Optimal estimates for the conductivity problem by Green's function method
aa r X i v : . [ m a t h . A P ] J un OPTIMAL ESTIMATES FOR THE CONDUCTIVITY PROBLEMBY GREEN’S FUNCTION METHOD
HONGJIE DONG AND HAIGANG LI
Abstract.
We study a class of second-order elliptic equations of divergenceform, with discontinuous coefficients and data, which models the conductivityproblem in composite materials. We establish optimal gradient estimates byshowing the explicit dependence of the elliptic coefficients and the distancebetween interfacial boundaries of inclusions. The novelty of these estimates isthat they unify the known results in the literature and answer open problem( b ) proposed by Li-Vogelius (2000) for the isotropic conductivity problem. Wealso obtain more interesting higher-order derivative estimates, which answersopen problem ( c ) of Li-Vogelius (2000). It is worth pointing out that theequations under consideration in this paper are nonhomogeneous. Introduction and main results
In this paper, we establish optimal gradient and higher derivative estimates forsolutions of the isotropic conductivity problem. The problem is modeled by a classof divergence form second-order elliptic equations with discontinuous coefficientsand data L ε ; r ,r u := D i ( a ( x ) D i u ) = D i f i in D , (1.1)where the Einstein summation convention in repeated indices is used, D is abounded open subset of R , a ( x ) = k χ B + k χ B + χ B ,k , k , r , r ∈ (0 , ∞ ) are constants, B := B r ( ε/ r , , B := B r ( − ε/ − r , , B = R \ ( B ∪ B ) , and χ is the indicator function. Here D models the cross-section of a fiber-reinforcedcomposite, with the disks B and B representing the cross-sections of the fibers; theremaining subdomain representing the matrix surrounding the fibers. The gradientof the potential u represents the electric field in the conductivity problem or thestress in anti-plane elasticity problem. Moreover, a ( x ) is the conductivity (for theconductivity problem) or the shear modulus (for the anti-plane shear problem),which is a constant on the fibers, and a different constant on the matrix. Theconstant ε is used to denote the distance between B and B . See Figure 1. It isimportant from a practical point of view to know whether | Du | can be arbitrarilylarge as the inclusions get closer to each other. It is also of interest to establishsimilar estimates for higher-order norms of solutions. The purpose of this paper isto investigate the explicit dependence of | D m u | ( m ≥
1) on ε, k , k , r , and r .In [7], Babu˘ska, Andersson, Smith, and Levin numerically analyzed the initiationand growth of damage in composite materials, in which the inclusions are frequently Date : November 13, 2018. a = 1 a = k a = k B B B " D Figure 1.
A domain with two closely spaced inclusionsspaced very closely and even touching. There have been many important work onthe gradient estimates for solutions of elliptic equations and systems arising fromcomposite materials. See, for instance, [10, 13, 15, 19, 20, 21] and the referencestherein.For two touching disks in 2D with k = k = k away from 0 and ∞ , Bonnetier andVogelius [13] proved that | Du | remains bounded, with an upper bound dependingon the value of k . Li and Vogelius [21] extended the result to general divergenceform second-order elliptic equations with piecewise H¨older continuous coefficientsin all dimensions, and they proved that | Du | remains bounded as ε ց
0. Actuallythey established stronger, ε -independent, piecewise C ,α estimates for solutions.This extension covers inclusions of arbitrary smooth shape. Later Li-Nirenberg[20] further extended these results to general divergence form second-order ellipticsystems including systems of elasticity. The estimates in [20] and [21] all depend onthe ellipticity of the coefficients. In [21, Page 94], Li and Vogelius proposed severalinteresting questions including the following two:(b ): How does the constant in the estimates depend on the ellipticity constants?(c ): Do similar estimates hold for higher order norms of the solution, assuming ofcourse all the data are appropriately smooth?On the other hand, if the ellipticity constants are allowed to partially deteriorate,the situation is very different. The perfect conductivity problem with two inclusionscan be described as follows ∆ u = 0 in D \ ( B ∪ B ) ,Du = 0 in B ∪ B ,u | + = u | − on ∂ B ∪ ∂ B , R ∂ B i ( ∂u/∂ν ) | + = 0 for i = 1 , . (1.2)Formally, the system (1.2) above can be obtained from (1.1) by setting f i = 0 andpassing to the limit as k , k ր ∞ . In contrast to the case when k and k arefinite and bounded below from zero, it was shown in [14] and [23] that the gradient STIMATES BY GREEN FUNCTION METHOD 3 of solutions may blow up as two inclusions approach each other with the blow-uprate ε − / for a special solution in the 2D case. Rigorous proofs were later carriedout by Ammari, Kang, and Lim [4], and Ammari, Kang, Lim, Lee, and Lee [5] forthe case of circular inclusions. Since then, the problem has been studied by manymathematician. It has been proved that for the two close-to-touching inclusionscase the generic rate of | Du | blow-up is ε − / in two dimensions, | ε log ε | − inthree dimensions, and ε − in dimensions greater than three. See Yun [24, 25], Bao,Li, and Yin [8], as well as Lim and Yun [22], and Ammari, Bonnetier, Triki, andVogelius [2]. We also mention that more detailed characterizations of the singularbehavior of | Du | have been obtained by Ammari, Ciraolo, Kang, Lee, and Yun [3],Ammari, Kang, Lee, Lim, and Zribi [6], Bonnetier and Triki [11, 12] and Kang,Lim, and Yun [17, 18].Similar to the perfect conductivity problem, the insulated conductivity problemcan also be derived from (1.1) by passing to the limit as k , k ց
0, which can beconsidered as the complementary problem to the perfect case. The correspondingsystem is then given by ( ∆ u = 0 in D \ ( B ∪ B ) ,∂u/∂ν = 0 on ∂ B ∪ ∂ B . (1.3)As in the perfect case, the gradient of solutions to (1.3) generally blows up as thedistance between the inclusions goes to zero. In 2D, the authors of [4, 5] obtained theoptimal bound for circular inclusions with comparable radii and the blow-up rateis ε − / . The proof uses harmonic conjugators to convert the insulated case to theperfect case, which no longer works when d ≥
3. For the higher dimensional case,Bao, Li, and Yin [9] established upper bound ε − / by using a flipping techniqueand a variant of Li-Nirenberg’s result. We point out that in all these papers, theequation is assumed to be homogeneous, i.e., f i ≡ i = 1 , L ε ; r ,r with piecewise constant coefficients in order to express the solution explic-itly. We then establish unified gradient estimates for the solutions of (1.1), whichinvolve a precise dependence on k , k , r , r , and ε . This answers the open prob-lem ( b ) proposed by Li-Vogelius [21] in the case of circular inclusions. Furthermore,regarding the open problem ( c ), we obtain an upper bound for higher derivatives,which extends the results of Li-Vogelius [21] and Dong-Zhang [16], where the casewhen ε = 0 was studied.Let k = 1, α = k − k + 1 , and β = k − k + 1 . To illustrate the main ideas of the proof, we first assume that r = r = 1 and showthe dependence of | Du | and | D m u | with respect to k , k , and ε . The general caseis treated in Theorem 1.5.In the first theorem below, we consider a weak solution to (1.1) in a neighborhoodof the origin, which may not entirely contain the balls B and B . Theorem 1.1.
