A Carleman-Type Inequality in Elliptic Periodic Homogenization

Since T. Carleman’s pioneer work [8], Carleman estimates have been indispensable tools for obtaining a three-ball (or three-cylinder) inequality and proving the unique continuation property for partial differential equations. In general, the Carleman estimates are weighted integral inequalities with suitable weight functions satisfying some convexity properties. The three-ball inequality is obtained by applying the Carleman estimates by choosing a suitable function. For Carleman estimates and the unique continuation properties for the elliptic and parabolic operators, we refer readers to [3, 4, 12, 21, 10, 11, 22, 17] and their references therein for more results.

Over the last forty years, there is a vast and rich mathematical literature on homogenization. Most of these works are focused on qualitative results, such as proving the existence of a homogenized equation. However, until recently, nearly all of the quantitative theory, such as the convergence rates in $L^{2}$ and $H^{1}$, the $W^{1,p}$-estimates, the Lipschitz estimates and the asymptotic expansion of the Green functions and fundamental solutions, were confined in periodic homogenization. There are many good expositions on this topic, see for instance the books [7, 13, 20] for periodic case, see also the book [1] for the stochastic case.

Recently, authors in [2, 16, 15, 18] care about the propagation of smallness in homogenization theory, such as the approximate three-ball inequality in [2, 15] in elliptic periodic homogenization and the approximate two-sphere one-cylinder inequality in [24] in parabolic case, and the nodal sets and doubling conditions in [16, 18] in elliptic homogenization, which are all related to the Carleman inequality in classical elliptic and parabolic theory and encourage us to deduce a Carleman-type inequality in elliptic periodic homogenization and left for further for the parabolic case.

In this paper, we would like to deduce a Carleman-type inequality in elliptic periodic homogenization. According to the author’s knowledge, this is the first attempt in homogenization theory. More precisely, We consider a family of second-order elliptic equations in divergence form with rapidly oscillating periodic coefficients,

$$\mathcal{L}_{\varepsilon}u_{\varepsilon}=:-\operatorname{div}\left(A(x/\varepsilon)\nabla u_{\varepsilon}\right)=0,$$

(1.1)

where $1>\varepsilon>0$ and $A(y)=(a_{ij}(y))$ is a real symmetric $d\times d$ matrix-valued function in $\mathbb{R}^{d}$ for $d\geq 2$. Assume that $A(y)$ satisfies the following assumptions:

(i) Ellipticity: For some $0<\mu<1$ and all $y\in\mathbb{R}^{d}$, $\xi\in\mathbb{R}^{d}$, it holds that

$$\mu|\xi|^{2}\leq A(y)\xi\cdot\xi\leq\mu^{-1}|\xi|^{2}.$$

(1.2)

(ii) Periodicity:

$$A(y+z)=A(y)\quad\text{for }y\in\mathbb{R}^{d}\text{ and }z\in\mathbb{Z}^{d}.$$

(1.3)

(iii)Lipschitz continuity: There exist constants $\tilde{M}>0$ such that

$$|A(x)-A(y)|\leq\tilde{M}|x-y|,\ \text{for any }x,y\in\mathbb{R}^{d}.$$

(1.4)

Let $B(x,r)=\left\{y\in\mathbb{R}^{d}:|y-x|<r\right\}$ and $B_{r}=B(0,r)$. For positive constants $M$, $N_{1}\geq 1$ and $N_{2}\geq 1$, let $u_{\varepsilon}\in H^{2}(B_{3})$ be a solution of, and satisfies the following growth conditions,

$$\int_{B_{3}}|u_{\varepsilon}|^{2}\leq M\max\left\{\left(\int_{B_{2}}|u_{\varepsilon}|^{2}\right)^{N_{1}},\left(\int_{B_{2}}|u_{\varepsilon}|^{2}\right)^{1/N_{2}}\right\},$$

(1.5)

then we could obtain the following result:

Throughout this paper, we always assume that $\eta$ is a fixed cutoff function defined in Theorem 1.1. We prove this theorem by compactness argument. For the compactness method used in homogenization theory, we refer readers to [5, 6] for more details.

The most trivial application of Theorem 1.1 is the following three-ball inequality at a macroscopic scale without an error term.

The Corollary above only implies the three-ball inequality at a macroscopic scale, in the following theorem, we could obtain the three-ball inequality at every scale by using Corollary 1.4 and the uniform doubling conditions proved in [18] in elliptic homogenization.

