Approximate Two-Sphere One-Cylinder Inequality in Parabolic Periodic Homogenization

Quantitative propagation of smallness is one of the central issues in the quantitative study of solutions of elliptic and parabolic equations. It can be stated as follows: a solution $u$ of a PDE $Lu=0$ on a domain $X$ can be made arbitrarily small on any given compact subset of $X$ by making it sufficiently small on an arbitrary given subdomain $Y$. There are many important applications in quantitative propagation of smallness, such as the stability estimates for the Cauchy problem [1] and the Hausdorff measure estimates of nodal sets of eigenfunctions [17], [18].

For solutions of second order parabolic equations

$$\left(\partial_{t}-\mathcal{L}\right)u=\partial_{t}u-\partial_{i}\left(a_{ij}(x,t)\partial_{j}u\right)+b_{i}\partial_{i}u+cu=0,$$

(1.1)

(the summation convention is used throughout the paper). There exists a large literature on the three cylinder inequality for solutions to parabolic equations. If $A(x,t)=\left(a_{ij}(x,t)\right)$ is three times continuously differentiable with respect to $x$ and one time continuously differentiable with respect to $t$, and $b$ and $c$ are bounded, then a not optimal three-cylinder inequality has been obtained in [11] and [20]. In [11], the three-cylinder inequality is derived by the Carleman estimates proved in [19]. In 2003, Vessella[21] has obtained the following optimal three-cylinder inequality:

$$\left\|u\right\|_{L^{2}(Q_{R}^{T/2})}\leqslant C\left(||u||_{L^{2}(Q_{\rho}^{T})}\right)^{\kappa_{\rho}}\left(||u||_{L^{2}(Q_{R_{0}}^{T})}\right)^{\kappa_{\rho}},$$

(1.2)

for every $0<\rho<R<R_{0}$, under the assumptions that the derivatives $\partial A/\partial t$, $\partial A/\partial x^{i}$, $\partial^{2}A/(\partial t\partial x^{j})$, $\partial^{2}A/(\partial x^{i}\partial x^{j})$, for every $i,j\in\{1,\cdots,d\}$, $b=(b_{1},\cdots,b_{d})$ and $c$ are bounded, where $Q_{r}^{t}=B_{r}\times(-t,t)$ with $B_{r}$ the $d$-dimensional open ball centered at $0$ of radius $r$ and $\kappa_{\rho}\sim|\log\rho|^{-1}$ as $\rho\rightarrow 0$. The optimality of Equation $(1.2)$ consists in the growth rate of the exponent $\kappa_{\rho}$. Later on, Escauriaza and Vessella [8] have obtained the inequality $(1.2)$ under the assumptions that $A\in C^{1,1}(\mathbb{R}^{n+1})$ , $b$ and $c$ are bounded. Moreover, Vessella [22] has obtained the following two-sphere one-cylinder inequality:

$$\left\|u\left(.,t_{0}\right)\right\|_{L^{2}\left(B_{\rho}\left(x_{0}\right)\right)}\leqslant C\left\|u\left(.,t_{0}\right)\right\|_{L^{2}\left(B_{r}\left(x_{0}\right)\right)}^{\theta}\|u\|_{L^{2}\left(B_{R}\left(x_{0}\right)\times\left(t_{0}-R^{2},t_{0}\right)\right)}^{1-\theta},$$

(1.3)

under the assumptions that $A$ satisfies the Lipschitz continuity: $|A(y,s)-A(x,t)|\leq C\left(|x-y|+|t-s|^{1/2}\right)$, $b=0$ and $c=0$, where $\theta=\left(C\log{\frac{R}{Cr}}\right)^{-1}$, $0<r<\rho<R$ and $C$ depends neither on $u$ nor on $r$ but may depend on $\rho$ and $R$, see also [6], where the two-sphere one-cylinder inequality $(1.3)$ for time-dependent parabolic operators was first established. And the estimate $(1.3)$ has first been obtained by Landis and Oleinik [15], when $A$ does not depend on $t$.

In general, the Carleman estimates are tools often used to obtain a three-cylinder inequality and the unique continuation properties for solutions. The Carleman estimates are weighted integral inequalities with suitable weight functions satisfying some convexity properties. The three-cylinder inequality is obtained by applying the Carleman estimates by choosing a suitable function. For Carleman estimates and the unique continuation properties for the parabolic operators, we refer readers to [4, 5, 7, 9, 22, 14] and their references therein for more results.

