Carleman estimates for the parabolic transmission problem and H\"older propagation of smallness across an interface

TL;DR

This paper establishes a H"older propagation of smallness for solutions to second order parabolic equations with jump discontinuities at an interface, using a novel local Carleman estimate for anisotropic operators.

Contribution

It introduces a new Carleman estimate for parabolic operators with anisotropic coefficients having jumps at interfaces, enabling propagation of smallness results.

Findings

Proved a local Carleman estimate for parabolic operators with interface jumps.

Established H"older propagation of smallness across interfaces.

Extended techniques to anisotropic, Lipschitz continuous coefficients.

Abstract

In this paper we prove a H\"older propagation of smallness for solutions to second order parabolic equations whose general anisotropic leading coefficient has a jump at an interface. We assume that the leading coefficient is Lipschitz continuous with respect to the parabolic distance on both sides of the interface. The main effort consists in proving a local Carleman estimate for this parabolic operator.

Equations

$$\mathcal{L}(u)=0\text{mbox}in\Omega \text{mbox}andu=0\text{mbox}in\tilde{\Omega }\text{mbox}implyu=0\text{mbox}in\Omega .$$

$$\mathcal{L}(u)=0\text{mbox}in\Omega \text{mbox}andu\text{mbox}smallin\tilde{\Omega }\text{mbox}implyu\text{mbox}smallin\Omega .$$

$$\mathcal{L}(u)=\text{div}(A(x,t)Du)-{\partial }_{t}u,$$

$$\mathcal{L}(u)=0\text{mbox}in\Omega \times (0,T)\text{mbox}andu=0\text{mbox}on\tilde{\Omega }\times J\text{mbox}implyu=0\text{mbox}on\Omega \times J.$$

$$u(x,t_0)=O\left(|x-x_0{|}^k\right)\forall k\in \mathbb{N}\text{mbox}impliesu(\cdot ,t_0)=0\text{mbox}on\Omega ,$$

$$|A(x,t)-A(y,s)|\le M{\left(|x-y{|}^2+|t-s|\right)}^{1/2}\text{mbox}forevery(x,t),(y,s)\in {\mathbb{R}}^{n+1},$$

$$\text{div}(A(x)Du)=0,\text{mbox}in{B}_4(0)$$

$$u=0,\text{mbox}in{B}_{1/2}(e_n),$$

$$u_{-}(\cdot ,0)=A_{-}(\cdot ,0)D{u}_{-}(\cdot ,0)\cdot e_n=0\text{mbox}forx\in \Gamma \cap B_4(0),$$

$$u=H_{+}{u}_{+}+H_{-}{u}_{-}=\sum _{\pm }{H}_{\pm }{u}_{\pm },$$

$$\mathcal{F}(f)=\hat{f}(\xi ,\eta )={\int }_{{\mathbb{R}}^n}f(x,t)e^{-ix\cdot \xi -it\eta }dxdt,\xi \in {\mathbb{R}}^{n-1},\eta \in \mathbb{R}.$$

$$[f{]}_{\frac{1}{2},0,{\mathbb{R}}^n}={\left[{\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^{n-1}}{\int }_{{\mathbb{R}}^{n-1}}\frac{|f(x_1,t)-f(x_2,t){|}^2}{|x_1-x_2{|}^n}d{x}_{1}d{x}_{2}dt\right]}^{1/2},$$

$$[f{]}_{0,\alpha ,{\mathbb{R}}^n}={\left[{\int }_{{\mathbb{R}}^{n-1}}{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}\frac{|f(x,t_1)-f(x,t_2){|}^2}{|t_1-t_2{|}^{1+2\alpha }}d{t}_{1}d{t}_{2}dx\right]}^{1/2}.$$

$$[f{]}_{\frac{1}{2},0,{\mathbb{R}}^n}<\infty ,$$

$$[f{]}_{0,\alpha ,{\mathbb{R}}^n}<\infty ,$$

$$||f|{|}_{H^{\frac{1}{2},0}({\mathbb{R}}^n)}=||f|{|}_{L^2({\mathbb{R}}^n)}+[f{]}_{\frac{1}{2},0,{\mathbb{R}}^n}$$

$$||f|{|}_{H^{0,\alpha }({\mathbb{R}}^n)}=||f|{|}_{L^2({\mathbb{R}}^n)}+[f{]}_{0,\alpha ,{\mathbb{R}}^n}.$$

$$C_n^{-1}{\int }_{{\mathbb{R}}^n}|\xi ||\hat{f}(\xi ,\eta ){|}^{2}d\xi d\eta \le [f{]}_{\frac{1}{2},0,{\mathbb{R}}^n}^2\le C_n{\int }_{{\mathbb{R}}^n}|\xi ||\hat{f}(\xi ,\eta ){|}^{2}d\xi d\eta ,$$

$$C_{n,\alpha }^{-1}{\int }_{{\mathbb{R}}^n}|\eta {|}^{2\alpha }|\hat{f}(\xi ,\eta ){|}^{2}d\xi d\eta \le [f{]}_{0,\alpha ,{\mathbb{R}}^n}^2\le C_{n,\alpha }{\int }_{{\mathbb{R}}^n}|\eta {|}^{2\alpha }|\hat{f}(\xi ,\eta ){|}^{2}d\xi d\eta ,$$

$$[D_{x}f{]}_{0,\frac{1}{4},{\mathbb{R}}^n}^2\le C_{n,\frac{1}{4}}{\int }_{{\mathbb{R}}^n}|\eta {|}^{1/2}|\xi {|}^2|\hat{f}(\xi ,\eta ){|}^{2}d\xi d\eta$$

$$\le$$

$$\mathcal{L}(u):=\sum _{\pm }{H}_{\pm }\left(-{\partial }_t{u}_{\pm }+{\text{div}}_{x,y}(A_{\pm }(x,t,y)D_{x,y}{u}_{\pm })\right),$$

$$A_{\pm }(x,t,y)=\{a_{ij}^{\pm }(x,t,y){\}}_{i,j=1}^n,x\in {\mathbb{R}}^{n-1},t\in \mathbb{R},y\in \mathbb{R}$$

$${\lambda }_0|z{|}^2\le A_{\pm }(x,t,y)z\cdot z\le {\lambda }_0^{-1}|z{|}^2,\forall (x,t,y)\in {\mathbb{R}}^{n+1},\forall z\in {\mathbb{R}}^n$$

$$|A_{\pm }(x^{\prime },t^{\prime },y^{\prime })-A_{\pm }(x,t,y)|\le M_0{\left(|x^{\prime }-x{|}^2+|y^{\prime }-y{|}^2+|t^{\prime }-t|\right)}^{1/2}.$$

$$h_0(u):=u_{+}(x,t,0)-u_{-}(x,t,0),\forall (x,t)\in {\mathbb{R}}^n$$

$$h_1(u):=A_{+}(x,t,0)D_{x,y}{u}_{+}(x,t,0)\cdot \nu -A_{-}(x,t,0)D_{x,y}{u}_{-}(x,t,0)\cdot \nu ,\forall (x,t)\in {\mathbb{R}}^n,$$

$${\partial }_{t}u={\text{div}}_{x,y}\left(A(x,t,y)\nabla u\right)\text{mbox}in{\mathbb{R}}^{n-1}\times \mathbb{R}\times \mathbb{R},$$

$$\left\{\begin{array}[]{rcl}\mathcal{L}(u)&=&0\mbox{ in }\mathbb{R}^{n-1}\times\mathbb{R}\times\mathbb{R},\\ h_{0}(u)&=&0\mbox{ in }\mathbb{R}^{n-1}\times\mathbb{R},\\ h_{1}(u)&=&0\mbox{ in }\mathbb{R}^{n-1}\times\mathbb{R}.\end{array}\right.$$

$$\varphi(y)=\begin{cases}\begin{array}[]{l}\varphi_{+}(y):=\alpha_{+}y+\beta y^{2}/2,\quad y\geq 0,\\ \varphi_{-}(y):=\alpha_{-}y+\beta y^{2}/2,\quad y<0,\end{array}\end{cases}$$

$${\varphi }_{\delta }(x,y):=\phi ({\delta }^{-1}y)-\frac{|x{|}^2}{2\delta }.$$

Full text

1. Introduction

The main purpose of this paper is to study unique continuation properties and propagation of smallness for solutions to second order parabolic equations whose anisotropic leading coefficients have jumps at an interface. Although there exist good general books and papers about Carleman estimates, unique continuation properties and related propagation of smallness (see [BLR1], [Cal], [H1], [H3], [I1], [Tr], [Z]) and a lot of surveys and introductory papers on the subject and on its several applications (see [FI], [Kl], [KlTi], [KSU], [I2], [LRLeb], [V2], [V3], [Yam]), we would like to give to the non expert reader some basic notions and quick historic panorama on the subject.