Let ε ∈ (0 , / and γ ∈ (0 , be constants. Assume that u is aweak solution of (1.1) in B := B (0) with r = r = 1 and f is piecewise C γ in B , which satisfy for some constant C > , k u k L ( B ) ≤ C , k f k C γ ( B ∩B j ) ≤ C k j , j = 0 , , . (1.4) H. DONG AND H.G. LI
Then we have | Du | ≤ CC − (1 − √ ε ) | αβ | in B / , (1.5) where C > is a constant depending only on γ . Furthermore, for any integer m ≥ , if f is piecewise C m − ,γ in B , and for some constant C m > , k u k L ( B ) ≤ C m , k f k C m − ,γ ( B ∩B j ) ≤ C m k j , j = 0 , , , (1.6) then we have | D m u | ≤ CC m (cid:0) − (1 − √ ε ) | αβ | (cid:1) m in B / , (1.7) where C > is a constant depending only on m and γ . In the next theorem, we obtain more precise estimates in B and B by assumingthat u satisfies (1.1) in a domain which contains both B and B . Theorem 1.2.
Let ε ∈ (0 , / and γ ∈ (0 , be constants. Assume that B ∪ B ⋐ D ⋐ D ⋐ D for some domains D and D , u is a weak solution of (1.1) in D with r = r = 1 , and f is piecewise C γ in D , which satisfy for some constant C > , k u k L ( D ) ≤ C , k f k C γ ( D∩B j ) ≤ C min { , k j } , j = 0 , , . (1.8) Then we have | Du | ≤ CC − (1 −√ ε ) | αβ | in D ∩ B , k +1 · CC − (1 −√ ε ) | αβ | in B , k +1 · CC − (1 −√ ε ) | αβ | in B , (1.9) where C > is a constant depending only on γ , D , D , and D . Furthermore, forany integer m ≥ , if f is piecewise C m − ,γ in D , and for some constant C m > , k u k L ( D ) ≤ C m , k f k C m − ,γ ( D∩B j ) ≤ C m min { , k j } , j = 0 , , , (1.10) then we have | D m u | ≤ CC m ( − (1 −√ ε ) | αβ | ) m in D ∩ B , k +1 · CC m ( − (1 −√ ε ) | αβ | ) m in B , k +1 · CC m ( − (1 −√ ε ) | αβ | ) m in B , (1.11) where C > is a constant depending only on m , γ , D , D , and D . In the case of circular inclusions, the estimates (1.5) and (1.9) unify the knownresults in the literature:(1) boundedness of | Du | for finite k and k regardless of the distance ε obtainedin [13, 20, 21];(2) blow-up of | Du | with rate of 1 / √ ε for k = k = + ∞ established in [3, 4,5, 8, 24, 25];(3) blow-up of | Du | with rate of 1 / √ ε for k = k = 0 established [4, 5].This is the first main contribution of this paper. It is worth pointing out that theestimate in D ∩ B of (1.9) was achieved by using a completely different method,single layer potential method, in [4, 5, 8, 24, 25]. Compared to the referencesmentioned above, we also consider more general non-homogeneous equations. Thesecond main contribution of our paper is the higher-order derivative estimates (1.7) STIMATES BY GREEN FUNCTION METHOD 5 and (1.11), extending the results of Li-Vogelius [21] and more recent work of Dong-Zhang [16], where the boundedness of the higher derivatives was proved when twoballs are assumed to touch each other at the origin. This in particular answers inthe affirmative the conjecture in Li-Vogelius [21, Remark 8.2, pp 137]: “We do feel, however, that for < a < (as is the case here) thesmoothness exhibited by u makes it quite likely that the u ε havepiecewise defined, uniformly bounded derivatives of any order...” Here u stands for the solution when ε = 0, i.e., the two balls touch each other. Remark 1.3.
The following example gives a lower bound of the gradient, whichshows that (1.5) is optimal when k , k ≥
1. Choose f to be a nonnegative functionsupported in B / ( − ,
0) which is a even function in x with unit integral, and f ≡
0. Assume that u is a weak solution to (1.1) with f defined above and B ∪ B ⊂ D . Then we have | Du (0) | ≥ C − (1 − √ ε ) αβ (1.12)for some constant C > ε , k , and k . See Section 3.2 for moredetails. For the special case when k = k = k , we can rewrite (1.5) and (1.12) asfollows to get more transparent dependence of | Du | on k and ε : when k ≫ | Du | ≤ CC √ ε + 1 /k in B / and | Du (0) | ≥ C √ ε + 1 /k ;when k ≪ | Du | ≤ CC √ ε + k in B / and | Du (0) | ≥ C √ ε + k . Remark 1.4.
When B ∪ B ⊂ D and ε is fixed small, it is easy to see from (1.9)that as k , k → ∞ , α, β →
1. Therefore, we not only have the blow-up rate 1 / √ ε in the narrow region D ∩ B , but also show that the upper bounds in B and B tend to zero as k , k → ∞ , which are consistent with the condition in [8] that Du = 0 in B ∪ B , derived by a variational argument. See [8, the Appendix].Finally we give the following theorem for the general case that r and r are notnecessarily equal to 1. Theorem 1.5.