The first result about the approximate three-ball inequality was obtained by Kenig and Zhu in [15] with the help of the asymptotic behavior of Green functions and the Lagrange interpolation technique, under the assumptions that the coefficient matrix $A$ is only Hölder continuous. Later on, an improvement of a sharp exponential error term (in the sense that if $A$ is only Hölder continuous, then the multiplicative factor must be at least exponential) in the error bound for the approximate three-ball inequality (under certain extra conditions) was discovered by Armstrong, Kuusi, and Smart in [2], as a consequence of the large-scale analyticity . Meanwhile, an approximate two-sphere one-cylinder inequality in parabolic periodic homogenization was obtained by the first author in [24], under the assumptions tha the coefficient matrix $A(x,t)$ is only Hölder continuous, with the help of the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.

Recently, the three-ball inequality without an error term was discovered by Kenig, Zhu and Zhuge in [16] under the assumptions that $A\in C^{0,1}$ and $u_{\varepsilon}$ satisfies a doubling condition at a macroscopic scale, with the help of the approximate three-ball inequality with a sharp exponential error term obtained in [2]. At this stage, we should compare the three-ball inequality obtained in [16] with the result proved in Theorem 1.5 in this paper. The result reads that:

Theorem. Assume that the coefficient matrix $A$ satisfies $(1.2)$-$(1.4)$. For every $\tau>0$, there exists $C>1$ and $\theta\in(0,1/2)$ depending only on $d$, $\mu$ and $\lambda$ such that if $u_{\varepsilon}$ is a weak solution of $\mathcal{L}_{\varepsilon}(u_{\varepsilon})=0$ in $B_{1}$ satisfying

$$\int_{B_{1}}|u_{\varepsilon}|^{2}dx\leq M\int_{B_{\theta}}|u_{\varepsilon}|^{2}dx,$$

then for any $r\in(0,1)$,

$$\int_{B_{\theta}}|u_{\varepsilon}|^{2}dx\leq\exp(\exp(CM^{\tau}))\int_{B_{\theta r}}|u_{\varepsilon}|^{2}dx$$

and

$$\int_{B_{\theta r}}|u_{\varepsilon}|^{2}dx\leq\exp(\exp(CM^{\tau}))\left(\int_{B_{\theta^{2}r}}|u_{\varepsilon}|^{2}dx\right)^{\tau_{1}}\left(\int_{B_{r}}|u_{\varepsilon}|^{2}dx\right)^{1-\tau_{1}}$$

for any $0<\tau_{1}<1$. It is clear that the authors in [16] have found an explicit estimate for the constant $C(M)$ in the doubling condition and in the three-ball inequality, with an unknown $\theta$. However, in our Theorem 1.5, we could state $\theta$ explicitly and obtain the three-ball inequality more directly with an unknown constant $C(M)$. Throughout this paper, with $Y=[0,1)^{d}\cong\mathbb{R}^{d}/\mathbb{Z}^{d}$, we use the following notation

$$H^{m}_{\text{per}}(Y)=:\left\{f\in H^{m}(Y)\text{ and }f\text{ is 1-periodic with }\fint_{Y}fdy=0\right\},$$

and we will write $\partial_{x_{i}}$ as $\partial_{i}$, $F^{\varepsilon}=F(x/\varepsilon)$ for a function $F$ and $\mathcal{M}(G)=\int_{Y}G(y)dy$ for a 1-periodic function $G$ if the context is understand.

Assume that $A=A(y)$ satisfies the conditions $(1.2)$-$(1.3)$. Let $\chi(y)\in H^{1}_{\text{per}}(Y;\mathbb{R}^{d})$ denote the first order corrector for $\mathcal{L}_{\varepsilon}$, where $\chi_{j}$ for $j=1,\cdots,d$ is the unique 1-periodic function in $H^{1}_{\text{per}}(Y)$ such that

$$\begin{cases}\mathcal{L}_{1}(\chi_{j})=-\mathcal{L}_{1}(y_{j})\quad\text{in }Y,\\ \fint_{Y}\chi_{j}dy=0.\end{cases}$$

(2.1)

By the classical Schauder estimates, $\chi\in C^{1,\alpha}$ if $A\in C^{0,\alpha}$. The homogenized operator for $\mathcal{L}_{\varepsilon}$ is given by $\mathcal{L}_{0}=-\operatorname{div}(\widehat{A}\nabla)$, where $\widehat{A}=(\widehat{a}_{ij})_{d\times d}$ and

$$\widehat{a}_{ij}=\fint_{Y}\left[a_{ij}+a_{ik}\frac{\partial}{\partial y_{k}}\left(\chi_{j}\right)\right](y)dy.$$

(2.2)

It is well-known that the homogenized matrix $\widehat{A}$ also satisfies the ellipticity condition $(1.2)$ with the same $\mu$. What’s more, if $A$ is symmetric, the same is also true for $\widehat{A}$. We refer the readers to [20] for the proofs.