We say that a linear partial differential equation $\mathcal{L}(u)=0$, enjoys a unique continuation property (UCP) in a connected open set $\Omega\subset\mathbb{R}^{N}$ if the following property holds true [T]: for any open subset $\widetilde{\Omega}$ of $\Omega$

[TABLE]

We call quantitative estimate of unique continuation (QEUC) or stability estimate related to the UCP property (1.1) the following type of result:

[TABLE]

Of course, the research in these topic is of some interest if either the function $u$ is not analytic or if the operator $\mathcal{L}$ has nonanalytic coefficients in $\Omega$.

In this sense Carleman, in his paper [Car] in 1939, marked a true milestone, because he proved that the 2D elliptic operator $\mathcal{L}(u)=\Delta u+a(x)u$, where $a$ is a bounded function, enjoys the UCP. At the same time Carleman conceived a highly constructive method that wide opened the doors to quantitative estimates of unique continuation for equations with nonanalytic coefficients. Since the 1950s the investigation on UCP has been extended to more general differential operators with a special attention to the regularity, in the first place, of the coefficients of the principal part of the operators. For instance, it was proved in [Pl], see also [Ma], [Mil], that the UCP for the second order elliptic equations doesn’t hold true if the coefficients of principal part is in $C^{0,\alpha}(\mathbb{R}^{n})$ for $\alpha<1$ and $n>2$. On the other side, the UCP applies when the coefficients of the principal part are Lipschitz continuous (see [AKS], [H2]) and, consequently, a Hölder type propagation of smallness can be proved in the form of three-sphere inequality ([La]). We refer to [ARRV] for an extensive and detailed analysis of connection between the UCP and propagation of smallness for second order elliptic equation. We should mention that the UCP for the second order elliptic equations with two variables with $L^{\infty}$ coefficients can be deduced from the theory of quasiconformal mappings ([BJS], [Sc], [AM] ).

In the parabolic context, broadly speaking, the investigation about UCP is focused on two main topics: (i) backward uniqueness and backward stability estimates, (ii) spacelike unique continuation properties (which include the noncharacteristic Cauchy problem) and their quantitative versions. In this paper we concentrate on the second issue. For backward uniqueness and stability we refer to [I3], [V3], [Yam].

Let us consider the operator

[TABLE]

where $A(x,t)$, $(x,t)\in\mathbb{R}^{n+1}$, is a symmetric $n\times n$ matrix which we assume uniformly elliptic. The spacelike UCP has the following formulation: let $\widetilde{\Omega}$ be any open subset in $\Omega$ and let $J\subset(0,T)$ an interval or a single point; we say that $\mathcal{L}$ enjoys the spacelike UCP if

[TABLE]

There exists a broad literature about property (1.3), so that we mention here the most meaningful papers (in the authors’ opinion) in this topic and we refer to the survey paper [V3] for more extensive references.

The first result of spacelike UCP was proved in [N] for $n=1$, $J$ an interval, and for the solutions to the equation $u_{xx}-u_{t}=a(x,t)u_{x}+b(x,t)u$, $a,b\in L^{\infty}$. For $n\geq 1$, $J$ is a single point, and the matrix $A$ in class $C^{2}$ and independent of $t$, the spacelike UCP was proved in [IY] (see also [Ya]). If $A(x,t)$ belongs to $C^{\infty}$ and $J$ is an interval, the spacelike UCP was proved in [Miz]; such a result is still valid if $u$ is a solution to $\tilde{\mathcal{L}}u=0$ where $\tilde{\mathcal{L}}$ is a first order perturbation of $\mathcal{L}$, namely $\tilde{\mathcal{L}}(u)=\mathcal{L}(u)+W(x,t)\cdot\nabla u+V(x,t)\,u$, with $W,V\in C^{\infty}$. The result in [Miz] was substantially improved in [SS] and in [So]: in [SS] the matrix $A(x,t)$ belongs to $C^{1}$, $W\in L^{\infty}$, $V\in L^{\infty}$ and in [So] $V\in L_{loc}^{\frac{n+2}{2}}$, $W\in L^{\infty}$, but again $A(x,t)\in C^{\infty}$. Lees and Protter, ([LeP], [Pr]) proved uniqueness for the Cauchy problem for the equation $\widetilde{\mathcal{L}}(u)=0$, when $A\in C^{2}$. A stability estimate of Hölder type (far from initial and final times) for the Cauchy problem under the same hypotheses of [LeP], [Pr], was proved (perhaps for the first time) in [AS], see also [A] and [LRS]. We mention the quite recent estimate of log type up to the initial and final time proved in [CY]. We refer to [EV], [V1] for additional improvements on the regularity of the leading coefficients.

The following strong unique continuation property (SUCP) for parabolic equations whose coefficients are smooth and time independent was proved in [LO]: if $(x_{0},t_{0})\in\Omega\times(0,T)$ and $\mathcal{L}(u)=0$ in $\Omega\times(0,T)$, then

[TABLE]

(see also [Li] for weaker assumptions on the regularity). Notice that (1.4) trivially implies (1.3).

It is rather obvious that, for the validity of (1.3) for $n\geq 3$, the minimum of regularity required on $A(x,t)$ with respect to space variables should be the same as the corresponding elliptic UCP. For this reason we assume that $A(x,t)$ is Lipschitz continuous with respect to the parabolic distance, namely

[TABLE]

for some given constant $M$.

As a matter of fact, under assumption (1.5), the SUCP holds true, see [AV1], [EFV], [EF], [F], [KoTa]. Some quantitative versions of this result were proved in [EFV]. Nevertheless, to the authors’ knowledge, there is no counterexample in literature that forbid a substantial reduction of the regularity of $A(x,t)$ with respect to $t$.

To get closer to the main theme of the present paper, we emphasize that whenever the matrix $A(x,t)$ has a jump discontinuity at a smooth enough interface QEUC is much more interesting then UCP. In order to make this point clear let us restrict for a moment to the elliptic case. Let us consider a symmetric $n\times n$ matrix $A(x)$, $x=(x^{\prime},x_{n})\in\mathbb{R}^{n}$, whose entries have a jump discontinuity at the interface $\Gamma=\{(x^{\prime},0)|x^{\prime}\in\mathbb{R}^{n-1}\}$ and are Lipschitz continuous on both the sides of $\Gamma$, that is on $\mathbb{R}^{n}_{\pm}=\{(x^{\prime},x_{n})|x_{n}\gtrless 0\}$. We assume that $A(x)$ is also uniformly elliptic. Denote by $B_{r}(\widetilde{x})$ the ball centered at $\widetilde{x}$ with radius $r>0$, $B^{\pm}_{r}(\widetilde{x})=B_{r}(\widetilde{x})\cap\mathbb{R}^{n}_{\pm}$ and let $u$ be a weak solution to

[TABLE]

which satisfies

[TABLE]

then, by the unique continuation property we have $u_{+}\equiv 0$ (here $u_{\pm}=u_{|B^{\pm}_{4}(0)}$) and by the homogeneous transmission conditions on $\Gamma$ we have

[TABLE]

hence the uniqueness for Cauchy problem gives $u_{-}\equiv 0$. Now, broadly speaking, if we translate in a quantitative form the above procedure assuming that $|u|\leq\varepsilon$ in $B_{1/2}(e_{n})$ instead of (1.7), and $|u|\leq 1$ in $B_{4}(0)$, we would obtain a logarithmic estimate of $u$ even in $B^{+}_{r}$ for $r<4$, see [AV2]. Clearly, this is a wrong way to perform propagation of smallness across the interface because in this way we treat equation (1.6) like two different equations, one in $B_{4}^{+}$ and the other one in $B_{4}^{-}$, and the interface as part of the boundary on which only logarithmic estimates can be obtained [ARRV, Sect. 1.1].

The right way to perform the smallness propagation estimate was provided for the first time, for isotropic coefficients (that is $A=aI$ for scalar $a$), in [LRR1, Sect. 3.1] where a Hölder type propagation of smallness across the interface (”interpolation inequality” in the terminology of [LRR1]) was proved in the form of three-region inequality. This result was extended in [FLVW] to the case of general anisotropic and Lipschitz continuous matrix $A$. In both the papers [LRR1], [FLVW] the three-region inequality was derived by Carleman estimates, see also [BDLR], [BDT] for some improvements of the results of [LRR1]. In [LRLer] the Carleman estimate was proved for $A\in C^{\infty}$, whereas in [DCFLVW] it was proved when the matrix $A$ is Lipschitz continuous. We refer to [BLR2] for general results about elliptic transmission problems with complex coefficients across an interface.

Undoubtedly the investigation about Carleman estimates for the transmission problem was driven not only by its the intrinsic interest but also by the interest in the issue of exact controllability for parabolic equation and for inverse problems. Here we should mention the paper [DOP] in which, perhaps for the first time, a Carleman estimate was proved in the parabolic context under the assumption that the leading coefficient it is independent on $t$ and it satisfies some monotonicity condition on the interface. Such a monotonicity condition was overcome in [LRR2], as well as the time independence of coefficients, for $C^{\infty}$ isotropic coefficients. For some application to an inverse source parabolic transmission problem we refer to [BY].

