , D and D are as assumed in Theorem 1.4. As assumedbefore, h = Ψ[ D , ( D ∪ D )]. Let w = Ψ[ D , D ]. By the same way as Theorem 1.2, wehave w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D ≃ h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D . Here, we use the monotonic property of Lemma 3.4 to estimate the difference betweendomains. Choosing two pairs of proper disks containing D and D , and contained D and D , respectively, we can obtain h (cid:12)(cid:12) ∂D − h (cid:12)(cid:12) ∂D ≃ r ǫ r under the assumption that r is small.Let w = Ψ[ D , D ]. Choosing two pairs of proper disks containing D and D , andcontained D and D , respectively, Then, we have w (cid:12)(cid:12) ∂D − w (cid:12)(cid:12) ∂D ≃ √ r . By the same argument as Theorem 1.2, we have Z ∂D H∂ ν h dS & r ǫ r ≥ . D ⊂ R − × R and D ∪ D ⊂ R + × R . Continuing to follow the proof of Theorem1.2, we can obtain u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D ≥ C r ǫ r + O ( √ ǫ ) ( O ( √ ǫ ) + O ( √ ǫ )) . Therefore, we have done the proof.
We consider the optimal upper bounds presented in Theorem 1.1, 1.2, 1.3 and 1.4. Theseproofs have essential thing in common. In this respect, we prove only the optimal upperbound presented in Theorem 1.2. As have assumed them before, we set h = Ψ[ D , D ∪ D ] h = Ψ[ D , D ] h = Ψ[ D , D ] h = Ψ[ D , D ] . Here, the domain D is given in Lemma 4.4, which is a disk containing D and D withdist( D , D ) = dist( D , D ) , and the diameter of D is in proportion as r , because r is sufficiently small. Then, wecompare h with h , h and h . The proof of Lemma 4.2 contains0 ≤ ∂ ν h ≤ h (cid:12)(cid:12)(cid:12) ∂ ( D ∪ D ) − h (cid:12)(cid:12)(cid:12) ∂D h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D ∂ ν h on ∂D , the proof of Lemma 4.3 yields0 ≤ ∂ ν h ≤ h (cid:12)(cid:12)(cid:12) ∂ ( D ∪ D ) − h (cid:12)(cid:12)(cid:12) ∂D h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D ∂ ν h on ∂D and the proof of Lemma 4.4 implies0 ≤ − ∂ ν h ≤ − h (cid:12)(cid:12)(cid:12) ∂ ( D ∪ D ) − h (cid:12)(cid:12)(cid:12) ∂D h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D ∂ ν h on ∂D . In the same way as Lemma 4.5, we can consider e H by choosing the point in D . In thisrespect, without any loss of generality, we can assume that ∂ x H (0 ,
0) = 0 . The reason why we assumed above is because the integration representation for the potentialdifference is not good enough, refer to [15]. The geometrical assumption of Case (B) impliesthat D and D ∪ D are separated by x = 0 and they are approaching to (0 , (cid:12)(cid:12)(cid:12) u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D (cid:12)(cid:12)(cid:12) + O ( √ ǫ ) ( O ( √ ǫ ) + O ( √ ǫ ))= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂ ( ∪ i =1 D i ) H∂ ν h dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . h (cid:12)(cid:12)(cid:12) ∂ ( D ∪ D ) − h (cid:12)(cid:12)(cid:12) ∂D h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D r r r r + r ǫ + h (cid:12)(cid:12)(cid:12) ∂ ( D ∪ D ) − h (cid:12)(cid:12)(cid:12) ∂D h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D r r r r + r r + h (cid:12)(cid:12)(cid:12) ∂ ( D ∪ D ) − h (cid:12)(cid:12)(cid:12) ∂D h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D r r r r + r ǫ and h (cid:12)(cid:12)(cid:12) ∂ ( D ∪ D ) − h (cid:12)(cid:12)(cid:12) ∂D ≈ h (cid:12)(cid:12)(cid:12) ∂D − h (cid:12)(cid:12)(cid:12) ∂D . Here, note that the