Denote the so-called flux correctors $b_{ij}$ by

$$b_{ij}(y)=\widehat{a}_{ij}-a_{ij}(y)-a_{ik}(y)\frac{\partial\chi_{j}(y)}{\partial{y_{k}}},$$

(2.3)

where $1\leq i,j\leq d$.

The following theorem states the existence of $u_{0}$ in Theorem 1.1 and is used to control the second term on the left hand side of $(1.6)$.

Next, we introduce the following well-known Div-Curl lemma whose proof may be found in [20].

The following interior Caccioppoli’s inequality with weights will be used in the proof of Theorem 1.1.

At the end of this section, we introduce the following Carleman inequality for the homogenized operator $\mathcal{L}_{0}=-\operatorname{div}(\widehat{A}\nabla)$, whose proof may be found in [9].

To proceed further, we first need to calculate the term $\mathcal{L}_{\varepsilon}(u_{\varepsilon}\eta+\varepsilon\chi_{j}^{\varepsilon}\partial_{j}\eta u_{\varepsilon})$ on the right hand side of $(1.6)$, which will be stated in the following lemma.

Now we are ready to give the proof of Theorem 1.1.
Proof of Theorem 1.1. We prove the result by contraction. Suppose that there exist sequence $\{\varepsilon_{k}\}\subset\mathbb{R}^{+}$, $\{A_{k}\}$ being symmetric and satisfying $(1.2)$-$(1.4)$, $\{u_{k}\}\subset H^{2}(B_{3})$, $\{u_{k,0}\}\in H^{1}(B_{11/4})$ given by Theorem 2.2, $\{\lambda_{k}\}$ satisfying $\lambda_{k}\geq\lambda_{0}$ and $\{\tau_{k}\}$ satisfying $\tau_{0}\leq\tau_{k}\leq C(\lambda_{0},\tau_{0})\frac{||u_{k}||_{L^{2}(B_{3})}}{||u_{k}||_{L^{2}(B_{1})}}+100\tau_{0}$, such that $\varepsilon_{k}\rightarrow 0$, and

$$\operatorname{div}(A_{k}(x/\varepsilon_{k})\nabla u_{k})=0\text{ in }B_{3},$$

(3.5)

$$\int_{B_{3}}|u_{k}|^{2}\leq M\max\left\{\left(\int_{B_{2}}|u_{k}|^{2}\right)^{N_{1}},\left(\int_{B_{2}}|u_{k}|^{2}\right)^{1/N_{2}}\right\}$$

(3.6)

$$\operatorname{div}(\widehat{A_{k}}\nabla u_{k,0})=0\text{ in }B_{11/4},$$

(3.7)

with

$$||u_{k}-u_{k,0}||_{L^{2}(B_{11/4})}\leq C\sqrt{\varepsilon_{k}}||u_{k}||_{L^{2}(B_{3})},$$

(3.8)

and

		$\displaystyle\frac{C_{0}}{2}\int_{B_{3}}\left(\lambda_{k}^{4}\tau_{k}^{3}\varphi_{k}^{3}(u_{k}\eta)^{2}+\lambda_{k}^{2}\tau_{k}\varphi_{k}\left|\nabla\left[(u_{k}-\varepsilon_{k}\chi_{k,j}^{\varepsilon_{k}}\partial_{j}u_{k,0})\eta\right]\right|^{2}\right)e^{2\tau_{k}\varphi_{k}}dx$		(3.9)
		$\displaystyle\quad\quad>\int_{B_{3}}\left[\operatorname{div}\left(A_{k}^{\varepsilon_{k}}\nabla(u_{k}\eta+\varepsilon_{k}\chi_{k.j}^{\varepsilon_{k}}\partial_{j}\eta u_{k})\right)\right]^{2}e^{2\tau_{k}\varphi_{k}}dx,$