The main effort in the present paper consists in proving a local Carleman estimate for the operator (1.2) when the matrix $A(x,t)$ have jumps at a flat interface orthogonal to the time direction and is Lipschitz continuous with respect to the parabolic distance on both the sides of the interface (see Theorem 2.2 for the exact statement of the main result). Then in Section 6, this Carleman estimate is applied to prove a Hölder type estimate for propagation of smallness across the interface. It is easy to check that, by using standard change of coordinates, this propagation of smallness continues to be true in the more general case of a $C^{1,1}$ interface whose normal vectors are never parallel to $t$ axis.

In order to prove our Carleman estimate (Theorem 2.2) we follow an approach similar to the one of [DCFLVW]. More precisely, in Section 3, we consider the case in which the coefficients of the operator $\mathcal{L}$ depend only on the variable $x_{n}$ normal to the interface ($x_{n}$ is renamed $y$ in the rest of the paper). The simpler structure of operator allows us, at a first stage, to prove a Carleman estimate (Theorem 3.1) with a weight function $\phi$ that is linear in all the space variables except the normal one. In this first step we perform the Fourier transform with respect to the tangential variables $x^{\prime}$ and $t$ of the conjugate operator $e^{\tau\phi}\mathcal{L}(e^{-\tau\phi}\,\cdot\,)$ and, inspired by [LRLer], we factorize this conjugate operator into two first order operators. By this approach we avoid the techniques of pseudodifferential operators (known to be ”greedy” of regularity) used in [LRLer], and allows us to assume weak regularity on the leading matrix. Nevertheless, the Carleman estimate obtained in the first step, for the features of the level set of the weight function $\phi$, doesn’t allow to find a smallness propagation estimate across the interface even in the most simple case of a constant matrix $A$. To get such a smallness propagation, say from an open set contained on one side of the interface to a set in the opposite side, we need a weight of type $-k|x^{\prime}|^{2}+\phi$, $k>0$, or the more comfortable weight $-k|x^{\prime}|^{2}-bt^{2}+\phi$, $k,b>0$. For this reason, in the second step of the proof of Carleman estimate (Section 4), we consider an operator with general coefficients and a weight that is quadratic in $x^{\prime}$. In order to treat this general case we use a suitable partition of unity. Finally in the third step of the proof (Section 5) we add in the weight the dependence on $t$. Once the Carleman estimate has been proved, we show in Section 6 a three-region inequality. In the Appendix (Section 7) we prove a regularity result for the parabolic transmission problem that, although quite standard, is not present in the literature (to the authors’ knowledge).

2. Notations and statement of the main theorem

2.1. General notations and norms

The functions we are interested in depend on $n$ space variables and one time variable. We denote the space variables as $(x,y)\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}$ and the time variable as $t\in{\mathbb{R}}$. We assume the flat interface to be $\{y=0\}$. Since the variable $y$, that is orthogonal to the interface, is the most important one, we use, instead of the usual notation $(x,y,t)$ the notation $(x,t,y)$. Also, the Fourier variables are denote by $(\xi,\eta,y)\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}\times{\mathbb{R}}$. Sometimes we denote $X=(x,t)$ for $x\in\mathbb{R}^{n-1}$ and $t\in\mathbb{R}$. For sake of shortness we also use the notation $dX=dxdt$.

For any $r>0$ and $\widetilde{x}\in\mathbb{R}^{n-1}$, $\widetilde{t}\in\mathbb{R}$ we denote by $B_{r}(\widetilde{x})$ the ball centered at $\widetilde{x}$ with radius $r>0$ and we define $Q^{\prime}_{r}(\widetilde{x})=\{x\in\mathbb{R}^{n-1}:|x_{j}-\widetilde{x}_{j}|\leq r,\,j=1,2,\cdots,n-1\}$, $Q_{r}(\widetilde{X})=Q^{\prime}_{r}(\widetilde{x})\times I_{r}(\widetilde{t})$, where $I_{r}(\widetilde{t})=[-r^{2}+\widetilde{t},r^{2}+\widetilde{t}]$. Whenever $x=0$ we denote $B_{r}=B_{r}(0)$, $Q^{\prime}_{r}=Q^{\prime}_{r}(0)$ and $Q_{r}=Q_{r}(0)$.

By $H_{\pm}=\chi_{{\mathbb{R}}^{n-1}\times{\mathbb{R}}\times{\mathbb{R}}_{\pm}}$ we denote the characteristic function of ${\mathbb{R}}^{n-1}\times{\mathbb{R}}\times{\mathbb{R}}_{\pm}=\{(x,t,y)\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}\times{\mathbb{R}}\,|\,y\gtrless 0\}$.

In places we use equivalently the symbols $D$, $\partial$ to denote the gradient of a function and we add the index $x$ or $y$ to denote the gradient in $\mathbb{R}^{n-1}$ and the derivative with respect to $y$, respectively.

Let $u_{\pm}\in C^{\infty}({\mathbb{R}}^{n-1}\times{\mathbb{R}}\times{\mathbb{R}})$; we define

[TABLE]

where $\sum_{\pm}a_{\pm}=a_{+}+a_{-}$.

For a function $f\in L^{2}(\mathbb{R}^{n})$, we define

[TABLE]

Moreover we define ([LM])

[TABLE]

and, for $\alpha\in[0,1)$,

[TABLE]

As usual we denote by $H^{\frac{1}{2},0}(\mathbb{R}^{n})$ and $H^{0,\alpha}(\mathbb{R}^{n}$ the spaces of the functions $f\in L^{2}(\mathbb{R}^{n})$ satisfying, respectively,

[TABLE]

and

[TABLE]

with the norms

[TABLE]

and

[TABLE]

We also use the notation $H^{\frac{1}{2},\alpha}(\mathbb{R}^{n})=H^{\frac{1}{2},0}(\mathbb{R}^{n})\cap H^{0,\alpha}(\mathbb{R}^{n})$ where we consider the norm $||f||_{H^{\frac{1}{2},\alpha}(\mathbb{R}^{n})}=||f||_{L^{2}(\mathbb{R}^{n})}+[f]_{\frac{1}{2},\alpha,\mathbb{R}^{n}}$ where $[f]_{\frac{1}{2},\alpha,\mathbb{R}^{n}}=[f]_{\frac{1}{2},0,\mathbb{R}^{n}}+[f]_{0,\alpha,\mathbb{R}^{n}}$.

Recall that there exist $C_{n}$, depending only on $n$, and $C_{n,\alpha}$, depending only on $n$ and $\alpha$, such that

[TABLE]

[TABLE]

so that the norm $||f||_{H^{\frac{1}{2},\alpha}(\mathbb{R}^{n})}$ is equivalent to $\left(\int_{\mathbb{R}^{n}}(1+|\eta|^{\alpha}+|\xi|)|\hat{f}(\xi,\eta)|^{2}d\xi d\eta\right)^{1/2}$.

Moreover,

[TABLE]

2.2. Differential operator, weight and trace operators

Let us define

[TABLE]

where

[TABLE]

are Lipschitz symmetric matrix-valued functions with real entries that satisfy, for given constants $\lambda_{0}\in(0,1]$ and $M_{0}>0$,

[TABLE]

and

[TABLE]

We also define

[TABLE]

[TABLE]

where $\nu=-e_{n}$.

Remark 2.1**.**

Notice that, if $A(x,t,y)=\sum_{\pm}A_{\pm}(x,t,y)$ and $u$ is a solution to the equation

[TABLE]

then

[TABLE]

Let us now introduce the weight function. Let $\varphi$ be

[TABLE]

where $\alpha_{+}$, $\alpha_{-}$ and $\beta$ are positive numbers which will be determined later. In what follows we denote by $\varphi_{+}$ and $\varphi_{-}$ the restriction of the weight function $\varphi$ to $[0,+\infty)$ and to $(-\infty,0)$ respectively. We use similar notation for any other weight functions. Let, for $\delta>0$,

[TABLE]

The more general weight function that we will use is of the form

[TABLE]

for some positive $b$.

Let us introduce a notation for the operators on the interface that appears in the Carleman estimates:

[TABLE]

[TABLE]

where $h_{0}(u)$ and $h_{1}(u)$ are defined as in (2.10) and (2.11).

We use the letters $C,C_{0},C_{1},\cdots$ to denote constants. The value of the constants may change from line to line, but it is always greater than $1$.

Let us now state our main result.

