POINTWISE ESTIMATE FOR ELLIPTIC EQUATIONS IN PERIODIC PERFORATED DOMAINS

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Introduction. Pointwise estimate for the solutions of elliptic equations in perforated domains with periodic microstructure is presented. This problem arises from contaminant transport in the subsurface, heat transfer in two-phase media, the stress in composite materials, and so on (see [3,6,15,16]). Let Ω ⊂ R n (n = 2 or 3) be a bounded smooth simply-connected domain with boundary ∂Ω; Y ≡ (0, 1) n consists of a smooth sub-domain Y m completely surrounded by another connected sub-domain Y f (≡ Y \ Y m ); ∈ (0, 1); Ω( ) ≡ {x ∈ Ω|dist(x, ∂Ω) > }; Ω m ≡ {x|x ∈ (Y m + j) ⊂ Ω( ) for some j ∈ Z n } with boundary ∂Ω m ; Ω f ≡ Ω \ Ω m is a connected region. The problem that we consider is In (1), K (x) = K( x ) is a symmetric positive definite matrix; T (x) = T( x ) and P (x) = P( x ) are vector functions; E (x) = E( x ) is a scalar non-negative function; K, T, P, E are smooth periodic functions in R n with period Y ; Q , F are given functions; n is a unit normal vector on ∂Ω m . Problem (1) consists of an elliptic equation with oscillatory periodic coefficients. One example shows that even the given functions Q , F of the elliptic equation are bounded uniformly in , the C 1,α norm and the W 2,p norm of the elliptic solution of (1) may not be bounded uniformly in (see Remark 1). It is also known that, to obtain an accurate approximation of the solution of (1) by classical numerical methods (see [7,10]), the computational mesh size in each space direction should be less than [14]. So a direct numerical simulation of the solution of (1) requires large computational memory and computing time when is small.
If T , P are small and Q , F are bounded, (1) is uniquely solvable in H 1 and where c is a constant independent of [13]. In addition that Q is bounded in H 1 (Ω f ), by tracing the proof of Theorem 2.7 [3], there exists a subsequence of {U , Q , F } (same notation for subsequence) satisfying In (3),Ẍ Ω f denotes the characteristic function on Ω f ;K is a symmetric positive definite matrix depending on K and Y f ;T 1 andP are constant vectors;Ë 1,1 is a constant function; |Y f | is the volume of Y f ; U ∈ H 1 (Ω);Q,F ,Q * ,F * ∈ L 2 (Ω).Q depends on K,Q * ;F depends on K, P,Q * ,F * . Explicit form ofK,T 1 ,P ,Ë 1,1 ,Q,F can be found in (8) below. The function U in (3) satisfies −∇ · (K∇U ) + (P −T 1 )∇U +Ë 1,1 U = |Y f |(∇ ·Q +F ) in Ω, U = 0 on ∂Ω.
Similar results as (3)-(4) can also be found in [5,11,15,19]. So it seems that the solution of (4) is a good approximation of the solution of (1), especially when is small. Indeed in those works [4,14,18], the solution of (1) was regarded as the perturbation of the homogenized solution of (4). Furthermore, the homogenized solution of (4) was used to construct the solution of (1). It is also known that the solution of (1) converges to the solution of (4) with convergence rate in L 2 norm and with convergence rate √ in H 1 norm as closes to 0 (see [5,11,15,19] and references therein). In [8,17], higher order asymptotic expansion for the solutions of elliptic equations in perforated domains was given. Higher order convergence estimate for the solution of (1) for T = P = E = 0 case was also derived in Hilbert spaces (see [5,9,19]).
In this work, we show that although the C 1,α norm and the W 2,p norm of the elliptic solution of (1) may not be bounded uniformly in , the Hölder uniform bound in and the Lipschitz uniform bound in for the elliptic solution can be obtained. We also present pointwise convergence estimate for the solution of (1). Precisely, the L ∞ convergence estimate with convergence rate for the solution of (1) is derived. For problem (1) with periodic boundary conditions, Lipschitz convergence estimate with higher order convergence rate k for k ≥ 1 is also proved.
The rest of the work is organized as follows: Notation and main results are stated in section 2. In section 3, we derive a priori W 2,p estimates for the solutions of elliptic equations around the Neumann boundary. In section 4, a uniform Hölder estimate in for the elliptic solution of (1) is proved. In section 5, L ∞ convergence estimate for the solution of (1) is given. Uniform Lipschitz bound in and Lipschitz convergence estimate for the solution of (1) with periodic boundary conditions are obtained in section 6.
Proof of Theorem 2.3 is given in section 5. From Theorem 2.3, we note that if F is close toF * in Ω f , then the right hand side of (18) is small. In this case, the U of (15) converges uniformly to the U of (17) in Ω f .
Next we state Lipschitz uniform bound results. Consider the following problem: Find a scalar function U ∈ H 1 (D f ) satisfying on ∂D m , U satisfies periodic boundary conditions on ∂D.
where µ ≡ δ n+δ and c is a constant independent of . Consider the following problem: Find a scalar function See Lemma 2.1 for Π . Then we have (20) exists uniquely in H 1 (D f ) and satisfies where µ ≡ δ n+δ and c is a constant independent of . Proofs of Theorem 2.4 and Theorem 2.5 are in subsection 6.1. Now we state Lipschitz convergence estimates. We find a scalar function U ∈ H 1 (D f ) satisfying on ∂D m , U satisfies periodic boundary conditions on ∂D.
(24) (8), equation (24) is an elliptic equation with constant coefficients and its solution ϕ j exists uniquely in . See (7) and (9) where c is a constant independent of . See (23) for F .
Let us also consider the following problem: Find a scalar function on ∂D m , U satisfies periodic boundary conditions on ∂D, F (x)dx = 0, (26) is solvable uniquely. By Lemma 2.1 and tracing the argument of Theorem 2.7 [3], there is a subsequence of {U , F } (same notation for subsequence) satisfying where Letφ 0 = U (i.e., the solution of (28)) and letφ j for j ≥ 1 be the solution of Theorem 2.7. Besides A6-A7 and A10, if (26) and theV k in (30) satisfy where c is a constant independent of . See (28) forF .
Proofs of Theorem 2.6 and Theorem 2.7 are given in subsection 6.2.
where n = (n (1) , , then any solution of (33) satisfies Proof. By assumption, at each point x 0 ∈ ∂Y m there exist a neighborhood B 1/8 (x 0 ) of x 0 and a C 1,1 diffeomorphismL that straightens the boundary ∂Y m in B 1/8 (x 0 ).
Combining the above two results, we prove Lemma 4.3.
Then these functions satisfy (53) with ν = 1. (62) for k = 1 is deduced from Lemma 4.4. Suppose (62) holds for some k satisfying /θ k ≤˜ 0 , then we define Then these functions satisfy where n /θ k is a unit vector normal to ∂Ω m /θ k . By induction, we see Then we follow the argument of Lemma 4.2 and employ Lemma 4.4 with ν =θ k to obtain (62) with k + 1 in place of k.
Combining the above two results, we prove Lemma 4.6.
Proof. Denote by c a constant independent of and letθ,˘ 0 be those in Lemma 6.2. Let k ∈ N such that /θ k ≤˘ 0 < /θ k+1 . By Lemma 6.2,
By Theorem 2.4, where c is a constant independent of . Which implies Theorem 2.6.