Homogenization for non-self-adjoint
locally periodic elliptic operators
We study the homogenization problem for matrix strongly elliptic operators on of the form . The function is Lipschitz in the first variable and periodic in the second. We do not require that , so need not be self-adjoint. In this paper, we provide, for small , two terms in the uniform approximation for and a first term in the uniform approximation for . Primary attention is paid to proving sharp-order bounds on the errors of the approximations.
Key words and phrases:homogenization, operator error estimates, locally periodic operators, effective operator, corrector
2010 Mathematics Subject Classification:Primary 35B27; Secondary 35J15, 35J47
Homogenization dates back to the late 1960s, and for more than fifty years it has become a well-established theory. In the simplest case, homogenization deals with asymptotic properties of solutions to differential equations with oscillating coefficients. Given a periodic (with period in each variable) uniformly bounded and uniformly positive definite function , consider the differential equation
where , and . The coefficients of the equation are -periodic and hence rapidly oscillate if is small. In homogenization theory one is interested in studying the asymptotic behavior of as becomes smaller. It is a basic fact that, after passing to a subsequence if necessary, converges to the solution of the differential equation
with constant . Since, in applications, the elliptic operator on the left side of (1.1) usually describes a physical process in a highly heterogeneous medium, this means that, in certain aspects, the process evolves very similar to that in a homogeneous medium.
It is a basic fact about homogenization theory that converges to in ; we refer the reader to [BLP78], [BP84] or [ZhKO93] for the details. Stated differently, the resolvent of converges in the strong operator topology to the resolvent of . In [BSu01] (see also [BSu03]), Birman and Suslina proved that, in fact, the resolvent converges in norm. Moreover, they found a sharp-order bound on the rate of convergence. Since that time there have been a number of interesting further results in this direction – see [Gri04], [Gri06], [Zh05], [ZhP05], [B08], [KLS12], [Su131], [Su132], [ChC16] and [ZhP16], to name a few.
Here we focus on a more general problem than the periodic one in (1.1). Let with being uniformly bounded functions that are Lipschitz in the first variable and periodic in the second (see Section 3 for a precise definition). Consider the operator on the complex space given by
The coefficients now depend not only on the “fast” variable, , but also on the “slow” one, . Assume that, for all in some neighborhood of , the operator is coercive and furthermore the constants in the coercivity bound are independent of . Then is strongly elliptic for such and there is a sector containing the spectrum of . In this paper, we will obtain approximations for and (with outside the sector) in the operator norm and prove that
the estimates being sharp with respect to the order (see Theorems 6.1 and 6.2). The effective operator is of the same form as , but its coefficients depend only on the slow variable. In contrast, the correctors and involve rapidly oscillating functions as well. The first of these plays the role of the traditional corrector and differs from the latter in that it involves a smoothing operator. The idea of using a smoothing to regularize the traditional corrector is due to Griso, see [Gri02]. The other corrector has no analogue in classical theory and was first presented in [BSu05] for purely periodic operators. Assume for simplicity that . Then has the form
(see Section 5). What is interesting here is that an analog of for periodic operators, while looking similar to this one, does not include the term , see [Se171]. In fact, one cannot remove from if the estimate (1.4) is to remain true, see Remark 5.9 for examples. So this term is a special feature of non-periodic problems.
The results of the present paper extend the author’s work [Se171] on periodic elliptic problems, where we studied non-self-adjoint scalar operators whose coefficients were periodic in some variables and Lipschitz in the others. Put differently, the fast and slow variables were separated in the sense that , where . We proved analogs of the estimates (1.3)–(1.5), yet the correctors were slightly different, see Remark 6.3 below. It should be pointed out that the operators in [Se171] were allowed to involve lower-order terms with quite general coefficients.
Previous results on uniform approximations for locally periodic elliptic operators are due to, on the one hand, Borisov and, on the other hand, Pastukhova and Tikhomirov. In [B08] Borisov established the estimates (1.3) and (1.5) for certain matrix self-adjoint operators with smooth coefficients. In the paper [PT07] of Pastukhova and Tikhomirov, similar results were proved for scalar self-adjoint operators with rough coefficients (although their techniques also apply to non-self-adjoint problems). As far as I know, the estimate (1.4) in the locally periodic settings was not obtained even for the simplest cases.