Theorem 2.2**.**

Let $A_{\pm}(x,t,y)$ satisfy (2.7)-(2.9). There exist $\alpha_{+}$, $\alpha_{-}$, $\beta$, $b$, $\delta_{0}$, $r_{0}$, $\tau_{0}$ and $C$ depending on $\lambda_{0},M_{0}$ such that if $\tau\geq\tau_{0}$, then

[TABLE]

where $u=H_{+}u_{+}+H_{-}u_{-}$, $u_{\pm}\in C^{\infty}(\mathbb{R}^{n+1})$ and ${\rm supp}\,u\subset Q_{\delta_{0}/2}\times(-\delta_{0}r_{0},\delta_{0}r_{0})$, $\Phi(x,t,y)=\phi_{\delta_{0}}(x,y)-\frac{b}{2}t^{2}$ with $\phi_{\delta_{0}}$ given by (2.13), and $Y^{L}(u,\Phi)$ and $Y^{R}(u,\Phi)$ are defined in (2.14) and (2.15), respectively.

Remark 2.3**.**

It is clear that estimate (2.16) remains true for large enough $\tau_{0}$ if the operator $\mathcal{L}$ is substituted by

[TABLE]

for bounded functions $W$ and $V$.

We divide the proof of Theorem 2.2 in 3 main steps. In Step 1, we consider a leading coefficient $A_{\pm}$ depending only on $y$ and a weight function linear in $x$ and independent of $t$. In Step 2, we take a general leading coefficient and a weight quadratic in $x$ but independent of $t$. In Step 3 we add the dependence on $t$ of the weight function.

3. Step 1: leading coefficient depending on $y$ only

In this section we consider the simple case of the leading matrices (2.7) independent of $x$ and $t$. Moreover, the weight function that we consider is linear with respect to $x$ variable, so that, as explained above, the Carleman estimates we get here are only preliminary to the one that we will get in the general case.

Assume that

[TABLE]

are symmetric matrix-valued functions satisfying (2.8) and (2.9), i.e.,

[TABLE]

[TABLE]

From (3.1), we have

[TABLE]

In the present case the differential operator (2.6) becomes

[TABLE]

and (2.11) becomes

[TABLE]

where $u=\sum_{\pm}H_{\pm}u_{\pm}$, $u_{\pm}\in C^{\infty}({\mathbb{R}}^{n+1})$ and $\nu=-e_{n}$.

We also set, for any $s\in[0,1]$ and $\gamma\in\mathbb{R}^{n-1}$ with $|\gamma|\leq 1$

[TABLE]

where $\varphi$ is defined in (2.12). Notice that this weight function is linear in $x$, hence this is not a special case of (2.13).

Our aim in this step is to prove:

Theorem 3.1**.**

Let $\mathcal{L}$ be the operator (3.3). There exist $\alpha_{\pm}$, $\beta$, $\tau_{0}$, $s_{0}$, $r_{0}$ and $C$ depending only on $\lambda_{0}$, $M_{0}$, such that for $\tau\geq\tau_{0}$, $0<s\leq s_{0}<1$, and for every $w=\sum_{\pm}H_{\pm}w_{\pm}$ with ${\rm supp}\,w\subset Q_{1}\times[-r_{0},r_{0}]$, we have that

[TABLE]

*where $\phi(x,y)$ is given by (3.4) and $Y^{L}$ and $Y^{R}$ where defined in (2.14) and (2.15), respectively. *

3.1. Fourier transform of the conjugate operator and its factorization

To proceed further, we introduce some operators and find their properties. We use the notation $\partial_{j}=\partial_{x_{j}}$ for $1\leq j\leq n-1$.

Let us define

[TABLE]

and set

[TABLE]

Let us define the operator

[TABLE]

In view of (3.1) we have

[TABLE]

[TABLE]

where $\lambda_{1}\leq\lambda_{0}$ depends only on $\lambda_{0}$, and $M_{1}$ depends on $\lambda_{0}$ and $M_{0}$ only.

It is easy to show, by direct calculations ([LRLer]), that

[TABLE]

In order to derive the Carleman estimate (3.5) we conjugate the operator $\mathcal{L}$ with $e^{\tau\phi}$ for $\phi$ given by (3.4) and get (see [DCFLVW] for further details)

[TABLE]

Now, we focus on the analysis of $e^{\tau\phi}\mathcal{L}(e^{-\tau\phi}v)$ and introduce some notations:

[TABLE]

for $b_{jk}^{\pm}$ as in (3.6),

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

By (3.1), we have

[TABLE]

where $\mathcal{F}(\cdot)$ is the Fourier transform with respect to $(x,t)$ and

[TABLE]

Our aim is to estimate $\mathcal{F}\left(e^{\tau\phi}\mathcal{L}(e^{-\tau\phi}v)\right)$ from below. In order to do this, we want to factorize the operators $P_{s,\pm}$.

For any $z=a+ib$ with $(a,b)\neq(0,0)$, we define the square root of $z$,

[TABLE]

It should be noted that $\Re\sqrt{z}\geq 0$.

Let us set

[TABLE]

and

[TABLE]

where

[TABLE]

We denote by

[TABLE]

and

[TABLE]

Notice that, since $|\gamma|\leq 1$ and $0\leq s\leq 1$, we have:

[TABLE]

[TABLE]

[TABLE]

where $C$ depends only on $\lambda_{0}$.

Moreover, for $s=0$ we have,

[TABLE]

We always assume that the constants $\alpha_{+}$ and $\alpha_{-}$ in the weight (2.12) are fixed in such a way that

[TABLE]

where $L$ was given in (3.11).

Proposition 3.2**.**

There exist $\tau_{0}$, $\overline{s}$, $\rho$, $\beta$ and $C$, depending only on $\lambda_{0}$ and $M_{0}$ such that for $\tau\geq\tau_{0}$, ${\rm supp}\,\hat{v}_{\pm}(\xi,\cdot)\subset[-\rho,\rho]$, $s\leq\overline{s}<1$ we have:

[TABLE]

Proof of Theorem 3.1. We integrate (3.2) with respect to $\xi$ and $\eta$. Since $\Lambda_{1}$ is positive and 1-homogeneous with respect to $|\xi|$, $\tau$ and $|\eta|^{1/2}$, $\Lambda_{1}$ and its the powers appearing in (3.2) can be bounded from below and above by polynomials with the same degree (with respect to $|\xi|$, $\tau$ and $|\eta|^{1/2}$). We can then choose the suitable polynomials that, thanks to (2.3) and (2.4), give (3.5). $\square$

3.2. Proof of Proposition 3.2

Let us define two operators

[TABLE]

[TABLE]

With all the definitions given above, we thus obtain that

[TABLE]

[TABLE]

Similarly to the elliptic case, we distinguish three cases:

[TABLE]

where $s_{0}$ will be chosen later and

[TABLE]

notice that by (3.20) we have $\kappa>0$.

3.2.1. First case

In this case we assume

[TABLE]

Proposition 3.3**.**

There exist a constant $C$ depending on $\lambda_{0}$ and $M_{0}$ such that

[TABLE]

[TABLE]

[TABLE]

Proof.

The proof of estimate (3.25) follows the same lines of the proof of Lemma 3.2 in [DCFLVW]. Estimates (3.26) and (3.27) easily follow from (3.19), (3.1) and (3.8). $\square$

Lemma 3.4**.**

Let $\tau\geq 1$ and assume (3.24). There exists a positive constant $C$ depending only on $\lambda_{0}$ and $M_{0}$ such that, if $0<s\leq s_{0}\leq 1/C$, we have

[TABLE]

and

[TABLE]

for ${\rm supp}(\hat{v}_{-}(\xi,\cdot))\subset[-\frac{1}{C\beta},\frac{1}{C\beta}]$.

Proof.

Define

[TABLE]

We have

[TABLE]

where we omit the arguments $(\xi,\eta,y)$ for sake of shortness. Since, by (3.2) and (3.19), there exists a constant $C_{0}$, depeding only on $\lambda_{0}$ and $M_{0}$, such that

[TABLE]

and by (3.24) (recalling that $y>0$) we can estimate

[TABLE]

provided

[TABLE]

Combining (3.2.1), (3.32) and (3.33) and the fact that, by (3.24),

[TABLE]

yields

[TABLE]

where $C$ depends only on $\lambda_{0}$ and $M_{0}$ and provided (3.34) holds true.

Similarly, we have that

[TABLE]

The assumption (3.24) and (3.32) imply

[TABLE]

provided

[TABLE]

Thus, if (3.34) and (3.38) hold true, by (3.37) and (3.32) we have

[TABLE]

Also by (3.1), (3.12), and the fact that

[TABLE]

for some $C$ depending only on $\lambda_{0}$, we have that, for any $\varepsilon\leq 1$

[TABLE]

Choosing $\varepsilon$ sufficiently small, we obtain, from (3.2.1), (3.39) and (3.40),

[TABLE]

where $C$ depends only on $\lambda_{0}$ and $M_{0}$.

Recalling (3.30) and combining (3.35) and (3.41) yields

[TABLE]

where $C$ depends only on $\lambda_{0}$ and $M_{0}$.

By (3.42), (3.25), and by (3.26), for $s_{0}$ small enough (3.28) follows.

