Homogenization for non-self-adjoint
locally periodic elliptic operators

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 $1$ in each variable) uniformly bounded and uniformly positive definite function $A:R\to C^{d\times d}$, consider the differential equation

where $\epsilon >0$, $\mu \in C\setminus R_{+}$ and $f\in L_2\left(R\right)$. The coefficients of the equation are $\epsilon$-periodic and hence rapidly oscillate if $\epsilon$ is small. In homogenization theory one is interested in studying the asymptotic behavior of $u_{\epsilon }$ as $\epsilon$ becomes smaller. It is a basic fact that, after passing to a subsequence if necessary, $u_{\epsilon }$ converges to the solution $u_0$ of the differential equation

with constant $A^0$. 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 $u_{\epsilon }$ converges to $u_0$ in $L_2\left(R\right)$; we refer the reader to [BLP78], [BP84] or [ZhKO93] for the details. Stated differently, the resolvent of $-\mathrm{div}A\left({\epsilon }^{-1}x\right)\nabla$ converges in the strong operator topology to the resolvent of $-\mathrm{div}{A}^0\nabla$. 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.

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 $A^{\epsilon }\left(x\right)=A(x_1,{\epsilon }^{-1}{x}_2)$, where $x=(x_1,x_2)$. 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 $\parallel \cdot {\parallel }_U$ will stand for the norm on a normed space $U$. If $U$ and $V$ are Banach spaces, then $B(U,V)$ is the Banach space of bounded linear operators from $U$ to $V$. When $U=V$, the space $B\left(U\right)=B(U,U)$ becomes a Banach algebra with the identity map $I$. The norm and the inner product on $C^n$ are denoted by $\left|\cdot \right|$ and $⟨\cdot ,\cdot ⟩$, respectively. We shall often identify $B(C^n,C^m)$ and $C^{m\times n}$.

Let $\Sigma$ be a domain in $R$ and $U$ a Banach space. The space $C^{0,1}(\overline{\Sigma };U)$ consists of those uniformly continuous functions $u:\Sigma \to U$ for which

where $\parallel u{\parallel }_{C(\overline{\Sigma };U)}={sup}_{x\in \Sigma }\parallel u\left(x\right){\parallel }_U$ and

We will use the notation $\parallel \cdot {\parallel }_{C^{0,1}}$, $\parallel \cdot {\parallel }_C$ and $[\cdot {]}_{C^{0,1}}$ as shorthand for $\parallel \cdot {\parallel }_{C^{0,1}(\overline{\Sigma };U)}$, $\parallel \cdot {\parallel }_{C(\overline{\Sigma };U)}$ and $[\cdot {]}_{C^{0,1}(\overline{\Sigma };U)}$ when the context makes clear which $\Sigma$ and $U$ are meant.

The symbol $L_p(\Sigma ;U)$ stands for the $L_p$-space of strongly measurable functions on $\Sigma$ with values in $U$. In case $U=C^n$, we write $\parallel \cdot {\parallel }_{p,\Sigma }$ for the norm on $L_p(\Sigma {)}^n$ and $(\cdot ,\cdot {)}_{\Sigma }$ for the inner product on $L_2(\Sigma {)}^n$. We let $W_p^m(\Sigma {)}^n$ denote the usual Sobolev space of $C^n$-valued functions on $\Sigma$ and $\left(W_p^m\right(\Sigma {)}^n{)}^{\ast }$, its dual space under the pairing $(\cdot ,\cdot {)}_{\Sigma }$. If $C_c^{\infty }(\Sigma {)}^n$ is dense in $W_p^m(\Sigma {)}^n$, then $W_{\text{cramped}{p}^{+}}^{-m}(\Sigma {)}^n=(W_p^m(\Sigma {)}^n{)}^{\ast }$, where $p^{+}$ is the exponent conjugate to $p$.