To prove the estimates, we develop the ideas of [Se171]. In the first step we establish a variant of the resolvent identity that involves the resolvents of the original and the effective operators and a corrector (see Section 7). This combination comes as no surprise, for it is well known that the effective operator and a corrector form a first approximation to the original operator (see, e.g., [BLP78] or [ZhKO93]). When this is done, all the desired estimates will follow at once. However, we cannot use the same technique as in [Se171], so the identity is proved by different means. The point is that the technique depends heavily on the smoothing operator that has been chosen. In the case of periodic operators, the smoothing was based on the Gelfand transform; but it is not as convenient now. To my knowledge, no natural smoothing for operators with locally periodic coefficients is known, so we choose the Steklov smoothing operator, which is the most simple and has proved to be quite useful; see [Zh05] and [ZhP05], where that smoothing first appeared in the context of homogenization, as well as [PT07], [Su131] and [Su132]. We remark that a very similar smoothing had been used earlier in [Gri02] and [Gri04] (see also [Gri06]). Our technique is strongly influenced by all these works.
I believe that the same method can be of use for locally periodic problems on domains with Dirichlet or Neumann boundary conditions as well.
It is also worth noting that, once the estimates (1.3)–(1.5) are verified, a limiting argument will give similar results for operators whose coefficients are Hölder continuous in the first variable, see Remark 6.6. These results, together with the results stated here, have been announced in [Se172].
The plan of the paper is as follows. Section 2 contains basic definitions and notation. In Section 3 we introduce the original operator. We study the effective operator in Section 4 and correctors in Section 5. Section 6 states the main results. Section 7 is the core of the paper, where we first prove the identity and then complete the proofs.
The symbol will stand for the norm on a normed space . If and are Banach spaces, then is the Banach space of bounded linear operators from to . When , the space becomes a Banach algebra with the identity map . The norm and the inner product on are denoted by and , respectively. We shall often identify and .
Let be a domain in and a Banach space. The space consists of those uniformly continuous functions for which
We will use the notation , and as shorthand for , and when the context makes clear which and are meant.
The symbol stands for the -space of strongly measurable functions on with values in . In case , we write for the norm on and for the inner product on . We let denote the usual Sobolev space of -valued functions on and , its dual space under the pairing . If is dense in , then , where is the exponent conjugate to .
Let be the closed cube in with center and side length , sides being parallel to the axes. Then denotes the completion of in the -norm. Here is the class of -times continuously differentiable functions on whose periodic extension to enjoys the same smoothness. Notice that coincides with the space of all periodic functions in . The spaces and are defined in a similar fashion. If , we write for , for , etc. The symbol will stand for the subspace of functions in with mean value zero. Any satisfies the Poincaré inequality
as can be seen by using Fourier series. Here and below, .
3. Original operator
Let each be a function in . Then may be thought of as a bounded mapping that is Lipschitz in the first variable and periodic in the second. As is well known, for any function satisfying the Carathéodory condition (i.e., the requirement of continuity with respect to the first variable and measurability with respect to the second) the map defined for and by
is measurable (here ). Notice that, if is another function from to , then . We adopt the notation .
Consider the matrix operator given by
It is easy to see that is bounded, with bound :
for all . Now we impose a condition that will render elliptic. Namely, we assume that is coercive uniformly in , where with , that is, there are and such that
for every . It follows that is -sectorial with sector
independent of . Whenever , the operator is an isomorphism and hence is invertible; moreover, for any we have
so is strongly elliptic for all . The Legendre–Hadamard condition does not generally imply (3.4). If we restrict our attention to the real-valued case, then for scalar operators the two statements are equivalent. But this is no longer true for matrix operators, let alone the complex-valued case. A necessary and sufficient algebraic condition on that would guarantee (3.4) is not known.
It is worthwhile to point out that we have to be able to verify the coercivity bound for all in some interval , which may be rather difficult. A sufficient condition not involving is that the operator is strongly coercive on and furthermore there is so that for any and
This can be seen by noticing that, by change of variable, the above inequality remains true with in place of . Then a partition of unity argument will do the job, since is uniformly continuous in the first variable.