where $\chi_{k,j}$ is the $j$-th corrector defined in $(2.1)$ for the operator $A_{k}$. Since $\widehat{A_{k}}$ is symmetric and bounded in $\mathbb{R}^{d\times d}$, we may assume that

$$\widehat{A_{k}}\rightarrow H$$

(3.10)

for some symmetric matrix $H$ satisfying the ellipticity condition $(1.2)$. Due to $\lambda_{k}^{4}\varphi_{k}^{3}=\lambda_{k}^{4}e^{-3\lambda_{k}|x|^{2}}\rightarrow 0$ and $\lambda_{k}^{2}\varphi_{k}=\lambda_{k}^{2}e^{-\lambda_{k}|x|^{2}}\rightarrow 0$ as $\lambda_{k}\rightarrow\infty$ if $|x|>1/2$, then we could assume that

$$\limsup_{k\rightarrow\infty}\lambda_{k}=\lambda_{\infty}<+\infty.$$

(3.11)

By multiplying a constant to $u_{k}$, we may assume that

$$||u_{k}||_{L^{2}(B_{3})}=1.$$

(3.12)

By Caccioppoli’s inequality, this implies that $\{u_{k}\}$ is bounded in $H^{1}(B_{r})$ for any $0<r<3$. Therefore, by passing to a subsequence still denoted by $\{u_{k}\}$, we may further assume that

$$u_{k}\rightharpoonup u\text{ weakly in }H^{1}(B_{r}),$$

(3.13)

$$u_{k}\rightarrow u\text{ strongly in }L^{2}(B_{r}),$$

(3.14)

$$A_{k}(x/\varepsilon_{k})\nabla u_{k}\rightharpoonup F\text{ weakly in }L^{2}(B_{r})$$

(3.15)

for any $0<r<3$, where $u\in H^{1}_{\text{loc}}(B_{3})$ and $F\in L^{2}_{\text{loc}}(B_{3})$. It follows from the theory of homogenization (see e.g. [20]) that $F=H\nabla u$ and

$$\operatorname{div}(H\nabla u)=0\text{ in }B_{3}\text{ with }||u||_{L^{2}(B_{3})}\leq 1.$$

(3.16)

If we write $\chi_{k,j}(y)$ with $j=1,\cdots,d$ as the first order corrector for $-\operatorname{div}(A_{k}(x/\varepsilon_{k})\nabla)$, then it is easy to see that

		$\displaystyle\operatorname{div}[A_{k}(x/\varepsilon_{k})\nabla(u_{k}-u-\varepsilon_{k}\chi_{k,j}(x/\varepsilon_{k})\partial_{j}u)]$		(3.17)
	$\displaystyle=$	$\displaystyle\operatorname{div}(H\nabla u)-\operatorname{div}(A_{k}(x/\varepsilon_{k})\nabla u)-\operatorname{div}(A_{k}(x/\varepsilon_{k})\nabla\chi_{k,j}(x/\varepsilon_{k})\partial_{j}u)$
		$\displaystyle-\varepsilon_{k}\operatorname{div}(A_{k}(x/\varepsilon_{k})\chi_{k,j}(x/\varepsilon_{k})\nabla\partial_{j}u)$
	$\displaystyle=$	$\displaystyle\operatorname{div}((H-\widehat{A_{k}})\nabla u)-\operatorname{div}((A_{k}(x/\varepsilon_{k})-\widehat{A_{k}}+A_{k}(x/\varepsilon_{k})\nabla\chi_{k}(x/\varepsilon_{k}))\nabla u)$
		$\displaystyle-\varepsilon_{k}\operatorname{div}(A_{k}(x/\varepsilon_{k})\chi_{k,j}(x/\varepsilon_{k})\nabla\partial_{j}u)$
	$\displaystyle=$	$\displaystyle\operatorname{div}((H-\widehat{A_{k}})\nabla u)-\varepsilon_{k}\partial_{x_{l}}\left\{F_{k,lij}\partial^{2}_{ij}u\right\}-\varepsilon_{k}\operatorname{div}(A_{k}(x/\varepsilon_{k})\chi_{k,j}(x/\varepsilon_{k})\nabla\partial_{j}u)\text{ in }B_{3},$

where $F_{k,lij}=-F_{k,ilj}\in H^{1}_{\text{per}}(Y)\cap L^{\infty}(Y)$ given by Lemma 2.1 after replacing the coefficient matrix $A$ by $A_{k}$ in this lemma.