Recently, Guadie and Malinnikova [12] developed the three-ball inequality with the help of Poisson kernel for harmonic functions. This method also has been used in [13] to obtain an approximate three-ball inequality in elliptic periodic homogenization. Moreover, it is an interesting problem to extend the Carleman estimates to the homogenization equations and left for the future.

In this paper, we intend to develop an approximate two-sphere one-cylinder inequality for the $L^{\infty}$-norm in parabolic periodic homogenization equation, parallel to the inequality $(1.3)$, with the different exponent $\theta$.

We consider a family of second-order parabolic equations in divergence form with rapidly oscillating and time-dependent periodic coefficients,

$$\partial_{t}u_{\varepsilon}-\operatorname{div}\left(A(x/\varepsilon,t/\varepsilon^{2})\nabla u_{\varepsilon}\right)=0,$$

(1.4)

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

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

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

(1.5)

(ii) 1-Periodicity:

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

(1.6)

(iii)Hölder continuity: There exist constants $\tau>0$ and $0<\lambda<1$ such that

$$|A(x,t)-A(y,s)|\leq\tau\left(|x-y|+|t-s|^{1/2}\right)^{\lambda}$$

(1.7)

for any $(x,t),(y,s)\in\mathbb{R}^{d}\times\mathbb{R}$.

We are able to establish the following approximate two-sphere one-cylinder inequality in ellipsoids. The definition of ellipsoids $E_{r}$ depending on the coefficients $A(y,s)$ is given in Section 2.

A direct consequence of Theorem 1.1 is the following approximate two-sphere one-cylinder inequality in balls.

This interpolation method may also apply to the parabolic equation with potential. Namely, let $u_{\varepsilon}$ satisfies the following equation

$$\partial_{t}u_{\varepsilon}-\operatorname{div}\left(A(x/\varepsilon,t/\varepsilon^{2})\nabla u_{\varepsilon}\right)+V^{\varepsilon}u_{\varepsilon}=0,$$

(1.10)

where $V^{\varepsilon}=V(x/\varepsilon)$ with $V$ being 1-periodic and $V\in L^{\frac{d+2}{2}}(Z)$ with $Z=[0,1)^{d}$. Note that $V$ is independent of the variable $t$. Consequently, we are able to establish the following approximate two-sphere one-cylinder inequality in ellipsoids for the solution to $(1.10)$.

Let $\mathcal{L}_{\varepsilon}=-\operatorname{div}\left(A_{\varepsilon}(x,t)\nabla\right)$, where $A_{\varepsilon}(x,t)=A(x/\varepsilon,t/\varepsilon^{2})$. Assume that $A(y,s)$ is 1-periodic in $(y,s)$ and satisfies the ellipticity condition $(1.5)$. For $1\leq j\leq d$, the corrector $\chi_{j}=\chi_{j}(y,s)$ is defined as the weak solution to the following cell problem:

$$\left\{\begin{array}[]{l}\left(\partial_{s}+\mathcal{L}_{1}\right)\left(\chi_{j}\right)=-\mathcal{L}_{1}\left(y_{j}\right)\quad\text{ in }Y,\\ \chi_{j}=\chi_{j}^{\beta}(y,s)\text{ is }1\text{ -periodic in }(y,s),\\ \int_{Y}\chi_{j}^{\beta}=0,\end{array}\right.$$

(2.1)

where $Y=[0,1)^{d+1}.$ Note that

$$\left(\partial_{s}+\mathcal{L}_{1}\right)\left(\chi_{j}+y_{j}\right)=0\ \text{in }\mathbb{R}^{d+1}.$$

(2.2)

By the rescaling property of $\partial_{t}+\mathcal{L}_{\varepsilon}$, we obtain that

$$\left(\partial_{t}+\mathcal{L}_{\varepsilon}\right)\left(\varepsilon\chi_{j}(x/\varepsilon,t/\varepsilon^{2})+y_{j}\right)=0\ \text{in }\mathbb{R}^{d+1}.$$

(2.3)

Moreover, if $A=A(y,s)$ is Hölder continuous in $(y,s)$, then by standard regularity for $\partial_{s}+\mathcal{L}_{1}$, $\nabla\chi_{j}(y,s)$ is Hölder continuous in $(y,s)$, thus $\nabla\chi_{j}(y,s)$ is bounded.