Let ε ∈ (0 , / , γ ∈ (0 , be constants and ε ≪ r , r < .Under the assumptions of Theorem 1.2, we have, for any integer m ≥ , | D m u | ≤ CC m (cid:16) − (cid:0) − √ /r +1 /r ) ε (cid:1) | αβ | (cid:17) m in D ∩ B , CC m k +1 · (cid:16) − (cid:0) − √ /r +1 /r ) ε (cid:1) | αβ | (cid:17) m in B , CC m k +1 · (cid:16) − (cid:0) − √ /r +1 /r ) ε (cid:1) | αβ | (cid:17) m in B , where C > is a constant depending only on m , γ , D , D , and D . The remaining part of the paper is organized as follows. Section 2 is devoted tothe construction of a Green’s function of the operator L ε ; r ,r by using the inversionmaps Φ and Φ with respect to the circles ∂ B and ∂ B . In Section 3 we first derivethe derivative estimates of the maps Φ Φ and Φ Φ , as well as their compositions.We then prove our main results, Theorems 1.1 and 1.2. Finally, Theorem 1.5 isproved in Section 4. H. DONG AND H.G. LI Construction of a Green’s function
In this section, we construct a Green’s function of the operator L ε ; r ,r by adapt-ing an idea in [21]. It is well known that − π ∆ log | x − y | = δ ( x − y ) . For simplicity of exposition, we write ∆ log | x − y | = δ ( x − y ). In order to illustratethe main ideas, we assume that r = r = 1. The general case is similar. SeeSection 4. We use L to denote the operator L ε ;1 , .Define the inversion maps of a point x ∈ R with respect to ∂ B = ∂B (1+ ε/ , ∂ B = ∂B ( − − ε/ , ( x , x ) := (cid:18) x − (1 + ε/ x − − ε/ + x + 1 + ε/ , x ( x − − ε/ + x (cid:19) (2.1)and Φ ( x , x ) := (cid:18) x + 1 + ε/ x + 1 + ε/ + x − − ε/ , x ( x + 1 + ε/ + x , (cid:19) . Recall that α = k − k + 1 and β = k − k + 1 . It is clear that α, β ∈ ( − , G ( x, y ) as follows.(1) When y ∈ B , G ( x, y ) equals2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | − β log | (Φ Φ ) l Φ ( x ) − y | (cid:17) for x ∈ B ;log | x − y | + ∞ X l =1 h ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | + log | (Φ Φ ) l ( x ) − y | (cid:17) − ( αβ ) l − (cid:16) β log | (Φ Φ ) l − Φ ( x ) − y | + α log | (Φ Φ ) l − Φ ( x ) − y | (cid:17)i for x ∈ B ;2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | − α log | (Φ Φ ) l Φ ( x ) − y | (cid:17) for x ∈ B . (2) When y ∈ B , G ( x, y ) equals1 k (cid:0) log | x − y | + α log | Φ ( x ) − y | (cid:1) − β ( k + 1) ∞ X l =0 ( αβ ) l log | (Φ Φ ) l Φ ( x ) − y | for x ∈ B \ { (1 + ε/ , } ;2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | − β log | (Φ Φ ) l Φ ( x ) − y | (cid:17) for x ∈ B ;4( k + 1)( k + 1) ∞ X l =0 ( αβ ) l log | (Φ Φ ) l ( x ) − y | for x ∈ B . STIMATES BY GREEN FUNCTION METHOD 7 (3) When y ∈ B , G ( x, y ) equals4( k + 1)( k + 1) ∞ X l =0 ( αβ ) l log | (Φ Φ ) l ( x ) − y | for x ∈ B ;2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | − α log | (Φ Φ ) l Φ ( x ) − y | (cid:17) for x ∈ B ;1 k (cid:0) log | x − y | + β log | Φ ( x ) − y | (cid:1) − α ( k + 1) ∞ X l =0 ( αβ ) l log | (Φ Φ ) l Φ ( x ) − y | for x ∈ B \ { (1 + ε/ , } . Remark 2.1.
We note that in the above definition of G , for the case y ∈ B (or y ∈ B ), the point (1 + ε/ ,
0) (or (0 , − − ε ), respectively) is excluded, becauseΦ (or Φ ) has a singularity at (1 + ε/ ,
0) (or (0 , − − ε ), respectively). All theother terms appearing in the summations are regular. Lemma 2.2.
Let G ( x, y ) be defined above. Then we have ∆ x G ( x, y ) = δ ( x − y ) for y ∈ B and x / ∈ ∂ ( B ∪ B ); k ∆ x G ( x, y ) = δ ( x − y ) for y ∈ B and x / ∈ ∂ ( B ∪ B ) ∪ { (1 + ε/ , } ; k ∆ x G ( x, y ) = δ ( x − y ) for y ∈ B and x / ∈ ∂ ( B ∪ B ) ∪ { ( − − ε/ , } . Moreover, G ( x, y ) and a ( x ) D ν G ( x, y ) are continuous across ∂ B ∪ ∂ B , where ν isthe unit norm vector field of ∂ ( B ∪ B ) Proof.
We first consider the case when y ∈ B . When x ∈ B , (Φ Φ ) l ( x ) ∈ B and (Φ Φ ) l Φ ( x ) ∈ B . Therefore, G ( x, y ) is harmonic in B . In the same way,we can show that G ( x, y ) is harmonic in B as well. When x ∈ B , each term inthe expression of G ( x, y ), with the exception of log | x − y | , is harmonic in B by thesame argument. Hence, when x ∈ B , ∆ x G ( x, y ) = δ ( x − y ). Recall that, as wemention at the beginning of this section, we write ∆ x log | x − y | = δ ( x − y ).It remains to verify the continuity of G ( x, y ) and a ( x ) D ν G ( x, y ) across the twocircles ∂ B and ∂ B . We only present the calculations corresponding to the case ∂ B because the other case is similar. Using the simple fact thatΦ ( x ) = x on ∂ B , (2.2) H. DONG AND H.G. LI we have, for any y ∈ B , G ( x, y ) (cid:12)(cid:12)(cid:12) − ∂ B = log | x − y | + ∞ X l =1 h ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | + log | (Φ Φ ) l ( x ) − y | (cid:17) − ( αβ ) l − (cid:16) β log | (Φ Φ ) l − Φ ( x ) − y | + α log | (Φ Φ ) l − Φ ( x ) − y | (cid:17)i = log | x − y | + ∞ X l =1 h ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | + log | (Φ Φ ) l − Φ ( x ) − y | (cid:17) − ( αβ ) l − (cid:16) β log | (Φ Φ ) l − Φ ( x ) − y | + α log | (Φ Φ ) l − ( x ) − y | (cid:17)i =(1 − α ) ∞ X l =0 ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | − β log | (Φ Φ ) l Φ ( x ) − y | (cid:17) = G ( x, y ) (cid:12)(cid:12)(cid:12) + ∂ B , where we used 1 − α = 2 / ( k + 1) in the last equality.Next, we check the continuity of a ( x ) D ν G ( x, y ) on ∂ B . Notice that for anydifferentiable function f , ∂ ν (cid:0) f (Φ ( x )) (cid:1) = − ∂ ν f ( x ) on ∂ B . (2.3)Therefore, for any y ∈ B , a ( x ) D ν G ( x, y ) (cid:12)(cid:12)(cid:12) − ∂ B = D ν G ( x, y ) (cid:12)(cid:12)(cid:12) − ∂ B = D ν log | x − y | + ∞ X l =1 h ( αβ ) l (cid:16) D ν log | (Φ Φ ) l ( x ) − y | + D ν log | (Φ Φ ) l ( x ) − y | (cid:17) − ( αβ ) l − (cid:16) β D ν log | (Φ Φ ) l − Φ ( x ) − y | + α D ν log | (Φ Φ ) l − Φ ( x ) − y | (cid:17)i = D ν log | x − y | + ∞ X l =1 h ( αβ ) l (cid:16) D ν log | (Φ Φ ) l ( x ) − y |− D ν log | (Φ Φ ) l − Φ ( x ) − y | (cid:17) − ( αβ ) l − (cid:16) β D ν log | (Φ Φ ) l − Φ ( x ) − y |− α D ν log | (Φ Φ ) l − ( x ) − y | (cid:17)i =(1 + α ) ∞ X l =0 ( αβ ) l (cid:16) D ν log | (Φ Φ ) l ( x ) − y | − βD ν log | (Φ Φ ) l Φ ( x ) − y | (cid:17) = k D ν G ( x, y ) (cid:12)(cid:12)(cid:12) + ∂ B = a ( x ) D ν G ( x, y ) (cid:12)(cid:12)(cid:12) + ∂ B , where we used 1 + α = 2 k / ( k + 1) in the last but one equality.For the case when y ∈ B , the singularity appears in B . If x ∈ B , then(Φ Φ ) l ( x ) ∈ B and (Φ Φ ) l Φ ( x ) ∈ B . Therefore, G ( x, y ) is harmonic in B . Inthe same way, G ( x, y ) is harmonic in B as well. When x ∈ B \ { (1 + ε/ , } , eachterm in the expression of G ( x, y ), with the exception of the term k log | x − y | , is STIMATES BY GREEN FUNCTION METHOD 9 harmonic in B \ { (1 + ε/ , } because Φ ( x ) / ∈ B and (Φ Φ ) l Φ ( x ) ∈ B . Hence,when x ∈ B \ { (1 + ε/ , } , k ∆ x G ( x, y ) = δ ( x − y ).Now we verify the continuity of G ( x, y ) and a ( x ) D ν G ( x, y ) across the circle ∂ B .The proof of the continuity across ∂ B is similar. For x ∈ ∂ B , using (2.2) again,we have, for any y ∈ B , G ( x, y ) (cid:12)(cid:12)(cid:12) − ∂ B = 2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) log | (Φ Φ ) l ( x ) − y | − β log | (Φ Φ ) l Φ ( x ) − y | (cid:17) = 2 k + 1 (log | x − y | − β log | Φ ( x ) − y | )+ 2 k + 1 ∞ X l =1 ( αβ ) l (cid:16) log | (Φ Φ ) l − Φ ( x ) − y | − β log | (Φ Φ ) l Φ ( x ) − y | (cid:17) = 1 + αk log | x − y | − β (1 − α )( k + 1) ∞ X l =0 ( αβ ) l log | (Φ Φ ) l Φ ( x ) − y | = 1 k (cid:0) log | x − y | + α log | Φ ( x ) − y | (cid:1) − β ( k + 1) ∞ X l =0 ( αβ ) l log | (Φ Φ ) l Φ ( x ) − y | = G ( x, y ) (cid:12)(cid:12)(cid:12) + ∂ B . Next, we check the continuity of a ( x ) D ν G ( x, y ) on ∂ B . In view of (2.3) again,for any y ∈ B , a ( x ) D ν G ( x, y ) (cid:12)(cid:12)(cid:12) − ∂ B = D ν G ( x, y ) (cid:12)(cid:12)(cid:12) − ∂ B = 2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) D ν log | (Φ Φ ) l ( x ) − y | − β D ν log | (Φ Φ ) l Φ ( x ) − y | (cid:17) = 2 k + 1 (cid:16) D ν log | x − y | − β D ν log | Φ ( x ) − y | (cid:17) − k + 1 ∞ X l =1 ( αβ ) l (cid:16) D ν log | (Φ Φ ) l − Φ ( x ) − y | + β D ν log | (Φ Φ ) l Φ ( x ) − y | (cid:17) =(1 − α ) D ν log | x − y | − k + 1 β D ν log | Φ ( x ) − y |− k + 1 ∞ X l =1 ( αβ ) l (cid:16) D ν log | (Φ Φ ) l − Φ ( x ) − y | + β D ν log | (Φ Φ ) l Φ ( x ) − y | (cid:17) = (cid:0) D ν log | x − y | + α D ν log | Φ ( x ) − y | (cid:1) − β k ( k + 1) ∞ X l =0 ( αβ ) l D ν log | (Φ Φ ) l Φ ( x ) − y | = k D ν G ( x, y ) (cid:12)(cid:12)(cid:12) + ∂ B = a ( x ) D ν G ( x, y ) (cid:12)(cid:12)(cid:12) + ∂ B , where we used 1 + α = 2 k / ( k + 1) in the last but one equality.The case when y ∈ B is similar, and thus omitted. The proof is finished. (cid:3) With Lemma 2.2, we are ready to construct a Green’s function G ( x, y ) of theoperator L . Define G ( x, y ) = G ( x, y ) for y ∈ B , G ( x, y ) + α − α G ( x, (1 + ε/ , y ∈ B , G ( x, y ) + β − β G ( x, ( − − ε/ , y ∈ B . When defining G , (1 + ε/ ,
0) and ( − − ε/ ,
0) are removed, since Φ and Φ havesingularity at these points. Nevertheless, in the following proposition we prove thatfor fixed y , G ( x, y ) is well-defined in R \ { y } . In particular,lim x → (1+ ε/ , G ( x, y ) (cid:0) or lim x → ( − − ε/ , G ( x, y ) (cid:1) exists when y = (1 + ε/ ,
0) (or y = ( − − ε/ , Proposition 2.3.
The function G ( x, y ) defined above is a Green’s function of L .Proof. From Lemma 2.2, G ( x, y ) satisfies the compatibility conditions. Thus bylinearity, G ( x, y ) satisfies the compatibility conditions as well. It remains to prove a ( x )∆ x G ( x, y ) = δ ( x − y ) for x / ∈ ∂ B ∪ ∂ B . For y ∈ B , this is proved in Lemma 2.2. It remains to treat the case when y ∈ B ∪ B . We only consider the case when y ∈ B because the case when y ∈ B is similar. As we mentioned in Remark 2.1, Φ has a singularity at (1 + ε/ , , (2.1), we have, for fixed y = (1 + ε/ , x is near(1 + ε/ , | Φ ( x ) − y | ∼ − log | x − (1 + ε/ , | , which implies that k G ( x, y ) ∼ − α log | x − (1 + ε/ , | . Moreover, the singular part in k G ( x, (1 + ε/ , − α ) log | x − (1 + ε/ , | . Therefore, G ( x, y ) is bounded around (1 + ε/ , ε/ ,
0) is aremovable singularity of G ( · , y ) for y = (1 + ε/ , k ∆ x G ( x, y ) = δ ( x − y ) . When y = (1 + ε/ , x ∈ B . Note thatlog | Φ ( x ) − y | = − log | x − y | and G ( x, y ) = 11 − α · G ( x, y ) . Thus, by the definition of G ( x, y ), k G ( x, y ) − log | x − y | is harmonic in B . Therefore, G ( x, y ) is well defined and k ∆ x G ( x, y ) = δ ( x − y ) . The proof is finished. (cid:3)
STIMATES BY GREEN FUNCTION METHOD 11 Derivative estimates
In order to establish the derivative estimates for solutions of (1.1), we first derivederivative estimates of the maps Φ Φ , Φ Φ , and their compositions. Lemma 3.1.