The proof of(3.29) follows the same path, the only difference is that, in the proof of (3.37) the assumption that ${\rm supp}(\hat{v}_{-}(\xi,\cdot))\subset[-\frac{1}{C\beta},\frac{1}{C\beta}]$ comes into play. $\square$

In order to conclude the proof of Proposition 3.2 in this case, we need to connect the traces of the function for $y=0$ to the transmission conditions $\sigma_{0}$ and $\sigma_{1}$. This is done in next Lemma.

Lemma 3.5**.**

Let $\tau\geq 1$ and assume (3.24). There exists a positive constant $C$ depending only on $\lambda_{0}$ and $M_{0}$ such that if $s_{0}\leq 1/C$ then

[TABLE]

if ${\rm supp}(\hat{v}_{\pm}(\xi,\cdot))\subset[-\frac{1}{C\beta},\frac{1}{C\beta}]$.

Proof.

It follows from (3.28) that, for some $C$ depending only on $\lambda_{0}$ and $M_{0}$,

[TABLE]

Since by (3.15) we have

[TABLE]

and since by (3.16)

[TABLE]

hence, using (3.44),

[TABLE]

where $C$ depends only on $\lambda_{0}$ and $M_{0}$.

By (3.14) and (3.45),

[TABLE]

In a similar way, By (3.13) and (3.45), we have that

[TABLE]

By putting together (3.45), (3.46) and (3.47) , we obtain

[TABLE]

that, together with (3.28) and (3.29) of Lemma 3.4, gives (3.43). $\square$

Remark 3.6**.**

Since $\tau\geq 1$, we can write (3.43) in the following weaker form

[TABLE]

where $C$ depends on $\lambda_{0}$ and $M_{0}$ only.

3.2.2. Some useful estimates

In this section we write down some estimates that will be useful in second and third cases of the main proof. In both these cases we have

[TABLE]

for $C_{1}$ independent on $s$. Notice that

[TABLE]

Lemma 3.7**.**

If (3.48) holds, there are two constants $C$ and $s_{1}$ depending on $\lambda_{0}$ and $C_{1}$ only, such that, if $s\leq s_{1}$, then

[TABLE]

Proof.

For a fixed $y$ two cases occur

[TABLE]

In case (3.50a), by (3.7) and (3.10), we have

[TABLE]

Moreover by (3.7) we have

[TABLE]

Now either

[TABLE]

or

[TABLE]

If case (i) occurs then we have

[TABLE]

hence by (3.48) and (3.51) we get (3.49) where $C$ depends on $\lambda_{0}$ only.

If case (ii) occurs then, by (3.7) and (3.50a) we have

[TABLE]

hence

[TABLE]

so that

[TABLE]

this inequality combined with (3.48) yields again (3.49).

Now, let us consider case (3.50b). By this condition and by (3.7) we have

[TABLE]

hence

[TABLE]

This inequality and (3.48) give

[TABLE]

by this inequality we have, for $s\leq\frac{\lambda_{1}}{2C_{1}}$,

[TABLE]

that combined with (3.52) gives, for $s\leq\frac{\sqrt{\lambda_{0}}\lambda_{1}}{4C_{1}}$

[TABLE]

By (3.52), (3.53), (3.54) we get again (3.49). $\square$

Lemma 3.8**.**

If (3.48) holds, there are two constants $C$ and $s_{1}$ depending on $\lambda_{0}$, $M_{0}$ and $C_{1}$ only, such that, if $s\leq s_{1}$, then

[TABLE]

[TABLE]

[TABLE]

Proof.

By (3.7) and (3.8) we have

[TABLE]

by this inequality, (3.48) and (3.49) we obtain (3.55).

Inequalities (3.56) follow immediately by (3.55).

In order to prove (3.57) we denote

[TABLE]

so that we have

[TABLE]

For a fixed $y$ we distinguish two cases

[TABLE]

If case (3.58a) occurs then by (3.49) we have

[TABLE]

Hence, if (3.58a) is satisfied then (3.57) is true.

Now, let us consider case (3.58b). First, let us notice that in such a case (3.58b), by (3.7) we have

[TABLE]

By (3.48) and (3.59), we get

[TABLE]

so that, for $s\leq\frac{\lambda_{1}}{2C_{1}}$, we have

[TABLE]

Now, by (3.59) and (3.60) we have, for $s\leq\frac{\sqrt{\lambda_{0}}\lambda_{1}}{4C_{1}}$,

[TABLE]

this inequality combined with (3.59) and (3.60) gives (3.57) whenever (3.58b) is satisfied. The proof is completed. $\square$

Lemma 3.9**.**

If (3.48) holds, there is a constant $C$ depending on $\lambda_{0}$ and $C_{1}$ only, such that

[TABLE]

[TABLE]

[TABLE]

Proof.

By (3.10) and (3.48) we have

[TABLE]

and (3.61) follows. Inequalities (3.62) (3.63) are immediate consequences of (3.57) and (3.61). $\square$

Lemma 3.10**.**

If (3.48) holds, there are two constants $C$ and $s_{1}$ depending on $\lambda_{0}$, $M_{0}$ and $C_{1}$ only, such that, if $s\leq s_{1}$, then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

Inequality (3.64) is an immediate consequence of (3.7), (3.8) and (3.48). Inequality (3.65) follows by (3.62) and (3.64). Inequalities (3.66) and (3.67) follow by (3.55) and (3.61). $\square$

3.2.3. Second case

In this case we assume

[TABLE]

Let us point out that in this case by (3.19)

[TABLE]

By (3.22), (3.23), (3.16), (3.55) and (3.69), we have

[TABLE]

[TABLE]

where $C$ depends on $\lambda_{0}$ and $M_{0}$ only.

Lemma 3.11**.**

Assume (3.68). There exist $C$, $s_{2}\leq s_{1}$ depending only on $\lambda_{0}$ and $M_{0}$ such that, if $0\leq s\leq s_{2}$, $\beta\geq C$ and $\tau\geq C$, then we have

[TABLE]

and

[TABLE]

provided ${\rm supp}(\hat{v}_{+}(\xi,\eta,\cdot))\subset[0,\frac{1}{C}]$.

Proof.

Define

[TABLE]

For sake of shortness we omit arguments unless they are necessary.

We have, by integration by parts,

[TABLE]

Since $R_{s,+}\geq 0$, for $s$ small enough

[TABLE]

and, hence, by (3.56) and (3.68),

[TABLE]

Moreover, for $s$ small enough

[TABLE]

hence

[TABLE]

that is (3.72) because

[TABLE]

Let us now consider (3.73). Let us write

[TABLE]

By (3.68) and (3.67) we have

[TABLE]

for $s$ small enough and $C$ depending on $\lambda_{0}$ and $\alpha_{\pm}$ only.

Moreover, by (3.68) and (3.67) ,

[TABLE]

for positive and small enough $y$ and $s$.

Hence, by (3.56), (3.69) and (3.76) we have

[TABLE]

for $\beta\geq C$.

By putting together (3.74), (3.75), (3.77) and (3.17), we finally get

[TABLE]

that, combined with (3.70) and (3.72) gives (3.73). $\square$

Lemma 3.12**.**

Assume (3.68). There exist $C$ depending only on $\lambda_{0}$ and $M_{0}$ such that, if $0\leq s\leq s_{1}$, $\beta\geq C$ and $\tau\geq C$, then we have

[TABLE]

and

[TABLE]

provided ${\rm supp}(\hat{v}_{-}(\xi,\eta,\cdot))\subset[-\frac{\alpha_{-}}{2\beta},0]$.

Proof.

Define

[TABLE]

We have

[TABLE]

By (3.56), (3.18) and (3.69) we have

[TABLE]

for $\beta$ sufficiently large, and

[TABLE]

and, since

[TABLE]

we finally get (3.78).

Let us now estimate

[TABLE]

Then, since if ${\rm supp}(\hat{v}_{-}(\xi,\eta,\cdot))\subset[-\frac{\alpha_{-}}{2\beta},0]$, provided $s$ small enough, we have

[TABLE]

and estimating the remaing terms as we did before, by (3.78) and (3.71) we finally get (3.79). $\square$

Lemma 3.13**.**

Assume (3.68). there exist $C$, $s_{2}\leq s_{1}$ depending on $\lambda_{0}$ and $M_{0}$ such that if $0\leq s\leq s_{0}$, $\beta\geq C$, $\tau\geq C$ and ${\rm supp}(\hat{v}(\xi,\eta,\cdot))\subset[-\frac{1}{C},\frac{1}{C}]$ then we have

[TABLE]

Proof.

The proof is the same of Lemma 3.5. $\square$

Remark 3.14**.**

Since, by (3.68), $\frac{\Lambda_{1}}{\tau}\leq C$, we can write (3.80) in the following more convenient form

[TABLE]

where $C$ depends on $\lambda_{0}$ and $M_{0}$ only.

3.2.4. Third case

In this case we assume

[TABLE]

Notice that, by (3.32), in this case condition (3.48) is verified. Moreover

[TABLE]

Lemma 3.15**.**

Assume (3.81). There exists a positive constant $C$ depending on $\lambda_{0}$, $M_{0}$ such that, if $0\leq s\leq s_{1}$ and $\tau\geq C$, then

[TABLE]

Furthermore, if ${\rm supp}(\hat{v}_{-}(\xi,\eta,\cdot))\subset[-\frac{1}{C},0]$, then

[TABLE]

Proof.

We have, integrating by parts,

[TABLE]

By (3.57) and (3.56), for $s$ small, we can write

[TABLE]

for $\tau\geq 2C^{2}$.

The term $\int_{0}^{+\infty}\!\!\!\!\left|\partial_{y}\omega_{s,+}+i(T_{+}(\xi,y)-J_{s,+})\omega_{s,+}\right|^{2}$ can be estimated from velow by the usual trick. Therefore (3.85) and (3.86) yield (3.83). The proof of (3.84) is essentially the same provided ${\rm supp}(\hat{v}_{-}(\xi,\eta,\cdot))\subset[-\frac{1}{C},0]$. $\square$

Lemma 3.16**.**

Assume (3.81). There exists a positive constant $C$, depending on $\lambda_{0},M_{0}$, such that if $0\leq s\leq s_{1}$, $\tau\geq C$, and $\beta\geq C$, then, for ${\rm supp}(\hat{v}_{+}(\xi,\cdot))\subset[0,\frac{1}{\beta}]$, we have that

[TABLE]

Proof.

We have, integrating by parts,

[TABLE]

Notice that, for $0\leq y\leq\frac{1}{\beta}$ and $0\leq s\leq 1/C$,

[TABLE]

We want now to estimate

[TABLE]

We distinguish two cases:

if $\tau\leq\frac{\Lambda_{1}}{2C_{*}(\alpha_{+}+2)}$, we can write

[TABLE]

if $\tau\geq 2(2C_{*})^{2}\overline{C}$.

On the other hand, if $\tau\geq\frac{\Lambda_{1}}{2C_{*}(\alpha_{+}+2)}$, then

[TABLE]

for $\beta\geq 4C_{*}\overline{C}(\alpha_{+}+1)$ and since $\frac{\Lambda_{1}}{\tau}\leq C$.

In each of these two cases we have

[TABLE]

By (3.88) and (3.89) and by the fact that $\tau\alpha_{+}-R_{s,+}\geq-C\Lambda_{1}$, we have

[TABLE]

if $\varepsilon$ is small enough. $\square$

Lemma 3.17**.**

Assume (3.81). There exist positive constants $C$, $s_{3}$ depending on $\lambda_{0}$, $M_{0}$, such that if $0\leq s\leq s_{3}$, and for ${\rm supp}\omega_{s,-}(\xi,\eta,\cdot)\subset[-1/C,0]$, we have that

[TABLE]

Proof.

We first compute by integration by parts,

[TABLE]

Since $y\leq 0$ and by (3.18) and (3.67) we can write

[TABLE]

On the other hand, by (3.11) and (3.81),

[TABLE]

By (3.91), (3.92) and (3.93) and for sufficiently small $|y|$ and $s$ we have

[TABLE]

which implies (3.90) by (3.82). $\square$

Lemma 3.18**.**

Assume (3.81). There exist constants $C$ and $s_{4}=\min\{s_{1},s_{2},s_{3}\}$, depending only on $\lambda_{0}$ and $M_{0}$, such that if $0\leq s\leq s_{4}$, $\tau\geq C$, $\beta\geq C$, then for ${\rm supp}(\hat{v}(\xi,\cdot))\subset\left[-\frac{1}{C},\frac{1}{C}\right]$ we have

[TABLE]

and

[TABLE]

Proof.

Inequality (3.94) follows from (3.83) and (3.70). Similarly, (3.95) follows from (3.90) and (3.71). $\square$

Lemma 3.19**.**

Assume (3.81). There exist constants $C$ and $s_{4}$, depending only on $\lambda_{0}$ and $M_{0}$, such that if $0\leq s\leq s_{4}$, $\tau\geq C$, $\beta\geq C$, then for ${\rm supp}(\hat{v}(\xi,\eta,\cdot))\subset\left[-\frac{1}{C},\frac{1}{C}\right]$ we have

[TABLE]

Proof.

By recalling (3.13) and (3.14) and by (3.16), we have

[TABLE]

On the other hand, by (3.57)

[TABLE]

By (3.97) and (3.98) we have

[TABLE]

Recalling (3.13), by (3.99), (3.94), (3.95), (3.97) and (3.98) we have

[TABLE]

By triangle inequality

[TABLE]

and, by (3.14), (3.100) and Lemma 3.18, we have

[TABLE]

Now, by (3.84), (3.87), (3.94), (3.95) and (3.100), we get

[TABLE]

Now, by adding up (3.100), (3.101) and (3.102) and by absorbing the term $C\Lambda_{1}^{2}\sum_{\pm}\|\hat{v}_{\pm}\|$ by the left hand side, we finally get (3.96). $\square$

3.2.5. Conclusion

By putting together the three cases and noticing that, by definition of $P_{s,\pm}$, we can write

[TABLE]

we finally proved Proposition 3.2.

4. Step 2 - Carleman estimate for general coefficients with weight independent of $t$

In the previous section we have proved the Carleman estimate when $A_{\pm}=A_{\pm}(y)$. Now we want to derive it for $A_{\pm}(x,t,y)$. To achieve this purpose, similarly to the elliptic case, we approximate $A_{\pm}(x,y,t)$ with coefficients depending on $y$ only and we make use of a special kind of partition of unity introduced in the next section. At the same time we consider a weight function that is quadratic in $x$.

4.1. Partition of unity and auxiliary results

In this section we collect some results on a partition of unity that we use in our proof and we describe how this partition of unity behaves with respect to the function spaces that we use.

Let $\vartheta_{0},\widetilde{\vartheta}_{0}\in C^{\infty}_{0}(\mathbb{R})$ such that

[TABLE]

and

[TABLE]

Let $\vartheta_{n-1}(x)=\vartheta_{0}(x_{1})\cdots\vartheta_{0}(x_{n-1})$. Given $\mu\geq 1$ and $g\in\mathbb{Z}^{n}$, $g=(g^{\prime},g_{n})$, where $g^{\prime}\in\mathbb{Z}^{n-1}$ and $g_{n}\in\mathbb{Z}$, we define

[TABLE]

and

[TABLE]

Notice that

[TABLE]

[TABLE]

and

[TABLE]

where $C_{1}\geq 1$ depends only on $n$.

Since, for any $\overline{g}\in\mathbb{Z}^{n}$,

[TABLE]

we can define

[TABLE]

and

[TABLE]

Hence we have that

[TABLE]

where $C_{2}\geq 1$ depends on $n$. In Section 2 we have recalled the definition of $H^{\frac{1}{2},0}(\mathbb{R}^{n})$, $H^{0,\alpha}(\mathbb{R}^{n})$, $\alpha\in[0,1)$ and their seminorms $[\cdot]_{\frac{1}{2},0,\mathbb{R}^{n}}$, $[\cdot]_{0,\alpha,\mathbb{R}^{n}}$, respectively, in what follows we also need the seminorms

[TABLE]

[TABLE]

where $Q_{r}=Q_{r}(0)$.

In the rest of this subsection we give the statements of some lemmas and propositions that we use in the sequel, their proofs are the same of the ones given in [DCFLVW] with the obvious changes. Since the constants of various inequalities always depends on $n$ or on $\alpha$ we will omit such dependence.

Lemma 4.1**.**

Let $f\in C^{\infty}(\mathbb{R}^{n})$ and ${\rm supp}f\subset Q_{3r/4}$ for some $r\leq 1$. There exists a constant C such that we have

[TABLE]

[TABLE]

Proposition 4.2**.**

Let $\{\varsigma_{g}\}_{g\in\mathbb{Z}^{n}}$ be a family of smooth functions such that ${\rm supp}\,\varsigma_{g}$ in contained in the interior of $Q_{3/2\mu}(X_{g})$, then

[TABLE]

[TABLE]

Proposition 4.3**.**

Let $F\in C^{\infty}(\mathbb{R}^{n})$ with ${\rm supp}\,F\subset Q_{3/2\mu}(X_{g})$, let $\beta\in(\alpha,1]$ and let $a$ be a function satisfying

[TABLE]

for $(x,t),(x^{\prime},t^{\prime}),(x^{\prime\prime},t^{\prime\prime})\in Q_{3/2\mu}(X_{g})$ and $E_{a}$, $K_{a}$, $\widetilde{K}_{a}$ positive constants. Then we have

[TABLE]

and

[TABLE]

Proof.

The proof of (4.7) is exactly the same to the proof of [DCFLVW, Proposition 4.2], hence we limit ourselves to the proof of (4.8).