Let $Q$ be the closed cube in $R$ with center $0$ and side length $1$, sides being parallel to the axes. Then ${\tilde{W}}_p^m(Q{)}^n$ denotes the completion of ${\tilde{C}}^m(Q{)}^n$ in the $W_p^m$-norm. Here ${\tilde{C}}^m\left(Q\right)$ is the class of $m$-times continuously differentiable functions on $Q$ whose periodic extension to $R$ enjoys the same smoothness. Notice that ${\tilde{L}}_p(Q{)}^n$ coincides with the space of all periodic functions in $L_{p,\text{loc}}(R{)}^n$. The spaces ${\tilde{W}}_p^m(R\times Q{)}^n$ and ${\tilde{C}}^m(R\times Q{)}^n$ are defined in a similar fashion. If $p=2$, we write $H^m$ for $W_p^m$, $H^{-m}$ for $W_p^{-m}$, etc. The symbol ${\tilde{H}}_0^m(Q{)}^n$ will stand for the subspace of functions in ${\tilde{H}}^m(Q{)}^n$ with mean value zero. Any $u\in {\tilde{H}}_0^1(Q{)}^n$ satisfies the Poincaré inequality

as can be seen by using Fourier series. Here and below, $D=-i\nabla$.

We will often use the notation $\alpha \lesssim \beta$ to mean that that there is a constant $C$, depending only on some fixed parameters (these are listed in Theorems 6.1 and 6.2), such that $\alpha \le C\beta$.

Let each $A_{kl}$ be a function in $C^{0,1}({\overline{R}}^d;{\tilde{L}}_{\infty }(Q){)}^{n\times n}$. Then $A=\left\{A_{kl}\right\}$ may be thought of as a bounded mapping $A:R\times R\to B\left(C^{d\times n}\right)$ that is Lipschitz in the first variable and periodic in the second. As is well known, for any function $u:R\times R\to L_2\left(Q\right)$ 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 ${\tau }^{\epsilon }u:R\to L_2\left(Q\right)$ defined for $x\in R$ and $z\in Q$ by

is measurable (here $\epsilon >0$). Notice that, if $v$ is another function from $R\times R$ to $L_2\left(Q\right)$, then ${\tau }^{\epsilon }\left(uv\right)=\left({\tau }^{\epsilon }u\right)\left({\tau }^{\epsilon }v\right)$. We adopt the notation $u^{\epsilon }={\tau }^{\epsilon }u$.

Consider the matrix operator $A^{\epsilon }:H^1(R{)}^n\to H^{-1}(R{)}^n$ given by

It is easy to see that $A^{\epsilon }$ is bounded, with bound $C_{♭}=\parallel A{\parallel }_C$:

for all $u\in H^1(R{)}^n$. Now we impose a condition that will render $A^{\epsilon }$ elliptic. Namely, we assume that $A^{\epsilon }$ is coercive uniformly in $\epsilon \in E$, where $E=(0,{\epsilon }_0]$ with ${\epsilon }_0\in (0,1]$, that is, there are $c_A>0$ and $C_A\ge 0$ such that

for every $u\in H^1(R{)}^n$. It follows that $A^{\epsilon }$ is $m$-sectorial with sector

independent of $\epsilon$. Whenever $\mu \notin S$, the operator $A_{\mu }^{\epsilon }=A^{\epsilon }-\mu$ is an isomorphism and hence is invertible; moreover, for any $f\in H^{-1}(R{)}^n$ we have

Before proceeding, we make a few remarks about the coercivity condition. It follows from (3.4) (via Lemma 4.1) that $A$ satisfies the Legendre–Hadamard condition

so $A^{\epsilon }$ is strongly elliptic for all $\epsilon >0$. 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 $A$ 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 $\epsilon$ in some interval $(0,{\epsilon }_0]$, which may be rather difficult. A sufficient condition not involving $\epsilon$ is that the operator $D^{\ast }A(x,\cdot )D$ is strongly coercive on $H^1(R{)}^n$ and furthermore there is $c>0$ so that for any $x\in R$ and $u\in H^1(R{)}^n$

This can be seen by noticing that, by change of variable, the above inequality remains true with $A(x,{\epsilon }^{-1}y)$ in place of $A(x,y)$. Then a partition of unity argument will do the job, since $A$ is uniformly continuous in the first variable.