Let $\widehat{A}=(\widehat{a}_{ij})$, where $1\leq i,j\leq d$, and

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

(2.4)

It is known that the constant matrix $\widehat{A}$ satisfies the ellipticity condition,

$$\mu|\xi|^{2}\leq\widehat{a}_{ij}\xi_{i}\xi_{j},\quad\quad\text{for any }\xi\in\mathbb{R}^{d},$$

and $|\widehat{a}_{ij}|\leq\mu_{1}$, where $\mu_{1}$ depends only on $d$ and $\mu$ [2]. It is also true or easy to verify that $(\widehat{a_{ij}})$ is symmetric if $(a_{ij})$ is symmetric. Denote

$$\mathcal{L}_{0}=-\operatorname{div}(\widehat{A}\nabla).$$

Then $\partial_{t}+\mathcal{L}_{0}$ is the homogenized operator for the family of parabolic operators $\partial_{t}+\mathcal{L}_{\varepsilon}$, with $1>\varepsilon>0$. Since $\widehat{A}$ is symmetric and positive definite, there exists a $d\times d$ matrix $S$ with $\det(S)>0$ such that $S\widehat{A}S^{T}=I_{d\times d}$. Note that $\widehat{A}^{-1}=S^{T}S$ and

$$\langle\widehat{A}^{-1}x,x\rangle=|Sx|^{2}.$$

(2.5)

We introduce a family of ellipsoids as

$$E_{r}(\widehat{A})=\{x\in\mathbb{R}^{n}:\langle\widehat{A}^{-1}x,x\rangle<r^{2}\}.$$

(2.6)

It is easy to see that

$$B_{\sqrt{\mu}r}(0)\subset E_{r}(0)\subset B_{\sqrt{\mu_{1}}r}(0).$$

(2.7)

We will write $E_{r}(\widehat{A})$ as $E_{r}$ if the context is understood.

To move forward, let $\Gamma_{\varepsilon}(x,t;y,s)$ and $\Gamma_{0}(x,t;y,s)$ denote the fundamental solutions for the parabolic operators $\partial_{t}+\mathcal{L}_{\varepsilon}$, with $1>\varepsilon>0$ and the homogenized operator $\partial_{t}+\mathcal{L}_{0}$, respectively. Moreover, it is easy to see that

$$\Gamma_{0}(x,t;y,s)=\frac{1}{\left(2\sqrt{\pi}\right)^{d}}\left(t-s\right)^{-d/2}|S|\exp\left\{-\frac{|Sx-Sy|^{2}}{4(t-s)}\right\},$$

(2.8)

for any $x,y\in\mathbb{R}^{d}$ and $-\infty<s<t<\infty$ with the matrix $S$ defined in $(2.5)$.

The following lemmas state the asymptotic behaviors of $\Gamma_{\varepsilon}(x,t;y,s)$ with $1>\varepsilon>0$, whose proof could be found in [10].

The next lemma states the asymptotic behaviors of $\nabla_{x}\Gamma_{\varepsilon}(x,t;y,s)$ and $\nabla_{y}\Gamma_{\varepsilon}(x,t;y,s)$.

With the summation convention this means that for $1\leq i,j\leq d$,

$$\left|\frac{\partial\Gamma_{\varepsilon}(x,t;y,s)}{\partial x_{i}}-\frac{\partial\Gamma_{0}(x,t;y,s)}{\partial x_{i}}-\frac{\partial\chi_{j}\left(x/\varepsilon,t/\varepsilon^{2}\right)}{\partial x_{i}}\frac{\partial\Gamma_{0}(x,t;y,s)}{\partial x_{j}}\right|$$

(2.12)

is bounded by the RHS of $(2.10)$. And the similar result holds for $\nabla_{y}\Gamma_{\varepsilon}(x,t;y,s)$.

The next lemma will be frequently used in the proof of Theorem 1.1.

Following [12], we are going to apply the Lagrange interpolation method to obtain the approximate two-sphere one-cylinder inequality. Actually, the similar method in [12] has been used by the author in [13] to obtain the approximate three-ball inequality in elliptic periodic homogenization. First, let us briefly review the standard Lagrange interpolation method in numerical analysis. Set

$$\Phi_{m}(z)=(z-p_{1})(z-p_{2})\cdots(z-p_{m})$$

(3.1)

for $z,p_{j}\in\mathcal{C}$ with $j=1,\cdots,m$. Let $\mathcal{D}$ ba a simply connected open domain in the complex plane $\mathcal{C}$ that contains the nodes $\tilde{p},p_{1},\cdots,p_{m}$. Assume that $f$ is an analytic function without poles in the closure of $\mathcal{D}$. By well-known calculations, it holds that

$$\frac{1}{z-\tilde{p}}=\sum_{j=1}^{m}\frac{\Phi_{j-1}(\tilde{p})}{\Phi_{j}(z)}+\frac{\Phi_{m}(\tilde{p})}{(z-\tilde{p})\Phi_{m}(z)}.$$

