of Green and Neumann Functions
For a family of second-order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann maps as well as near optimal convergence rates in for solutions with Dirichlet or Neumann boundary conditions.
The main purpose of this paper is to study the asymptotic behavior of the Green and Neumann functions for a family of elliptic operators with rapidly oscillating coefficients. More precisely, consider
(the summation convention is used throughout the paper). We will assume that with and is real and satisfies the ellipticity condition
where , and the periodicity condition
We will also impose the smoothness condition
Let and denote the Green and Neumann functions respectively, for in a bounded domain , with pole at . We are interested in the asymptotic behavior, as , of , , and as well as an . We shall use and to denote the Green and Neumann functions respectively, for the homogenized (effective) operator in .
Let with in the position for and . To state our main results, we need to introduce the matrix of Dirichlet correctors in , defined by
as well as the matrix of Neumann correctors in , defined by
Here denotes the conormal derivative associated with for .
The following are the main results of the paper.
Suppose that satisfies the same conditions as in Theorem 1.1. Also assume that , i.e., for and . Then for any ,
if is a bounded domain, where depends only on , , , , and . Moreover, if is a bounded domain for some ,
for any and , where and depends only on , , , , , and .
A few remarks are in order.
The matrix of Dirichlet correctors was introduced in  to establish the boundary Lipschitz estimates for solutions with Dirichlet conditions, while the matrix of Neumann correctors was introduced in  to establish the same estimates for solutions with Neumann boundary conditions. It is known that and . Under the condition for some , we also have (see Propositions 2.4 and 2.5).
Estimates (1.7) and (1.9) in Theorems 1.1 and 1.2 allow us to establish estimates for () for solutions with Dirichlet or Neumann boundary conditions (see Theorems 3.4 and 4.5). More importantly, estimates (1.8) and (1.10) yield near optimal convergence rates in for any . In fact, let in and on . Then
In the case of Neumann boundary conditions we obtain
for any , where in , on and . See subsections 3.2 and 4.2 for details. Let , where denotes the matrix of correctors for in . Estimates (1.11) and (1.12) should be compared to the well known estimate: (see e.g. ), and to the following estimate,
proved in [20, Theorems 3.4 and 5.2]. Due to the presence of a boundary layer, both the estimate and (1.13) are more or less sharp. The Dirichlet and Neumann correctors are introduced precisely to deal with boundary layer phenomena in periodic homogenization.
The asymptotic expansion of the fundamental solutions as well as the heat kernels for in has been studied, using the method of Bloch waves; see e.g. [26, 11] and their references (also see  for results obtained by the method of -convergence). In the presence of boundary, the Bloch representation is no longer available. In a series of papers [2, 4, 3], M. Avellaneda and F. Lin introduced the compactness methods, which originated in the regularity theory in the calculus of variations and minimal surfaces, to homogenization problems. In particular, they established an asymptotic expansion for in , using Dirichlet correctors. As a result, it was proved in  that if is for some ,
where () denotes the Poisson kernel for in ,
and the remainder satisfies
for any compact subset of (the results are stated for the case ; however, the argument in  works equally well for elliptic systems). In (1.15) we have used to denote the Dirichlet correctors for , the adjoint of . Also, and is the (constant) coefficient matrix of . The expansion (1.14) was used in  to identify the limit, as , of solutions to a problem of exact boundary controllability for the wave operator .
Besides its applications to boundary control problems, estimate (1.17) may also be used to investigate the Dirichlet problem
where is 1-periodic in . The Dirichlet problem (1.18) arises natually in the study of boundary layer phenomena and higher-order convergence in periodic homogenization (see e.g. [25, 1, 15, 14] and their references). Let be the solution to
(see Theorem 3.9). This effectively reduces the asymptotic problem (1.18) to the study of convergence properties of on , under various geometric conditions on . This line of research will be developed in a future work.
We now describe the main ideas in the proof of Theorems 1.1 and 1.2. The basic tools in our approach are representation formulas by Green and Neumann functions, uniform estimates for Green functions established in ,
where in and on . This is done by considering and using the Green representation formula and the observation that , where is a bounded periodic function. Estimate (1.7) follows from (1.21) by a more or less standard argument (see subsection 3.1). Next, we show in subsection 3.2 that
and represent in , using the Green function in , where is a domain such that .
Although a bit more complicated, the proof of Theorem 1.2 follows the same line of argument as Theorem 1.1. In subsections 4.1 and 4.2 we establish boundary and Lipschitz estimates similar to (1.21) and (1.22) for and satisfying in and on . The results rely on the uniform and Neumann function estimates obtained in [23, 21] under the additional symmetry condition .
The rest of the paper is organized as follows. Section 2 contains some basic formulas and estimates which are more or less known. The case of Dirichlet boundary conditions is treated in Section 3, while Section 4 is devoted to the case of Neumann boundary conditions. In Section 5 we prove two inequalities, which are used in subsection 4.3 and are of interest in their own right, for the Dirichlet-to-Neumann map .
for each and . Here and with in the position. The homogenized operator is given by , where and
It is known that the constant matrix is positive definite with an ellipticity constant depending only on , and (see ).
To see (2.4), one solves in with and , and let
(see e.g. ). Note that if is Hölder continuous, then and hence are Hölder continuous. It follows that is Hölder continuous. In particular, is bounded by a constant depending only on , , , and . In the case of the scalar equation with bounded measurable coefficients, the corrector is Hölder continuous by the De Giorgi -Nash estimates. This, together with Cacciopoli’s inequality and Hölder’s inequality, implies that there exist and , depending only on and , such that
In view of (2.3) we obtain
Since in and ,
The following proposition plays an important role in this paper. We mention that formula (2.8) with is known and may be used to show that , where in and on (see e.g. ). The proof of our main results on the first-order derivatives of Green and Neumann functions will rely on (2.8) with the matrices of Dirichlet and Neumann correctors respectively in the place of the functions .
Suppose that and in . Let
where and in for each and . Then
where if , and zero otherwise.
The next proposition will be used to handle the Neumann boundary condition (cf. ).
Let be given by (2.7). Suppose that . Then
The following two propositions provide the properties of the Dirichlet and Neumann correctors needed in this paper.