As an example of satisfying (3.7), let be a matrix first-order differential operator with symbol
where . Suppose that the symbol has the property that, for some ,
Let be a function in with uniformly positive definite. Now if we take , then application of the Fourier transform will yield
Homogenization for self-adjoint operators of this type was studied by Birman and Suslina in the purely periodic setting (see, e.g., [BSu01], [BSu03], [BSu05], [BSu06], [Su131] and [Su132]) and by Borisov in the locally periodic setting (see [B08]).
Observe that the more restrictive Legendre condition, which amounts to the uniform positive definiteness of , does ensure coercivity, but excludes some strongly elliptic operators with important applications – such as certain elasticity operators.
4. Effective operator
Given and , we let be the weak solution of
in . The function is well defined, since is a continuous linear functional on and the operator is strongly coercive on , as we shall now see.
For any and all , we have
for every . Thus, the definition of makes good sense.
Denote by the map sending to . Evidently, depends linearly on , so is simply an operator of multiplication by a function (still denoted by ). The next lemma shows that has the same regularity in the first variable as .
In what follows, we denote differentiation in the first variable by and differentiation in the second variable by . When no confusion can arise, we omit the subscript and write , as we did before.
We have .
Let be given by
Since and are continuous in the first variable, so is . In fact, we have . Indeed, the estimate
is immediate from the definition of , and that
follows by an easy calculation. Hence, is finite.
Now we define the effective operator by setting
Observe that is bounded and coercive (recall Gårding’s inequality) and thus -sectorial. It can be proved that satisfies an estimate similar to (3.4) with exactly the same constants, however the bound on its norm may be different from (3.3). Nevertheless, the sector for remains the same as for . We briefly sketch the argument; see [Se171, Section 2.3] for a related proof. First consider the two-scale effective system as in [A92] and check that the associated form, which is defined on by
is -sectorial with sector . We only remark that the coercivity is obtained by substituting (with sufficiently smooth and ) into (3.4) for and letting tend to ; cf. the proof of Lemma 4.1. Then notice that
provided (which is definitely in ). The claim is proved.
Thus, we see that the operator is an isomorphism as long as is outside . In addition, standard regularity theory for strongly elliptic systems (see, e.g., [McL00, Theorem 4.16]) implies that the pre-image of under is all of and for any
Let us return to our discussion of coercivity at the end of the previous section. As we have seen, (4.2) follows from (3.4), which in turn is a consequence of (3.7). On the other hand, (4.2) does not generally imply (3.7), and there are examples (for , of course) where (4.2) holds, but (3.7) is false, see [BF15]. In such cases, a subsequence of may still converge in the weak operator topology to , but will fail to be strongly elliptic, i.e., will not satisfy the Legendre–Hadamard condition.
Let the operator be given by
The very same argument shows that is bounded on as well:
Since we do not impose any extra assumptions on the coefficients, the traditional corrector will not even map into itself. So we must first appropriately regularize the traditional corrector, and a smoothing operator is used for exactly this purpose.
Let be the translation operator
where and . Certainly, for any satisfying we have . Next, the adjoint of is given by
Note that is defined on and as well, by way of identifying these spaces with the corresponding subspaces of . We define the Steklov smoothing operator to be the restriction of to . In other words,
The operator is plainly self-adjoint.
Here we collect some facts about and .
The restriction of to is an isometry.
By change of variable,
But since is periodic in the second variable, this equals . ∎
A related result for is the following.
The restriction of to is bounded, with bound at most .
This is immediate from Cauchy’s inequality and Lemma 5.1. ∎
It is easy to see that both and converge in the strong operator topology to the identity operator, yet they do not converge in norm. The uniform convergence will, however, take place if we restrict them to certain Sobolev spaces.
For any we have
where . Integrating out the and variables then yields (5.6). ∎
For any we have
Now we can prove the following result.
For any we have
We define the first corrector by
Because of the smoothing , this corrector is bounded with
Indeed, using Lemma 5.2, we see that
While the -norm of is merely uniformly bounded, the -norm of turns out to be of order .
For any and we have
To describe the second corrector, we need some additional notation. Let be the adjoint of . Then we construct the effective operator , the corrector and the other objects (which will be marked with “” as well) for just as we did for . (It may be noted in passing that is the adjoint of .) Of course, all results for will transfer to . We shall not explicitly formulate these results here, but refer to them by the numbers of the corresponding statements for with “” following the reference (for example, Lemma 5.7+ and the estimate (5.10)+).