Doubling inequalities and nodal sets in periodic elliptic homogenization

The paper is concerned with doubling inequalities and upper bounds of nodal sets of solutions in periodic elliptic homogenization. We consider a family of elliptic operators in divergence form with rapidly oscillating periodic coefficients

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

• •

  Strong ellipticity: there is $\Lambda>0$ such that

  $\displaystyle\Lambda|\xi|^{2}\leq\langle A(y)\xi,\ \xi\rangle\leq|\xi|^{2},\quad\quad\text{for any }y\in\mathbb{R}^{d},\xi\in\mathbb{R}^{d}.$

  (1.2)

• •

  Periodicity:

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

  (1.3)

• •

  Lipschitz continuity: There exists a constant $\gamma\geq 0$ such that

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

  (1.4)

The doubling inequality describes quantitative behavior to characterize the strong unique continuation property, which has important applications in inverse problems, control theory and the study of nodal sets of eigenfunctions. For harmonic functions or solutions of general elliptic equations in divergence form with Lipschitz coefficients, the doubling inequality is a consequence of a monotonicity formula or Carleman estimates; see [6, 8, 9, 26]. In periodic elliptic homogenization, the first doubling inequality was obtained recently by Lin and Shen [15] with an implicit dependence on the doubling index. Precisely, they proved that if $u_{\varepsilon}$ is a weak solution of $\mathcal{L}_{\varepsilon}(u_{\varepsilon})=0$ in $B_{2}=B_{2}(0)$ and

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

where $C(N)$ depends only on $d,\Lambda,\gamma$ and $N$. The point here is that the constant $C(N)$ is independent of the small parameter $\varepsilon$. This cannot be derived directly from the classical doubling inequality as the Lipschitz constant of the coefficients blows up as $\varepsilon$ approaches zero. However, it is not known that how the constant $C(N)$ in (1.6) depends on $N$, because (1.6) was proved by a compactness argument. We mention that if $\varepsilon=1$, the classical doubling inequality shows that $C(N)=CN^{K}$ for some $C,K\geq 1$; also see Lemma 3.2.

On the other hand, the Hadamard three-ball inequality also describes the quantitative unique continuation property. In periodic elliptic homogenization, two different versions of the three-ball inequality with error terms were discovered in [2] and [11]. In general, the three-ball inequalities with errors are weaker than the doubling inequalities, as they alone do not imply the strong unique continuation.

Our first goal of this paper is to find an explicit estimate for the constant $C(N)$ in the doubling inequality in periodic elliptic homogenization. The explicit doubling inequality not only provides more clear quantitative information for the solutions (such as the vanishing order), but also has more applications. We state the result as follows.

The double exponential growth $\exp(\exp(CN^{\tau}))$ for $d\geq 3$ in (1.8) and sub-exponential growth $\exp(C(\ln N)^{2})=N^{C\ln N}$ for $d=2$ in (1.10) seem to be the best we can obtain from the existing tools and results; see Remark 4.5. Our ultimate hope is for an estimate of the form $C(N)=N^{{C(\ln\ln N)}^{p}}$, for some $p>0$ depending on $d,\Lambda$ and $\gamma$. Such an estimate would have very important consequences for the study of long-standing open problems regarding the spectral properties of second order elliptic operators with periodic coefficients and their quantitative unique continuation properties (see for instance Conjecture 6.13, Theorem 6.15 and Conjecture 6.16 in [12]). This connection between the conjectured optimal doubling estimates and Conjecture 6.13 in [12] was observed by the first author, D. Mendelson and C. Smart in the fall of 2019. This motivated the current work.

As a straightforward corollary, Theorem 1.1 implies that the vanishing order of $u_{\varepsilon}$ at the origin does not exceed $\exp(CN^{\tau})$ for $d\geq 3$ and $C(\ln N)^{2}$ for $d=2$. Theorem 1.1 also implies a three-ball inequality without an error term, in contrast to the results in [2] and [11], namely (e.g., for $d\geq 3$),

for any $0<\tau_{1}<1$.

The proof of Theorem 1.1 breaks down into three steps:

• •

  Step 1: $\varepsilon/r\lesssim N^{-5}$. In this case, we take advantage of the convergence rate in homogenization theory and use the precise three-ball inequality of harmonic functions. The smoothness of the coefficients is not needed in this step.

• •

  Step 2: In this step, we need to use “analyticity”, which distinguishes between $d\geq 3$ and $d=2$. For $d\geq 3$, we let $N^{-5}\lesssim\varepsilon/r\lesssim N^{-\frac{1}{2}\tau}$, and use a three-ball inequality with a sharp exponential error term proved recently in [2] by Armstrong, Kuusi and Smart, which is a consequence of the “large-scale analyticity” from periodic homogenization. This will lead to a nontrivial improvement on the exponent so that $\tau>0$ in Theorem 1.1 can be arbitrarily small. Again in this case, the periodic structure will play a role; but the smoothness of coefficients is still not required. For $d=2$, we let $N^{-5}\lesssim\varepsilon/r\lesssim 1$, and apply a doubling inequality derived from quasi-regular mappings [1] (related to complex analyticity), which requires no smoothness or periodicity on the coefficients. Unfortunately, this method works only in two dimensions.