For any integers l ≥ and m ≥ , | D m (Φ Φ ) l ( x ) | ≤ Cl m − (1 + 2 √ ε ) l in B , (3.1) and | D m (Φ Φ ) l ( x ) | ≤ Cl m − (1 + 2 √ ε ) l in B , where C depends only on m .Proof. We identify a point x = ( x , x ) ∈ R with a complex number z = x + ix ∈ C . For convenience, here we take a = 1+ ε/
2. With respect to the complex variable,Φ ( z ) = a ¯ z − ( a − z − a and Φ ( z ) = − a ¯ z − ( a − z + a . (3.2)Thus, (Φ Φ )( z ) := Φ ◦ Φ ( z ) = − (2 a − z + 2 a ( a − az − (2 a − . Using a translation and dilation of coordinates2 az − (2 a − → z, and 2 a (Φ Φ )( z ) − (2 a − → (Φ Φ )( z ) , we obtain (Φ Φ )( z ) := − /z − a − . To find an expression of (Φ Φ ) l , we consider the fixed points of Φ Φ . Notice thatΦ Φ has two fixed points, the one in B given by λ := − (2 a −
1) + 2 a p a − ∼ − √ ε, and the one in B given by λ := − (2 a − − a p a − ∼ − − √ ε. Clearly, for z ∈ B ,(Φ Φ )( z ) − λ = − /z − a − − λ = − /z + 1 /λ = z − λ zλ , which implies that for any z = λ ,1(Φ Φ )( z ) − λ = λ z − λ + λ , and thus 1(Φ Φ )( z ) − λ − λ − λ = λ (cid:16) z − λ − λ − λ (cid:17) . By iteration, we have for any z = λ and l ≥ Φ ) l ( z ) − λ = λ l (cid:16) z − λ − λ − λ (cid:17) + λ − λ . Hence(Φ Φ ) l ( z ) = λ + (cid:16) λ l (cid:0) z − λ − λ − λ (cid:1) + λ − λ (cid:17) − = λ + λ − λ l +12 · z − λ z (1 − λ − l ) − ( λ − − λ − l +12 )= λ + λ − λ l +12 · − λ − l (cid:16) λ − − λ z (1 − λ − l ) − ( λ − − λ − l +12 ) (cid:17) . (3.3)Note that the identity above also holds when z = λ .Next, we differentiate (3.3) with respect to z . For m = 1, D (Φ Φ ) l ( z ) = ( λ − /λ ) λ l · (cid:16) z (1 − λ − l ) − ( λ − − λ − l +12 ) (cid:17) − = ( λ − /λ ) λ l · (cid:16) ( z − λ − )(1 − λ − l ) + ( λ − λ − ) λ − l (cid:17) − . (3.4)Since λ − λ − ∼ −√ ε and, for z ∈ B (in the new coordinates),Re ( z − λ − ) ≤ − a ( a − − (2 a − − λ − ≤ − − λ − . −√ ε, it follows that (cid:12)(cid:12)(cid:12) ( z − λ − )(1 − λ − l ) + ( λ − λ − ) λ − l (cid:12)(cid:12)(cid:12) & √ ε. (3.5)Thus, we obtain for l ≥ | D (Φ Φ ) l ( z ) | ≤ Cλ l ≤ C (1 + 2 √ ε ) l . For higher-order derivatives, from (3.4), we have for m ≥ D m (Φ Φ ) l ( z )= ( λ − /λ ) λ l ( − m − m !(1 − λ − l ) m − (cid:16) z (1 − λ − l ) − ( λ − − λ − l +12 ) (cid:17) − ( m +1) . For z ∈ B , using (3.5) and the simple inequality0 ≤ − λ − l ≤ Cl √ ε, we get D m (Φ Φ ) l ( z ) ≤ Cl m − λ l ≤ Cl m − (1 + 2 √ ε ) l . (3.6)Thus, (3.1) is proved.Similarly, write(Φ Φ )( z ) := Φ ◦ Φ ( z ) = (2 a − z + 2 a ( a − az + 2 a − . By a translation and dilation of coordinates2 az + 2 a − → z and 2 a (Φ Φ )( z ) + 2 a − → (Φ Φ )( z ) , we obtain (Φ Φ )( z ) := − /z + 2(2 a − . The map Φ Φ also has two fixed points, the one in B given by˜ λ := 2 a − a p a − ∼ √ ε, STIMATES BY GREEN FUNCTION METHOD 13 and the one in B given by˜ λ := 2 a − − a p a − ∼ − √ ε. For z ∈ B ,(Φ Φ )( z ) − ˜ λ = − /z + 2(2 a − − ˜ λ = − /z + 1 / ˜ λ = z − ˜ λ z ˜ λ . In the same way, D (Φ Φ ) l ( z ) = (˜ λ − / ˜ λ ) ˜ λ l · (cid:16) z (1 − ˜ λ − l ) − (˜ λ − − ˜ λ − l +11 ) (cid:17) − = (˜ λ − / ˜ λ ) ˜ λ l · (cid:16) ( z − ˜ λ − )(1 − ˜ λ − l ) + (˜ λ − ˜ λ − )˜ λ − l (cid:17) − . (3.7)Since ˜ λ − ˜ λ − ∼ √ ε and for z ∈ B (in the new coordinates),Re ( z − ˜ λ − ) ≥ a ( a −
1) + 2 a − − ˜ λ − ≥ − ˜ λ − & √ ε, we have (cid:12)(cid:12)(cid:12) ( z − ˜ λ − )(1 − ˜ λ − l ) + (˜ λ − ˜ λ − )˜ λ − l (cid:12)(cid:12)(cid:12) & √ ε. Thus, for l ≥ | D (Φ Φ ) l ( z ) | ≤ C ˜ λ l ≤ C (1 + 2 √ ε ) l . For higher-order derivatives, using0 ≤ − ˜ λ − l ≤ Cl √ ε, we obtain D m (Φ Φ ) l ( z ) ≤ Cl m − ˜ λ l ≤ Cl m − (1 + 2 √ ε ) l . This completes the proof of the lemma. (cid:3)
In the next lemma, we show a Schauder estimate for the elliptic equation (1.1)with piecewise constant coefficient in the upper and lower half balls. We allow thecoefficient to be partially degenerate and we give an explicit dependence of theconstant in the estimate with respect to the coefficient.
Lemma 3.2.
Let m ≥ , γ ∈ (0 , , and k ∈ (0 , ∞ ) be constants. Suppose that u ∈ W ( B ) is a weak solution to D i ( a ( x ) D i u ) = D i f i , where a ( x ) = kχ B +1 + χ B − and f i is C m − ,γ in B +1 and B − . Then u is C m,γ in B +1 and B − , and k u k C m,γ ( B +1 / ) + k u k C m,γ ( B − / ) ≤ C k u k L ( B ) + Ck − k f k C m − ,γ ( B +1 ) + C k f k C m − ,γ ( B − ) , where C is a constant depending only on d and m .Proof. We define v ( x , x ) = u ( − x , x ) on B +1 . Then when k ≥ u and v satisfythe elliptic system ∆ u = f /k, ∆ v = f ( − x , x ) on B +1 with the boundary conditions u − v = 0 and D u + k − D v = 0 on B ∩ { x = 0 } . When k ∈ (0 , u − v = 0 and kD u + D v = 0 on B ∩ { x = 0 } . In both cases, the boundary conditions are complementing boundary conditionsintroduced in [1] with uniformly bounded (constant) coefficients. Therefore, thedesired estimate follows immediately from the classical regularity theory for ellipticsystems. See, for instance, [1]. (cid:3)
Proof of Theorem 1.1.
With the help of Green’s function constructed inSection 2, and Lemmas 3.1 and 3.2, we are in the position to consider the non-homogeneous equation (1.1) in B . Proof of Theorem 1.1.