(3.2)

Multiplying the last identify by $\frac{1}{2\pi i}f(z)$ and integrating along the boundary of $\mathcal{D}$ leads to

$$\frac{1}{2\pi i}\int_{\partial\mathcal{D}}\frac{f(z)}{z-\tilde{p}}dz=\sum_{j=1}^{m}\frac{\Phi_{j-1}(\tilde{p})}{2\pi i}\int_{\partial\mathcal{D}}\frac{f(z)}{\Phi_{j}(z)}dz+\left(R_{m}f\right)(\tilde{p}),$$

(3.3)

where

$$\left(R_{m}f\right)(\tilde{p})=\frac{1}{2\pi i}\int_{\partial\mathcal{D}}\frac{\Phi_{m}(\tilde{p})f(z)}{(z-\tilde{p})\Phi_{m}(z)}dz.$$

(3.4)

By the residue theorem, there holds that

	$\displaystyle\left({R}_{m}f\right)(\tilde{p})$	$\displaystyle=\sum_{j=1}^{m}\frac{\Phi_{m}(\tilde{p})}{\left(p_{j}-\tilde{p}\right)\Phi_{m}^{\prime}\left(p_{j}\right)}f\left(p_{j}\right)+f(\tilde{p})$		(3.5)
		$\displaystyle=-\sum_{j=1}^{m}\prod_{i\neq j}^{m}\frac{\tilde{p}-p_{i}}{p_{j}-p_{i}}f\left(p_{j}\right)+f(\tilde{p}),$

where $\left(R_{m}f\right)(\tilde{p})$ is called the interpolation error. See chapter 4 in [3] for more information.

In order to obtain the approximate two-sphere one-cylinder inequality for the solution in $(1.4)$, we consider the Lagrange interpolation for $f(h)=\Gamma_{0}(hx_{0}\frac{r_{1}}{r_{2}},t_{0};y,s)$, where $0<r_{1}<r_{2}<{{r_{3}}}/{12}<R/8$ and $(x_{0},t_{0})$ is a fixed point such that $\sqrt{\langle\widehat{A}^{-1}x_{0},x_{0}\rangle}=|Sx_{0}|<r_{2}$. In view of $(3.5)$, we need to estimate the error term $(R_{m}\Gamma_{0})(x_{0},t_{0};y,s)$ of the approximation. Following the idea in [13], we choose points $x_{i}=h_{i}x_{0}\frac{r_{1}}{r_{2}}$ on the segment $[0,x_{0}\frac{r_{1}}{r_{2}}]$ with $h_{i}\in(0,1)$, then $x_{i}\in E_{r_{1}}$, $i=1,\cdots,m$. Select $p_{i}=h_{i}$ in the definition of $\Phi_{m}$ in $(3.1)$ and $\tilde{p}=r_{2}/r_{1}$. Define

$$c_{i}=\prod_{j\neq i}^{m}\frac{r_{2}r_{1}^{-1}-h_{j}}{h_{i}-h_{j}}.$$

(3.6)

Since $0<h_{i}<1$, direct computation shows that

$$|c_{i}|\leq\frac{\left(r_{2}r_{1}^{-1}\right)^{m-1}}{|\Phi_{m}^{\prime}(h_{i})|}.$$

(3.7)

To estimate $|c_{i}|$, we choose $h_{i}$ to be the Chebyshev nodes, which means, $h_{i}=\cos\left(\frac{(2i-1)\pi}{2m}\right)$, $i=1,\cdots,m$. Then we can write

$$\Phi_{m}(h)=2^{1-m}T_{m}(h),$$

where $T_{m}$ is the Chebyshev polynomial of the first kind. There also holds that

$$\Phi_{m}^{\prime}(h)=m2^{1-m}U_{m-1}(t),$$

(3.8)

where $U_{m-1}$ is the Chebyshev polynomial of the second kind. See e.g. section 3.2.3 in [3]. At each $h_{i}$, there hold

$$U_{m-1}(h_{i})=U_{m-1}\left(\cos\left(\frac{(2i-1)\pi}{2m}\right)\right)=\frac{\sin{\frac{(2i-1)\pi}{2}}}{\sin{\frac{(2i-1)\pi}{2m}}}=\frac{(-1)^{i-1}}{\sin\frac{(2i-1)\pi}{2m}}.$$