• •

  Step 3: $\varepsilon/r\gtrsim N^{-\frac{1}{2}\tau}$ for $d\geq 3$ or $\varepsilon/r\gtrsim 1$ for $d=2$. In this case, the classical doubling inequality for elliptic operators with Lipschitz coefficients can be handled by a monotonicity formula for the frequency function. If $d\geq 3$, the Lipschitz constant of the coefficients turns out to be $O(N^{\frac{1}{2}\tau})$ after rescaling. A careful calculation shows that the constant in (1.8) is at least $\exp(\exp(CN^{\tau}))$, if the periodicity is not used. If $d=2$, the Lipschitz constant of the coefficients after rescaling is bounded by $C$, independent of $\varepsilon$ and $N$. This allows us to obtain a much better estimate in two dimensions.

For $d\geq 3$, one will see in the proof that the estimate in Step 3 leads to the double-exponential growth of the constant in (1.8). What happens when $\varepsilon/r\gtrsim N^{-\frac{1}{2}\tau}$? To gain some intuition, consider a typical harmonic function $w_{k}=r^{k}\cos(k\alpha)$ in $\mathbb{R}^{2}$ (see [6] or [7]). Note that

By setting $N=C\theta^{-2k}$, we see that the intrinsic frequency of $w_{k}$ (i.e., the number of times that $w_{k}$ changes signs) is approximately $\ln N/(-\ln\theta)$. Now, let $u_{\varepsilon}$ be a weak solution of $\mathcal{L}_{\varepsilon}(u_{\varepsilon})=0$ whose limit is $w_{k}$ (the homogenized solution) as $\varepsilon\to 0$. In view of the interior first-order approximation $u_{\varepsilon}\sim w_{k}+\varepsilon\chi(x/\varepsilon)\nabla w_{k}$, the intrinsic frequency of $w_{k}$ will interact with the frequency of oscillation of the corrector $\chi(x/\varepsilon)$. Particularly, under rescaling, if $r/\varepsilon\approx\ln N$, the frequency of oscillation of the rescaled coefficients $A(rx/\varepsilon)$ (or correctors) is comparable to the intrinsic frequency of $w_{k}$. Note that the intrinsic frequency does not change under rescaling. It seems that the resonance between these two frequencies causes the failure of the arguments in Step 1 and Step 2 when ${\varepsilon/r}\approx(\ln N)^{-1}\gtrsim N^{-\frac{1}{2}\tau}$ (note that $\tau$ can be arbitrarily small and thus $N^{-\frac{1}{2}\tau}$ is close to the resonant situation), and we do not have a tool to handle this situation (except for $d=2$). We believe that an effective argument should take advantage of both the periodicity and the Lipschitz continuity of the coefficients.

Our second goal is to obtain an upper bound for the nodal sets of solutions in periodic elliptic homogenization. The study of the $(d-1)$-dimensional Hausdorff measure of nodal sets centers around Yau’s conjecture for Laplace eigenfunctions on smooth manifolds:

where $\mathcal{M}$ is a compact smooth Riemannian manifold without boundary. It was conjectured in [25] that the bounds of nodal sets of eigenfunctions in (1.12) are controlled by

where $C,c$ depend only on the manifold $\mathcal{M}$ and $H^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure. The conjecture (1.13) was shown for real analytic manifolds by Donnelly-Fefferman in [5]. Lin [14] also proved the upper bound for the analytic case, using an approach by frequency functions. We should mention that, by a lifting argument, Yau’s conjecture can be reduced to studying the nodal sets of harmonic functions on smooth manifolds. In recent years, there was an important breakthrough made by Logunov and Malinnikova [18], [16] and [17]. A polynomial upper bound was given in [16] and the sharp lower bound in the conjecture was shown in [17]. We are interested in the upper bound of nodal sets for $\mathcal{L}_{\varepsilon}(u_{\varepsilon})=0$ with rapidly oscillating periodic coefficients. The study of nodal sets in homogenization was initiated by Lin and Shen [15], where an implicit upper bound depending on the doubling index was shown. We are able to provide an explicit upper bound.