By dividing u and f by C , without loss of generality, wemay assume that C = 1. We take a cutoff function η ∈ C ∞ ( B ) such that η = 1in B / . Let v = uη , which satisfies D i ( aD i v ( x )) = D i ˜ f i + ˜ f in R , (3.8)where ˜ f i = f i η + auD i η, ˜ f = − f i D i η + aD i uD i η. Since supp( D i η ) ⊂ B \ B / , by (1.4) and Lemma 3.2 with a conformal map, weinfer that ˜ f , ˜ f , and ˜ f are compactly supported in B , piecewise C γ , and k ˜ f i k C γ ( B j ) ≤ Ck j , j = 0 , , , i = 1 , , . (3.9)Now define˜ u ( x ) = − Z B D y i G ( x, y ) ˜ f i ( y ) dy − Z B D y i G ( x, y ) ˜ f i ( y ) dy − Z B D y i G ( x, y ) ˜ f i ( y ) dy + Z B G ( x, y ) ˜ f ( y ) dy := − w ( x ) − w ( x ) − w ( x ) + w ( x ) , (3.10)which is a solution to (3.8) in R . Claim:
We have v = ˜ u + C for some constant C .Assuming for the moment that the claim above is proved, it suffices for us toestimate ˜ u in B / . Gradient estimates.
To estimate ˜ u , we define for j = 0 , , h j ( x ) = Z B j D y i log | x − y | ˜ f i ( y ) dy (3.11)and g j ( x ) = Z B j log | x − y | ˜ f ( y ) dy. (3.12)Since log | x − y | is the fundamental solution of the Laplace equation in R , h j and g j satisfy ∆ h j = − D i ( ˜ f i χ B j ) and ∆ g j = ˜ f χ B j in R . Because ˜ f i is piecewise C γ and the interfaces ∂ B and ∂ B are smooth, using Lemma3.2 we see that for j = 1 , h j and g j are piecewise C ,γ . Moreover, due to (3.9), k h j k C ,γ ( B ) + k g j k C ,γ ( B ) ≤ Ck j . (3.13) STIMATES BY GREEN FUNCTION METHOD 15
Using Lemma 2.1 in [16] and the same argument for g as in the proof of Theorem4.12 there, we see that h and g are also piecewise C ,γ , and k h k C ,γ ( B ) + k g k C ,γ ( B ) ≤ C. (3.14)Estimates in B ∩ B / : First we consider the narrow region between B and B .For x ∈ B ∩ B / , by the definition of G ( x, y ), we have w ( x ) = 2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) Z B D y i log | (Φ Φ ) l ( x ) − y | ˜ f i ( y ) dy − β Z B D y i log | (Φ Φ ) l Φ ( x ) − y | ˜ f i ( y ) dy (cid:17) = 2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) h ((Φ Φ ) l ( x )) − β h ((Φ Φ ) l Φ ( x )) (cid:17) . (3.15)By the definition (3.2), k D (Φ Φ ) k L ∞ ( B ) and k D Φ k L ∞ ( B ) are bounded by 1.For x ∈ B ∩ B / , Φ Φ ( x ) and Φ ( x ) are both in B . Using the chain rule and(3.15), we have | Dw ( x ) | = 2 k + 1 ∞ X l =0 | αβ | l (cid:12)(cid:12)(cid:12)(cid:16) Dh ((Φ Φ ) l ( x )) − β Dh ((Φ Φ ) l Φ ( x )) (cid:17)(cid:12)(cid:12)(cid:12) ≤ k + 1 (cid:16) k Dh k L ∞ ( B ) + | β |k Dh k L ∞ ( B ) k D Φ k L ∞ ( B ) (cid:17) + 2 k + 1 ∞ X l =1 | αβ | l (cid:16) k Dh k L ∞ ( B ) k D (Φ Φ ) l − k L ∞ ( B ) k D (Φ Φ ) k L ∞ ( B ) + | β |k Dh k L ∞ ( B ) k D (Φ Φ ) l k L ∞ ( B ) k D Φ k L ∞ ( B ) (cid:17) , which, thanks to Lemma 3.1 and (3.13), is bounded by ≤ Ck k + 1 (cid:16) ∞ X l =0 | αβ | l k D (Φ Φ ) l k L ∞ ( B ) (cid:17) ≤ Ck k + 1 | αβ | − | αβ | (1+2 √ ε ) ≤ Ck k + 1 · − (1 − √ ε ) | αβ | . Similarly, by the definition of G ( x, y ), w ( x ) = (1 + β ) ∞ X l =0 ( αβ ) l (cid:16) Z B D y i log | (Φ Φ ) l ( x ) − y | ˜ f i ( y ) dy − α Z B D y i log | (Φ Φ ) l Φ ( x ) − y | ˜ f i ( y ) dy (cid:17) , = 2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) h ((Φ Φ ) l ( x )) − α h ((Φ Φ ) l Φ ( x )) (cid:17) . Since for x ∈ B ∩ B / , (Φ Φ )( x ) and Φ ( x ), l ≥ B , using Lemma3.1 and (3.13), | Dw ( x ) | = 2 k + 1 ∞ X l =0 | αβ | l (cid:12)(cid:12)(cid:12)(cid:16) Dh ((Φ Φ ) l ( x )) − α Dh ((Φ Φ ) l Φ ( x )) (cid:17)(cid:12)(cid:12)(cid:12) ≤ Ck k + 1 (cid:16) ∞ X l =0 | αβ | l k D (Φ Φ ) l k L ∞ ( B ) (cid:17) ≤ Ck k + 1 · − (1 − √ ε ) | αβ | . Because w ( x )= Z B D y i log | x − y | ˜ f i ( y ) dy + ∞ X l =1 h ( αβ ) l (cid:16) Z B D y i log | (Φ Φ ) l ( x ) − y | ˜ f i ( y ) dy + Z B D y i log | (Φ Φ ) l ( x ) − y | ˜ f i ( y ) dy (cid:17) − ( αβ ) l − (cid:16) β Z B D y i log | (Φ Φ ) l − Φ ( x ) − y | ˜ f i ( y ) dy + α Z B D y i log | (Φ Φ ) l − Φ ( x ) − y | ˜ f i ( y ) dy (cid:17)i = h ( x ) + ∞ X l =1 h ( αβ ) l (cid:16) h ((Φ Φ ) l ( x )) + h ((Φ Φ ) l ( x )) (cid:17) − ( αβ ) l − (cid:16) β h ((Φ Φ ) l − Φ ( x )) + α h ((Φ Φ ) l − Φ ( x )) (cid:17)i , using the same argument as above, from Lemma 3.1 and (3.14), we have | Dw ( x ) | ≤ C + C ∞ X l =0 | αβ | l (cid:16) k D (Φ Φ ) l k L ∞ ( B ) + k D (Φ Φ ) l k L ∞ ( B ) (cid:17) ≤ C − (1 − √ ε ) | αβ | . The estimate of w is also similar by using (3.12), (3.13), and (3.14). Therefore,recalling ˜ u = − w − w − w + w , we have, for x ∈ B ∩ B / , | D ˜ u | ≤ C − (1 − √ ε ) | αβ | . (3.16)Estimates in B ∩ B / : In this case, we have (Φ Φ )( x ) ∈ B and Φ ( x ) ∈ B .By the definition (3.10) and the same argument as in the case x ∈ B ∩ B / , we STIMATES BY GREEN FUNCTION METHOD 17 have w ( x ) = Z B D y i G ( x, y ) ˜ f i ( y ) dy = 1 k Z B D y i log | x − y | ˜ f i ( y ) dy + αk Z B D y i log | Φ ( x ) − y | ˜ f i ( y ) dy − β ( k + 1) ∞ X l =0 ( αβ ) l Z B D y i log | (Φ Φ ) l Φ ( x ) − y | ˜ f i ( y ) dy = 1 k h ( x ) + αk h (Φ ( x )) − β ( k + 1) ∞ X l =0 ( αβ ) l h ((Φ Φ ) l Φ ( x )) . Thus, using Lemma 3.1 and (3.13), | Dw ( x ) | ≤ C + C ( k + 1) · − (1 − √ ε ) | αβ | in B ∩ B / . Since w ( x ) = Z B D y i G ( x, y ) ˜ f i ( y ) dy = 2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) Z B D y i log | (Φ Φ ) l ( x ) − y | ˜ f i ( y ) dy − β Z B D y i log | (Φ Φ ) l Φ ( x ) − y | ˜ f i ( y ) dy (cid:17) = 2 k + 1 ∞ X l =0 ( αβ ) l (cid:16) h ((Φ Φ ) l ( x )) − β h ((Φ Φ ) l Φ ( x )) (cid:17) , we have | Dw ( x ) | ≤ Ck + 1 · − (1 − √ ε ) | αβ | in B ∩ B / . Similarly, in B ∩ B / , | Dw ( x ) | ≤ Ck ( k + 1)( k + 1) · − (1 − √ ε ) | αβ | , | Dw ( x ) | ≤ C + Ck + 1 · − (1 − √ ε ) | αβ | . Therefore, we have (3.16) in B ∩ B / .Estimates in B ∩ B / : In this case, we have (Φ Φ )( x ) ∈ B and Φ ( x ) ∈ B .By exactly the same argument as in the case x ∈ B ∩ B / , we get (3.16). Hence,estimate (1.5) is proved. Higher derivative estimates.