We have

[TABLE]

$\square$

Proposition 4.4**.**

Let $f\in C^{\infty}(\mathbb{R}^{n})\cap H^{\frac{1}{2},\alpha}(\mathbb{R}^{n})$. Then

[TABLE]

and

[TABLE]

Proposition 4.5**.**

Let $f\in C^{\infty}(\mathbb{R}^{n-1})\cap H^{\frac{1}{2},\alpha}(\mathbb{R}^{n})$. Then

[TABLE]

[TABLE]

4.2. Estimate of the left hand side of the Carleman estimate, I

In the present subsection and in the next one we derive the Carleman estimate for general coefficients. In order to make clear the procedure that we follow let us introduce and recall some notation and some definitions. For any $0<\delta\leq 1$ we define

[TABLE]

[TABLE]

and

[TABLE]

where $\nu=-e_{n}$.

Next, for any $g=(g^{\prime},g_{n})\in\mathbb{Z}^{n}$ we define

[TABLE]

Notice that

[TABLE]

and

[TABLE]

Concerning the weight functions, let us introduce the following notation.

[TABLE]

where $\varphi(y)$ is defined in (2.12). In addition we assume that $\alpha_{+},\alpha_{-},\beta$ are fixed positive numbers such that Theorem 3.1 holds for the operator $\mathcal{L}_{\delta,g}$.

Notice that

[TABLE]

In order to estimate the left hand side of (2.16) we define

[TABLE]

[TABLE]

and

[TABLE]

Roughly speaking, $\Xi_{E}(w)$ behaves similarly to the corresponding elliptic term defined in [DCFLVW, (4.23)] and $\Xi_{N}(w)$ is the additional contribution that arises from the parabolic operator.

If we assume that ${\rm supp}\,w\subset\mathfrak{U}:=Q_{1/2}\times[-r_{0},r_{0}]$ and that

[TABLE]

then arguing as in [DCFLVW, Sect. 4.2] we have

[TABLE]

where

[TABLE]

and $C$ depends only on $\lambda_{0},M_{0}$.

In the rest of the present subsection we prove that

[TABLE]

where

[TABLE]

and $C$ depends only on $\lambda_{0},M_{0}$.

By (4.3), we can write

[TABLE]

From (4.2), (4.14) and (4.20), we get the following estimate from above of the first term at the righthand side of (4.15)

[TABLE]

In the next Lemma we give some estimates that will be useful in this subsection as well as in subsection 4.3

Lemma 4.6**.**

If $\alpha\in[0,1)$ and ${\rm supp}\,f\subset Q_{3/2\mu}(X_{g})$, then we have that

[TABLE]

[TABLE]

Proof.

For sake of shortness, we only show the proof the inequality on right of (4.22). The proof of inequality on the left is similar and the proof of (4.23) is the same of [DCFLVW, Lemma 4.2].

Denote by

[TABLE]

By (4.14) and by the second of (4.17) we have

[TABLE]

that is (4.22). $\square$

By (4.4), (4.20), (4.5) and (4.22) we have

[TABLE]

Similarly we estimate the third and the fourth term at the righthand side of (4.15) so that, taking into account (4.21), (4.19) follows.

Finally, (4.18) and (4.19) give

[TABLE]

where

[TABLE]

and $C$ depends only on $\lambda_{0},M_{0}$.

4.3. Estimate of the left hand side of the Carleman estimate, II

In this section, we continue to estimate $\Xi(w)$ from above using (4.24). To this aim we apply Theorem 3.1 to the function $w\eta_{g,\mu}$ with the weight function $\psi_{\varepsilon,g^{\prime}}=\varphi(y)-\varepsilon x_{g^{\prime}}\cdot x+\varepsilon|x_{g^{\prime}}|^{2}/2$. First we notice that if ${\rm supp}\,w\subset\mathfrak{U}:=Q_{1/2}\times[-r_{0},r_{0}]$ and $\mu\geq 4$ then either $|x_{g}^{\prime}|\leq 1$ and $|t_{g_{n}}|\leq 1$ or ${\rm supp}\,\eta_{g,\mu}\cap Q_{1/2}=\emptyset$ so that, in both the cases, we can apply Theorem 3.1.

By applying (3.5) and by adding up with respect to $g\in\mathbb{Z}^{n-1}$, we obtain that

[TABLE]

where

[TABLE]

where we set

[TABLE]

[TABLE]

In order to estimate from above the four terms of (4.25) we would like to point out that $\sum_{g\in\mathbb{Z}^{n}}d^{(0)}_{g,\mu}$ is the ”new” term that arises in this parabolic context, whereas the other terms are basically the same of the corresponding terms of the elliptic case, [DCFLVW, (4.36)] as soon as we notice that by (2.9) and (4.13) we have

[TABLE]

We begin to estimate from above the comparatively new term $\sum_{g\in\mathbb{Z}^{n}}d^{(0)}_{g,\mu}$. First we estimate $\sum_{g\in\mathbb{Z}^{n}}[e^{\tau\psi_{\varepsilon,g^{\prime}}(\cdot,0)}h_{0;g,\mu}(w)]^{2}_{0,\frac{3}{4},\mathbb{R}^{n}}$. By (4.4), (4.10), (4.22) and (4.26) we have

[TABLE]

Now we estimate $\sum_{g\in\mathbb{Z}^{n}}[e^{\tau\psi_{\varepsilon,g^{\prime}}(\cdot,0)}h_{1;g,\mu}(w)]^{2}_{0,\frac{1}{4},\mathbb{R}^{n}}$. By (4.27) it is easy to write $h_{1;g,\mu}(w)$ as

[TABLE]

where

[TABLE]

By (4.2), (4.10) and (4.22) we have

[TABLE]

From (4.2), (4.3) and (4.4) we have

[TABLE]

Now, in order to estimate the first term at the righthand side of (4.31), after using the triangle inequality, we apply (4.3), (4.8) (we choose $\beta=1/2$, $\alpha=1/4$) and (4.12) and we get

[TABLE]

Hence by (4.31) and (4.32) we have

[TABLE]

In order to estimate $\sum_{g\in\mathbb{Z}^{n}}[e^{\tau\psi_{\varepsilon,g^{\prime}}(\cdot,0)}J^{(2)}_{g,\mu}]^{2}_{0,\frac{1}{4},\mathbb{R}^{n}}$ we use (2.5), (4.2), (4.3), (4.4), (4.8), (4.10) and (4.28) we have

[TABLE]

Arguing as before it is simple to obtain

[TABLE]

Finally , combining (4.29), (4.30), (4.33), (4.34) and (4.35) we have

[TABLE]

where

[TABLE]

By (2.8), (4.3) and (4.28) we obtain that

[TABLE]

which, together with (4.2), (4.14) and (4.17), gives

[TABLE]

where

[TABLE]

By (4.2), (4.9), (4.23), (4.26) and (4.9) we have

[TABLE]

where

[TABLE]

Similarly, by (2.8), (4.2), (4.3), (4.11), (4.23) and (4.28) we have

[TABLE]

where

[TABLE]

Now we choose $\varepsilon=\delta$, so that by (4.24), (4.25), (4.36), (4.37), (4.38) and (4.39) we have

[TABLE]

where

[TABLE]

and $C$ depends on $\lambda_{0},M_{0},n$. Now it is easy to note that there exists a sufficiently small $\delta_{0}$ and a sufficiently large $\tau_{0}$, both depending on $\lambda_{0},M_{0},n$ such that if $\delta\leq\delta_{0}$ and $\tau\geq\tau_{0}$, then $R_{4}$ on the right hand side of (4.40) can be absorbed by $\Xi(w)$ (defined in (4.16)). In other words, we have proved that

[TABLE]

Now, applying (4.41) to the function $w(x,t,y)=u(\delta x,\delta^{2}t,\delta y)$, by a standard change of variable we obtain (2.16) with $b=0$.

5. Step 3 - Carleman estimate with weight depending on $t$

In the previous step we have proved that

[TABLE]

Let us now define

[TABLE]

and insert in (5.1) the function $u=ve^{\tau\varphi_{1}(t)}$. It is easy to see that

[TABLE]

[TABLE]

and

[TABLE]

Moreover, by (2.14) and (2.15), we have

[TABLE]

hence, for large enough $\tau$ the extra terms appearing in (5.3) and (5.5) can be absorbed and Theorem 2.2 is finally proved.

6. Three-region inequality

Theorem 6.1**.**

Let $A(x,t,y)=\sum_{\pm}A_{\pm}(x,t,y)$ where $A_{\pm}$ satisfy assumptions (2.8) and (2.9) and let $V$ be bounded function and $W$ a bounded vector valued function such that

[TABLE]

Let $\tau_{0}$, $\alpha_{+}$, $\alpha_{-}$, $\beta$, $b$ and $\delta_{0}$ given by Theorem 2.2, and let

[TABLE]

There exist $C$ and $R$ depending only on $\lambda_{0}$, $M_{0}$ and $n$, such that, if $u$ is a weak solution to the equation

[TABLE]

then, for $0<R_{1},R_{2}\leq R$,

[TABLE]

where

[TABLE]

for

[TABLE]

Notice that function $z$ coincides on ${\mathbb{R}}^{n-1}\times{\mathbb{R}}\times{\mathbb{R}}_{-}$ with the weight function $\Phi$ appearing in Theorem 2.2.