The strategy of the proof is as follows. For relatively large $\varepsilon$, we adapt a blow-up argument to obtain the upper bounds of nodal sets. For small $\varepsilon$, the solution $u_{\varepsilon}$ can be approximated by a harmonic function $u_{0}$, and thus the nodal set of $u_{\varepsilon}$ is a small perturbation of the nodal set of $u_{0}$. We then derive a quantitative estimate for the nodal set of $u_{\varepsilon}$ by carefully studying the small perturbations near the nodal set and critical set of $u_{0}$, which has its root in the analogous qualitative estimates obtained in [15]. By iterating such quantitative estimate, we are able to show the upper bound for the nodal sets of $u_{\varepsilon}$. The restriction $\alpha>8$ for $d\geq 3$ arises from the doubling inequality (5.2) for $\beta\in(\frac{3}{4},1)$. If we consider $N$ to be $\exp(CM)$ for some large constant $M$, which is the case for the doubling inequality of eigenfunctions, the upper bounds of nodal sets are double exponential functions $\exp(\exp(CM))$. In this sense, the restriction $\alpha>8$ only affects the constant $C$ in such upper bounds, which does not play an important role. For $d=2$, we point out that there is no misprint in the exponential (compared to $d\geq 3$). We still have the exponential, because in this situation, instead of the doubling inequality in (1.10), the suboptimal quantitative stratification of the critical set of harmonic functions [22] dominates the upper bound.

The paper is organized as follows. Section 2 is devoted to a doubling inequality at relatively large scales by the homogenization theory. In section 3, we derive the doubling inequality, using frequency functions, and show how it depends on the large Lipschitz constant of the coefficients. Then, Theorem 1.1 is proved in section 4 and Theorem 1.2 is proved in section 5. Throughout the paper, the letters $c$, $C$, $\hat{C}$, $\tilde{C}$, $C_{i}$, $c_{i}$ denote positive constants that do not depend on $\varepsilon$ or $u_{\varepsilon}$, and they may vary from line to line.

Acknowledgements: Parts of this work were carried out during the second author’s visit to the Department of Mathematics at the University of Chicago during January-March 2020. The first author would like to thank D. Mendelson and C. Smart for many insightful discussions on doubling inequalities in periodic homogenization. The second author would like to thank the Department of Mathematics at Chicago for the warm hospitality and the wonderful academic atmosphere. The second author also would like to thank F. Lin for helpful discussions on three-ball inequalities in periodic homogenization. The third author would like to thank D. Mendelson for insightful discussions during the early stages of this work.

In this section, we deal with the case $\varepsilon/r\gtrsim N^{-\frac{1}{\beta-\frac{3}{4}}}$ for all dimensions. Indeed, we will prove a quantitative version of [15, Theorem 3.1].

Let $\mathcal{L}_{0}=-\nabla\cdot(\widehat{A}\nabla)$ be the homogenized operator and $\widehat{A}$ be the homogenized coefficient matrix of $A$ (see, e.g., [24] for the general theory of periodic elliptic homogenization). Define the ellipsoid

The following is the main theorem of this section.

This follows from Lemma 2.2 and Lemma 2.3.

Now, if $\varepsilon<cN^{-1/(\beta-3/4)}$, the above lemma allows us to iterate (2.1) down to $r=c^{-1}N^{1/(\beta-3/4)}\varepsilon$. Precisely, if $r=\theta^{k}>CN^{1/(\beta-3/4)}\varepsilon$ and

with $A_{0}=N$, then

where

provided $A_{k}(\theta^{-k}\varepsilon)^{\beta-3/4}<c$.

In this section, we derive the doubling inequality with a large Lipschitz constant, which will be used in the Step 3 of the proof of Theorem 1.1. We aim to show how the Lipschitz character of the coefficients plays a role in quantitative unique continuation, which seems to be largely unexplored. Assume that

where $A(x)$ satisfies (1.2) and

for some large positive constant $L>1$. We emphasize that throughout this section, the constant $C$ will never depend on $L$. Since the $L^{\infty}$ norm and the $L^{2}$ norm of $u$ are comparable, parallel to the assumption (1.7), we may assume the following

for some large constant $M>1$.

In order to define the frequency function later, we need to construct the geodesic polar coordinates. The construction of polar coordinates has been obtained in [3]. We adopt a slightly different construction of the metric from [7, Chapter 3.1]. We follow the construction with an eye on the explicit dependence of the Lipschtiz constant $L$. For $d\geq 3$, we define the Lipschitz metric $\hat{g}=\hat{g}_{ij}(x)dx_{i}\otimes dx_{j}$ as follows

where $a^{ij}({x})$ is the entry of $A^{-1}({x})$. The case $d=2$ will be discussed in Remark 3.3. Note that $\hat{g}$ is Lipsthitz continuous and satisfies

Define

and

From (3.6), we can also write

Thus, we can check that $\psi(x)$ is a non-negative Lipschitz function satisfying

where $C$ depends only on $d$ and $\Lambda$. We introduce a new metric ${g}={g}_{ij}(x)dx_{i}\otimes dx_{j}$ by setting