As before, we may assume that C m = 1. ByLemma 3.2 and (1.6), ˜ f i are piecewise C m − ,γ , and k ˜ f i k C m − ,γ ( B ∩B j ) ≤ Ck j , j = 0 , , , i = 1 , , , which yields k h j k C m,γ ( B ) + k g j k C m,γ ( B ) ≤ Ck j . (3.17)Recall Fa`a di Bruno’s formula for composition d m dt m f ( g ( t )) = X m ! s ! s ! · · · s n ! f ( s ) ( g ( t )) (cid:16) g ′ ( t )1! (cid:17) s (cid:16) g ′′ ( t )2! (cid:17) s · · · (cid:16) g ( n ) ( t ) n ! (cid:17) s n , where the summation is over all positive integer solutions of the Diophantine equa-tion s + 2 s + · · · + ns n = m and s := s + s + · · · + s n . Using (3.6), we obtain | D mx h ((Φ Φ ) l ( x )) |≤ X C | D x (Φ Φ ) l ( x ) | s · | D x (Φ Φ ) l ( x ) | s · | D nx (Φ Φ ) l ( x ) | s n ≤ Cl m − λ l , where C > m . Therefore, for instance in B ∩ B / ,using (3.15) and (3.17), | D m w ( x ) | = 2 k + 1 ∞ X l =0 | αβ | l (cid:12)(cid:12)(cid:12)(cid:16) D mx h ((Φ Φ ) l ( x )) − β D mx h ((Φ Φ ) l Φ ( x )) (cid:17)(cid:12)(cid:12)(cid:12) ≤ Ck k + 1 (cid:16) ∞ X l =1 | αβ | l Cl m − (1 + 2 √ ε ) l (cid:17) ≤ Ck k + 1 · (cid:0) − (1 − √ ε ) | αβ | (cid:1) m . In the same way, we bound | D m w | , | D m w | , and | D m w | in all the three re-gions, then for | D m ˜ u | . Therefore, we obtain the upper bound (1.7) for higher-orderderivatives | D m ˜ u | . Proof of the claim.
We consider the growth property of ˜ u as x → ∞ . Firstwe estimate w . From the definition of h in (3.11) and the fact that ˜ f i ∈ C α ( B ),we have | h ( x ) | ≤ C/ (1 + | x | ). It then follows from (3.15) that k w k L ∞ ( R ) ≤ C forsome constant C depending only on k , k , and ε . Similarly, we have k w k L ∞ ( R ) + k w k L ∞ ( R ) ≤ C. On the other hand, from (3.12), we have | g j ( x ) | ≤ C log( | x | + e ), which gives that | w ( x ) | ≤ C log( | x | + e ). Therefore, | ˜ u ( x ) | ≤ C log( | x | + e ) , and because of the boundedness of v , | ˜ u ( x ) − v ( x ) | ≤ C log( | x | + e ) . (3.18)Since ˜ u − v satisfies the homogeneous equation D i ( aD i (˜ u − v )) = 0, by the DeGiorgi-Nash-Moser estimate, for any R >
0, we have for some α ∈ (0 , u − v ] C α ( B R/ ) ≤ CR − α k ˜ u − v k L ∞ ( B R ) , which goes to zero as R → ∞ thanks to (3.18). The claim is proved.The proof of Theorem 1.1 is completed. (cid:3) A lower bound for the gradient.
Assume that k , k ≥
1. Choose f to bea nonnegative function supported in B / ( −
3) even in x with unit integral, and f ≡
0. Define u ( x ) := − Z B / ( − D y G ( x, y ) f ( y ) dy and h ( x ) := − Z B / ( − D y log | x − y | f ( y ) dy. STIMATES BY GREEN FUNCTION METHOD 19
Then u ( x ) = h ( x ) + ∞ X l =1 h ( αβ ) l (cid:16) h ((Φ Φ ) l ( x )) + h ((Φ Φ ) l ( x )) (cid:17) − ( αβ ) l − (cid:16) βh ((Φ Φ ) l − Φ ( x )) + αh ((Φ Φ ) l − Φ ( x )) (cid:17)i . Denote S := { ( x ,
0) : x ∈ [ − ( ε + ε / , ( ε + ε / } . Note that h is also even with respect to x . By a simple calculation, we have D h ( x ) ≤ − c < D h ( x ) = 0 on S (3.19)for some constant c >
0. It is easily seen that for any l ≥ Φ ) l (0) , (Φ Φ ) l (0) , (Φ Φ ) l − Φ (0) , (Φ Φ ) l − Φ (0) ∈ S and from the proof of Lemma 3.1 (cf. (3.4) and (3.7)), D (Φ Φ ) l (0) ≥ C/λ l , D (Φ Φ ) l (0) ≥ C/λ l ,D (cid:0) (Φ Φ ) l − Φ (cid:1) (0) ≤ − C/λ l , D (cid:0) (Φ Φ ) l − Φ (cid:1) (0) ≤ − C/λ l for some C > l ( ≥ D u (0) = D h (0) + ∞ X l =1 h ( αβ ) l (cid:16) ( D h )((Φ Φ ) l (0)) D (Φ Φ ) l (0)+ ( D h )((Φ Φ ) l (0)) D (Φ Φ ) l (0) (cid:17) − ( αβ ) l − (cid:16) β ( D h )((Φ Φ ) l − Φ (0)) D (cid:0) (Φ Φ ) l − Φ (cid:1) (0)+ αD h ((Φ Φ ) l − Φ (0)) D (cid:0) (Φ Φ ) l − Φ (cid:1) (0) (cid:17)i . which, in view of (3.19), is less than − c multiplied by1 + C ∞ X l =1 h ( αβ ) l λ − l + ( αβ ) l − (cid:16) βλ − l + αλ − l (cid:17)i ≥ C − (1 − √ ε ) αβ . Therefore, (1.12) is proved.3.3.