radius of D can be choosen between r and 2 r . Lemma 2.2 impliesthat (cid:12)(cid:12)(cid:12) u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D (cid:12)(cid:12)(cid:12) . r r r + r √ r √ ǫ . Therefore, we establish the optimal upper bound for (cid:12)(cid:12)(cid:12) u (cid:12)(cid:12) ∂D − u (cid:12)(cid:12) ∂D (cid:12)(cid:12)(cid:12) .Based on this, the optimal upper bound on the gradient of u in the narrow region beobtained. Here, the main idea to get the gradient estimate from the potential difference hasalready been presented by Bao et al. (Theorem 1.3, Lemma 2.2 and 2.3 in [7]), and has beenmodified to fit our problem by Lim and Yun in [13]. Thurs, we give a brief description onthe method. We choose a large domain D containing D , D and D , where ∂D is at asufficient distance from D , D and D . Then, u can be decomposed as follows: u = C + v + C v + C v where for i = 0 , , v i is a harmonic function in D \ ( D ∪ D ∪ D ) with the boundarydata v i = δ j on ∂D j for i = 1 , v = δ ij u on ∂D j for any j = 0 , , ,
3. Thus, the constants C and C keep | C | . r r r + r √ r √ ǫ and | C | . r r r + r √ r √ ǫ .
24o estimate ∇ v , we consider a harmonic function ρ in D \ ( D ∪ D ∪ D ) with theboundary data ρ = δ j on ∂D j for any j = 0 , , ,
3. By comparing with the harmonic function ρ i in D \ D i with ρ i = 0on ∂D i and ρ i = 1 on ∂D , the Hopf’s Lemma yields k∇ ρ k L ∞ ( D \ ( D ∪ D ∪ D )) ≤ max {k∇ ρ k L ∞ ( ∂D ) , k∇ ρ k L ∞ ( ∂D ) , k∇ ρ k L ∞ ( ∂D ) , k∇ ρ k L ∞ ( ∂D ) } < C. Applying the Hopf’s Lemma again, we can have that the gradient of v is bounded indepen-dent of ǫ , refer to Lemma 2.2 in [7].We estimate C ∇ v in the narrow region between D and D . Since v is constat on theboundaries and the boundaries is smooth enough in the narrow region, the proof of Lemma4.3 implies that v can be extend into the interior areas of D and D by the distance almost ǫ from the boundaries in the narrow region, independently of r and r . By the gradientestimate for harmonic functions allows | C ∇ v | . r r r + r √ r √ ǫ in the narrow region between D and D . Note that the inequality above is a local propertyindependent of choosing D .Now, we consider C ∇ v in the narrow region between D and D . Let e ρ be a harmonicfunction in D \ ( D ∪ D ) with the boundary data e ρ = 0 on ∂ ( D ∪ D ) and e ρ = 1 on ∂D . By the maximum principle, we have0 ≤ v ≤ e ρ in D \ ( D ∪ D ∪ D )Considering the standard estimate for Ψ[ D , D ], we can obtain | C v | ≤ C √ ǫ . Similarly to the estimate for C ∇ v , the gradient estimate for harmonic functions yields | C ∇ v | . r ǫ ǫ in the narrow region between D and D .Therefore, we can obtain the desirable upper bound. Here, it is noteworthy that theupper bound is dominated only by the estimate for C ∇ v , which is independent of choosing D . In this respect, the constant C of the upper bound in Theorem 1.2 is independent of r , r , r , ǫ and ǫ . (cid:3) cknowledgements The authors would like to express his gratitude to Professor Hyeonbae Kang, who suggestedthe original problem studied in this paper. The authors is also grateful to Professor YanYanLi for his concern with their subject and suggestions. The second named author would alsolike to express his thanks to Professor Gang Bao, and gratefully acknowledges his hospitalityduring the visiting period at Michigan State University.
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