Proof.

We prove the Theorem with the additional assumption $u_{\pm}\in H^{2,1}(Q^{\pm}_{\delta_{0}/2}\times(-\delta_{0}r_{0},\delta_{0}r_{0}))$, where $Q^{\pm}_{\delta_{0}/2}\times(-\delta_{0}r_{0},\delta_{0}r_{0}))=Q_{\delta_{0}/2}\times(-\delta_{0}r_{0},\delta_{0}r_{0}))\cap({\mathbb{R}}^{n}\times{\mathbb{R}}_{\pm})$, in appendix we show that indeed the weak solution $u$ satisfy such an additional assumption. After performing a standard density argument we can apply Theorem 2.2 to the function $u\theta$ where $\theta$ is a cut off function such that ${\rm supp}\,\theta\subset Q_{\delta_{0}/2}\times(-\delta_{0}r_{0},\delta_{0}r_{0})$. We can assume, that $\alpha_{+}>\alpha_{-}$. By following the calculations in the proof of the three-region inequality in the elliptic case ([FLVW, Theorem 3.1]), let us choose

[TABLE]

Given $0<R_{1}<R_{2}\leq R$, let $\theta_{1}(s)\in C^{\infty}({\mathbb{R}})$ such that $0\leq\theta_{1}(s)\leq 1$ and

[TABLE]

and let $\theta_{2}(y)\in C^{\infty}({\mathbb{R}})$ such that $0\leq\theta_{2}(y)\leq 1$ and

[TABLE]

Define

[TABLE]

for $z$ as in (6.1).

The support of function $\theta$ is contained in the set

[TABLE]

notice that, for $R$ given by (6.4), the support of $\theta$ is contained in $Q_{\delta_{0}/2}\times[-\delta_{0}r_{0},\delta_{0}r_{0}]$ (see details in [FLVW]).

We can, hence, apply estimate (2.16) (see also remark 2.3) to the function $\theta(x,t,y)u(x,t,y)$. Let us calculate

[TABLE]

Since $\tilde{\mathcal{L}}(u)=0$ and since the derivatives of $\theta$ are nonzero only on the set

[TABLE]

we have

[TABLE]

where $\left|D_{x,y}u\right|^{2}=\sum_{\pm}H_{\pm}\left|D_{x,y}u_{\pm}\right|^{2}$.

Since $h_{0}(u)=h_{1}(u)=0$, we also have

[TABLE]

[TABLE]

for

[TABLE]

By explicit calculations it is easy to see that

[TABLE]

hence, $\eta(x,t)$ is different from zero only in

[TABLE]

By (2.16), (6.5), (6.6) and (6.7), we have

[TABLE]

Notice that $\tilde{U}\subset\tilde{U}_{1}\cup\tilde{U}_{3}$, where

[TABLE]

and

[TABLE]

The weight function $\Phi$ can be written as

[TABLE]

hence, by (6.4),

[TABLE]

By (6.9),

[TABLE]

A similar estimate can be written for $D_{x,y}u$ instead of $u$. We can, then, use a parabolic Caccioppoli-type inequality and, observing that $\tilde{U}_{1}\subset\subset U_{1}$ and $\tilde{U}_{3}\subset\subset U_{3}$, we can write

[TABLE]

In the set $\mathcal{G}$, that contains the support of $\eta$, we have

[TABLE]

hence

[TABLE]

and, by Proposition 4.3 (applying (4.7) and also (4.8) for $\alpha=1/4$ and $\beta=1/2$), we have

[TABLE]

By putting together (6.8), (6.10), (6.11), (6.12) and (6.13), we get

[TABLE]

On the other hand, since $\mathcal{G}\subset\subset U_{3}$, by using traces and regularity estimates (see [LSU, Theorem 5.1]), we have

[TABLE]

Let us now consider the left-hand side of (6.14). On the set

[TABLE]

we have $\theta\equiv 1$ and $\Phi\geq-R_{2}$, hence

[TABLE]

By (6.14), (6.15) and (6.16) we get, for $\tau\geq\tau_{0}$

[TABLE]

Now we want to choose $\tau$ in order to get (6.3). Let us denote by $\varepsilon^{2}=\int_{U_{1}}\left|u\right|^{2}dXdy$, $E^{2}=\int_{U_{3}}\left|u\right|^{2}dXdy$ so that (6.17) becomes

[TABLE]

Let $\tau_{1}$ be such that

[TABLE]

that is

[TABLE]

If $\tau_{1}>\tau_{0}$, then w can choose $\tau=\tau_{1}$ and (6.18) gives

[TABLE]

that is (6.3).

On the other hand, if $\tau_{1}\leq\tau_{0}$, it means that

[TABLE]

that is

[TABLE]

and, hence, we can write

[TABLE]

that is, again, (6.3).

$\square$

7. appendix

In what follows we assume that $A$, $V$, $W$ satisfy the same assumptions of Theorem 6.1. Moreover, for the sake of brevity, we denote $\Omega_{\delta_{0}}=Q_{\delta_{0}/2}\times(-\delta_{0}r_{0},\delta_{0}r_{0})$, $I_{\delta_{0}}=[-\delta_{0},\delta_{0}]$ and $\Omega_{\delta_{0}/2}^{\pm}=\Omega_{\delta_{0}/2}\cap({\mathbb{R}}^{n}\times{\mathbb{R}}_{\pm})$. We recall that $H^{1,\frac{1}{2}}(\Omega_{\delta_{0}})$ the space of functions $f\in L^{2}(\Omega_{\delta_{0}})$ satisfying $\|f\|_{H^{1,\frac{1}{2}}(\Omega_{\delta_{0}})}<\infty$ where

[TABLE]

and

[TABLE]

Proposition 7.1**.**

Let $u\in H^{1,\frac{1}{2}}(\Omega_{\delta_{0}})$ be a weak solution to the equation (6.2) then $u_{\pm}\in H^{2,1}(\Omega_{\delta_{0}/2}^{\pm})$.

We give a sketch of the proof. We limit ourselves to the case $W=0$, $V=0$. Let $\{A^{(m)}_{\pm}\}_{m\in\mathbb{N}}$ be a sequence of smooth symmetric matrix-valued functions that approximate $A_{\pm}$ in $L^{\infty}(\Omega_{\delta_{0}})$, and that satisfies

[TABLE]

and

[TABLE]

where $\widetilde{\lambda}_{0}$ and $\widetilde{M}_{0}$ depend only on $\lambda_{0}$ and $M_{0}$. For every $m\in\mathbb{N}$, let $u^{(m)}$ be a weak solution to

[TABLE]

where $\Gamma$ is the parabolic boundary of $\Omega_{3\delta_{0}/4}$. Now, see [LSU, Chapter III, Section 13], we have $u^{(m)}_{\pm}\in H^{2,1}(\Omega^{\pm}_{\delta_{1}})$ for every $0<\delta_{1}<\delta_{0}$ and

[TABLE]

where $C$ doesn’t depend by $m$ (it depends only by $\widetilde{\lambda}_{0}$). By (7.1) we have

[TABLE]

Now we apply the Carleman estimate (2.16), for a fixed $\tau$, to the operator $\mathcal{L}$. Since $u^{(m)}_{\pm}\in H^{2,1}(\Omega^{\pm}_{3\delta_{0}/4})$, for every $m\in\mathbb{N}$, after performing a standard density argument we can apply (2.16) to $u^{(m)}\zeta$ where $\zeta\in C^{\infty}_{0}(\Omega_{3\delta_{0}/4})$ is a cut off function such that $0\leq\zeta\leq 1$ and $\zeta=1$ in $\Omega_{\delta_{0}/2}$. It is easy to check that that

[TABLE]

where $C$ depends only on $\widetilde{\lambda}_{0}$ and $\widetilde{M}_{0}$. Moreover by trace inequality and (7.1) we have, for every $m\in\mathbb{N}$

[TABLE]

where $C$, $C^{\prime}$ and $\widetilde{C}$ are independent of $m$.

Now by (2.16), (7.3), (7.4) we get

[TABLE]

where $C$ is independent of $m$. Therefore there exists a subsequence $\{u^{(j_{m})}\}_{m\in\mathbb{N}}$ such that $\{D_{x,y}^{k}u^{(j_{m})}\}_{m\in\mathbb{N}}$ and $\{\partial_{t}u^{(j_{m})}\}_{m\in\mathbb{N}}$ weakly converge in $L^{2}(\Omega_{\delta_{0}/2})$ so that, taking into account (7.2), we conclude the proof. $\square$

Acknowledgement

The authors were partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). EF was partially supported by FIR Project Geometrical and Qualitative Aspects of PDE’s.