Proof of Theorem 1.2.
The proof is similar to that of Theorem 1.1 withsome modifications, which we shall point out. We assume that C = 1. Take acutoff function η ∈ C ∞ ( D ) such that η = 1 on D . Then v := uη satisfies (3.8),where ˜ f i = f i η + uD i η, ˜ f = − f i D i η + D i uD i η. By the interior estimates for elliptic equations with constant coefficients, instead of(3.9) we have k ˜ f i k C γ ( B j ) ≤ C min { k j , } , j = 0 , , , i = 1 , , . (3.20)Now we define ˜ u , w i , i = 0 , . . . , h j and g j in (3.11) and(3.12). Using (3.20), it holds that k h j k C ,γ ( B ) + k g j k C ,γ ( B ) ≤ C min { k j , } . (3.21) As in the proof of Theorem 1.1, in B ∩ B / , by using (3.21) we have | Dw ( x ) | ≤ C min { k , } k + 1 · − (1 − √ ε ) | αβ | , | Dw ( x ) | ≤ C min { k , } k + 1 · − (1 − √ ε ) | αβ | , | Dw ( x ) | + | Dw ( x ) | ≤ − (1 − √ ε ) | αβ | , which yield (3.16). While in B ∩ B / , we have | Dw ( x ) | ≤ C min { k , } k + C ( k + 1) · − (1 − √ ε ) | αβ | , | Dw ( x ) | ≤ C min { k , } ( k + 1)( k + 1) · − (1 − √ ε ) | αβ | , | Dw ( x ) | + | Dw ( x ) | ≤ Ck + 1 · − (1 − √ ε ) | αβ | , which yield | D ˜ u ( x ) | ≤ Ck + 1 · − (1 − √ ε ) | αβ | . Similarly, in B ∩ B / , | D ˜ u ( x ) | ≤ Ck + 1 · − (1 − √ ε ) | αβ | . The theorem is proved. 4.
Proof of Theorem 1.5
For general r and r , the inversion maps with respect to ∂B and ∂B in thecomplex variable are given byΦ ( z ) := r ¯ z − ( r + ε/
2) + r + ε/ , Φ ( z ) := r ¯ z + ( r + ε/ − ( r + ε/ . Then (Φ Φ )( z ) := Φ ◦ Φ ( z )= r ( z − ( r + ε/ r + r + ε ) z − ( r + ε/ r + r + ε ) + r − ( r + ε/ . Using a translation and dilation of coordinates( r + r + ε ) z − ( r + ε/ r + r + ε ) + r → z and ( r + r + ε )(Φ Φ )( z ) − ( r + ε/ r + r + ε ) + r → (Φ Φ )( z ) , we obtain (Φ Φ )( z ) := − r r z − (cid:16) r r + 2( r + r ) ε + ε (cid:17) . STIMATES BY GREEN FUNCTION METHOD 21
The two fixed points of the map (Φ Φ ) are given by λ = − ( r r + ( r + r ) ε + ε / r r r (cid:16) r + r ) ε + ε (cid:17) + (cid:16) ( r + r ) ε + ε / (cid:17) ∼ r r (cid:16) − p /r + 1 /r ) ε (cid:17) in B and λ ∼ r r (cid:16) − − p /r + 1 /r ) ε (cid:17) in B . Since for x ∈ B ∩ B / , (Φ Φ ) l ( x ) and (Φ Φ ) l − Φ ( x ), l ≥
1, are all in B ,we here choose λ . By a similar calculation as before,1(Φ Φ )( z ) − λ − λ r r − λ = λ r r (cid:16) z − λ − λ r r − λ (cid:17) . By iteration, we have for any z = λ and k ≥ Φ ) l ( z ) = λ + (cid:0) λ − r r /λ (cid:1)(cid:0) r r /λ (cid:1) l · − (cid:0) r r /λ (cid:1) l · (cid:16) r r λ − − λ z (cid:16) − (cid:0) r r /λ (cid:1) l (cid:17) − (cid:16) r r λ − − λ (cid:0) r r /λ (cid:1) l (cid:17) (cid:17) . Therefore, D (Φ Φ ) l ( z ) = ( λ − r r /λ ) (cid:0) r r /λ (cid:1) l · (cid:16)(cid:0) z − r r λ − (cid:1)(cid:16) − (cid:0) r r /λ (cid:1) l (cid:17) + ( λ − r r λ − ) (cid:0) r r /λ (cid:1) l (cid:17) − . Since λ − r r λ − ∼ − r r p /r + 1 /r ) ε and for z ∈ B (in the new coordinates),Re ( z − r r λ − ) ≤ − ( r + r + ε ) ε/ − ( r + ε/ r + r + ε ) + r − r r λ − = r r (cid:16)
11 + p /r + 1 /r ) ε − (cid:17) + O ( ε ) . − r r p (1 /r + 1 /r ) ε, we have (cid:12)(cid:12)(cid:12)(cid:0) z − r r λ − (cid:1)(cid:16) − (cid:0) r r /λ (cid:1) l (cid:17) + ( λ − r r λ − ) (cid:0) r r /λ (cid:1) l (cid:12)(cid:12)(cid:12) & r r p (1 /r + 1 /r ) ε. Thus, for l ≥ | D (Φ Φ ) l ( z ) | ≤ C (cid:0) p /r + 1 /r ) ε (cid:1) l . Similarly, we can bound D (Φ Φ ) as well as the higher derivatives of Φ Φ andΦ Φ . Thus we obtain a generalization of Lemma 3.1, which shows the dependenceof r and r . Lemma 4.1.
For any integers l ≥ and m ≥ , | D m (Φ Φ ) l ( z ) | ≤ Cl m − (cid:0) p /r + 1 /r ) ε (cid:1) l in B , and | D m (Φ Φ ) l ( z ) | ≤ Cl m − (cid:0) p /r + 1 /r ) ε (cid:1) l in B , where C depends only on m . Proposition 2.3 still holds with obvious modifications. Using Lemma 4.1 insteadof Lemma 3.1, by the same procedure as in the proof of Theorem 1.2, Theorem 1.5is proved.
Funding:
Hongjie Dong was partially supported by the NSF under agreementDMS-1056737. Haigang Li was partially supported by NSFC (11571042) (11371060),Fok Ying Tung Education Foundation (151003), and the Fundamental ResearchFunds for the Central Universities.
Conflict of Interest:
The authors declare that they have no conflict of interest.
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Division of Applied Mathematics, Brown University, 182 George Street,Providence, RI 02912, USA. Tel: 1-(401)8637297
E-mail address : Hongjie [email protected] (H.G. Li)
School of Mathematical Sciences, Beijing Normal University, Laboratoryof Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China.
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