(3.9)

According to $(3.8)$ and $(3.9)$, there holds

$$|\Phi_{m}^{\prime}(h_{i})|\geq m2^{1-m}.$$

(3.10)

Therefore, by $(3.7)$, we have

$$|c_{i}|\leq(m)^{-1}\left(\frac{2r_{2}}{r_{1}}\right)^{m-1}\text{ for }i=1,\cdots,m.$$

(3.11)

To estimate the error term $(R_{m}\Gamma_{0})(x_{0},t_{0};y,s)$, we do an analytic extension of the function $f(h)=\Gamma_{0}(hx_{0}\frac{r_{1}}{r_{2}},t_{0};y,s)$ to the disc $\mathcal{D}_{\frac{{r_{3}}}{3r_{1}}}$ of radius $\frac{{r_{3}}}{3r_{1}}$ centered at the origin in the complex plane $\mathcal{C}$. According to $(2.8)$, we have

$$f(z)=\frac{1}{\left(2\sqrt{\pi}\right)^{d}}\left(t-s\right)^{-d/2}|S|\exp\left\{-\frac{(z\frac{r_{1}}{r_{2}}Sx_{0}-Sy)^{2}}{4(t-s)}\right\},$$

(3.12)

where $(z\frac{r_{1}}{r_{2}}Sx_{0}-Sy)^{2}=(z\frac{r_{1}}{r_{2}}Sx_{0}-Sy)\cdot(z\frac{r_{1}}{r_{2}}Sx_{0}-Sy)=\sum_{i=1}^{d}(z\frac{r_{1}}{r_{2}}(Sx_{0})_{i}-(Sy)_{i})^{2}$. Note that $|z\frac{r_{1}}{r_{2}}Sx_{0}|\leq\frac{{r_{3}}}{3}$ in the disc $\mathcal{D}_{\frac{{r_{3}}}{3r_{1}}}$, then there holds

$$|f(z)|\leq\tilde{C}\left(t-s\right)^{-d/2}\exp\left\{-\frac{Cr_{3}^{2}}{t-s}\right\}\quad\text{for }y\in E_{4{r_{3}}/5}\backslash E_{3{r_{3}}/4},$$

(3.13)

where $C$ and $\tilde{C}$ depend only on $d$.

Similarly, with the notations above, consider the Lagrange interpolation for $g(h)=\nabla_{y}\Gamma_{0}(hx_{0}\frac{r_{1}}{r_{2}},t_{0};y,s)$, and we do an analytic extension of the $g(h)$ to the disc $\mathcal{D}_{\frac{{r_{3}}}{3r_{1}}}$. Then according to $(2.8)$ again, there holds

$$g(z)=\frac{S\left(z\frac{r_{1}}{r_{2}}Sx_{0}-Sy\right)}{2\left(2\sqrt{\pi}\right)^{d}}\left(t-s\right)^{-d/2-1}|S|\exp\left\{-\frac{(z\frac{r_{1}}{r_{2}}Sx_{0}-Sy)^{2}}{4(t-s)}\right\},$$

(3.14)

where $S(z\frac{r_{1}}{r_{2}}Sx_{0}-Sy)$ is a vector with $S_{ik}\cdot(z\frac{r_{1}}{r_{2}}S_{ij}x_{0,j}-S_{ij}y_{j})$ being its $k$-th position. Note that $|z\frac{r_{1}}{r_{2}}Sx_{0}|\leq\frac{{r_{3}}}{3}$ in the disc $\mathcal{D}_{\frac{{r_{3}}}{3r_{1}}}$, then we have

$$|g(z)|\leq\tilde{C}{r_{3}}\left(t-s\right)^{-d/2-1}\exp\left\{-\frac{Cr_{3}^{2}}{t-s}\right\}\quad\text{for }y\in E_{4{r_{3}}/5}\backslash E_{3{r_{3}}/4},$$

(3.15)

where $C$ and $\tilde{C}$ depend only on $d$.

The following lemma gives the interpolation error terms $(R_{m}(\nabla_{y}\Gamma_{0}))(x,t;y,s)$ and $(R_{m}\Gamma_{0})(x,t;y,s)$ for $\nabla_{y}\Gamma_{0}(x,t;y,s)$ and $\Gamma_{0}(x,t;y,s)$, respectively.