Observe that the more restrictive Legendre condition, which amounts to the uniform positive definiteness of $\mathrm{Re}A$, does ensure coercivity, but excludes some strongly elliptic operators with important applications – such as certain elasticity operators.

Given $\xi \in C^{d\times n}$ and $x\in R$, we let $N_{\xi }(x,\cdot )$ be the weak solution of

in ${\tilde{H}}_0^1(Q{)}^n$. The function $N_{\xi }$ is well defined, since $D^{\ast }A(x,\cdot )\xi$ is a continuous linear functional on ${\tilde{H}}^1(Q{)}^n$ and the operator $D^{\ast }A(x,\cdot )D$ is strongly coercive on ${\tilde{H}}^1(Q{)}^n$, as we shall now see.

It is clear from Lemma 4.1 and Poincaré’s inequality (2.1) that

for every $u\in {\tilde{H}}_0^1(Q{)}^n$. Thus, the definition of $N_{\xi }$ makes good sense.

Denote by $N$ the map sending $\xi$ to $N_{\xi }$. Evidently, $N_{\xi }$ depends linearly on $\xi$, so $N$ is simply an operator of multiplication by a function (still denoted by $N$). The next lemma shows that $N$ has the same regularity in the first variable as $A$.

Let $A^0:R\to B\left(C^{d\times n}\right)$ be given by

Since $A$ and $D_{2}N$ are continuous in the first variable, so is $A^0$. In fact, we have $A^0\in C^{0,1}\left({\overline{R}}^d\right)$. Indeed, the estimate

is immediate from the definition of $A^0$, and that

follows by an easy calculation. Hence, $\parallel A^0{\parallel }_{C^{0,1}\left({\overline{R}}^d\right)}$ is finite.

Now we define the effective operator $A^0:H^1(R{)}^n\to H^{-1}(R{)}^n$ by setting

Observe that $A^0$ is bounded and coercive (recall Gårding’s inequality) and thus $m$-sectorial. It can be proved that $A^0$ 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 $A^0$ remains the same as for $A^{\epsilon }$. 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 $H^1(R{)}^n\oplus L_2(R;{\tilde{H}}_0^1\left(Q\right){)}^n$ by

is $m$-sectorial with sector $S$. We only remark that the coercivity is obtained by substituting $u+\epsilon U^{\epsilon }$ (with sufficiently smooth $u$ and $U$) into (3.4) for $u$ and letting $\epsilon$ tend to $0$; cf. the proof of Lemma 4.1. Then notice that

provided $U=N{D}_{1}u$ (which is definitely in $L_2(R;{\tilde{H}}_0^1(Q){)}^n$). The claim is proved.

Thus, we see that the operator $A_{\mu }^0=A^0-\mu$ is an isomorphism as long as $\mu$ is outside $S$. In addition, standard regularity theory for strongly elliptic systems (see, e.g., [McL00, Theorem 4.16]) implies that the pre-image of $L_2(R{)}^n$ under $A_{\mu }^0$ is all of $H^2(R{)}^n$ and for any $f\in L_2(R{)}^n$

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 $n>1$, of course) where (4.2) holds, but (3.7) is false, see [BF15]. In such cases, a subsequence of $(A_{\mu }^{\epsilon }{)}^{-1}$ may still converge in the weak operator topology to $(A_{\mu }^0{)}^{-1}$, but $A^0$ will fail to be strongly elliptic, i.e., $A^0$ will not satisfy the Legendre–Hadamard condition.

Let the operator $K_{\mu }:L_2(R{)}^n\to {\tilde{H}}^1(R\times Q{)}^n$ be given by

Lemma 4.3, combined with the estimate (4.6), readily implies that $K_{\mu }$ is continuous:

The very same argument shows that $D_1{D}_2{K}_{\mu }$ is bounded on $L_2(R{)}^n$ as well: