In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In Part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the coefficients satisfy suitable conditions and taking the usual weight function $φ(x,t)={\rm e}^{λψ(x,t)}$, $ψ(x,t)=\left|x-x_{0}\right|^{2}-β\left(t-t_0\right)^{2}+βt_0^{2}$ for $x$ in a bounded domain in $\mathbb{R}^{n}$ with $C^{3}$-boundary and $t\in(0, T)$, where $t_0=T/2$. We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding Part II paper \cite{part II}.
We consider the inverse problem of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation.
We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of arbitrary radius. The novelty of this Carleman estimate is that we establish an explicit dependence on the Lipschitz and ellipticity constants, the dimension of the space and the radius of the ball. In particular we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and L…
In this paper we establish some new $L^{2}-L^{2}$ Carleman estimates for the Baouendi-Grushin operators $\mathscr{B}_γ$, in (1.1) below. We apply such estimates to obtain: (i) an extension of the Bourgain-Kenig quantitative unique continuation; (ii) the strong unique continuation property for some degenerate sublinear equations.
We study the problem of recovery the source $a(t,x)F(x)$ in the wave equation in anisotropic medium with $a$ known so that $a(0,x)\not=0$ with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery the sound speed in the equation $u_{tt} -c^2(x)Δu =0$ with one measurement. We give sharp conditions for stability, as well. An application to thermoacoustic tomography is also presented.
We study via Carleman estimates the sharpest possible exponential decay for {\it waveguide} solutions to the Laplace equation $$(\partial^2_t+\triangle)u=Vu+W\cdot(\partial_t,\nabla)u,$$ and find a necessary quantitative condition on the exponential decay in the spatial-variable of nonzero waveguides solutions which depends on the size of $V$ and $W$ at infinity.
We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the continuum setting in which the unique continuation property is known to hold under suitable regularity assumptions. As a key auxiliary result which might be of independent interest we present a Carleman estimate for these discrete operators.
We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for $Δ+ V$. That is, for any non-trivial $u$ that solves $Δu + V u = 0$ in some open, connected subset of $\mathbb{R}^n$, we estimate the vanishing order of solutions in terms of the $L^t$-norm of $V$. Our results apply to all $t > \frac n 2$ and $n \ge 3$. With these maximal order of vanishing estimates, we employ a scaling argument to produce quantitative unique con…
In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [9] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations. We also …
Carleman Estimates for Wave Equations on the Half-Line: AI-Assisted Weights and Applications
University of California, San Diego（加州大学圣地亚哥分校）；Southwest Jiaotong University（西南交通大学）；Worcester Polytechnic Institute（伍斯特理工学院）；University of Missouri-Kansas City（密苏里大学堪萨斯城分校）
Quantitative unique continuation principles for multiscale structures are an important ingredient in a number applications, e.g. random Schrödinger operators and control theory. We review recent results and announce new ones regarding quantitative unique continuation principles for partial differential equations with an underlying multiscale structure. They concern Schrödinger and second order elliptic operators. An important feature is that the estimates are scale free and with quantitative dependence on parameters. These unique continuation principles apply to functions satisfying certain `r…
We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established.
We investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth manifolds. Quantitative uniqueness is a quantitative form of strong unique continuation property. We characterize quantitative uniqueness by the rate of vanishing. We can obtain the vanishing order of solutions by $C^{1, 1}$ norm of the potential functions, as well as the $L^\infty$ norm of the coefficient functions. Some quantitative Carleman estimates and three cylinder inequalities are established.
Carleman estimates for backward Cahn-Hilliard-reaction-diffusion problems
University of Pavia（帕维亚大学）；New York University Abu Dhabi（纽约大学阿布扎比分校）；University of Milan（米兰大学）；Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, CNR（恩里科·马格内斯应用数学与信息技术研究所，意大利国家研究委员会）；The University of Tokyo（东京大学）
In this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials. We derive lower bounds of decay rate at infinity for any nontrivial solution under some general assumptions. The lower bounds depend on asymptotic behaviors of magnetic and electric potentials. The proof is carried out by the Carleman method and the bootstrapping arguments.
We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.
We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a $C{^1}$ potential $V$. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.
In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.
In this paper we study the local behavior of a solution to the Stokes system with singular coefficients. One of the main results is the bound on the vanishing order of a nontrivial solution to the Stokes system, which is a quantitative version of the strong unique continuation property. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial \emph{optimal} three-ball inequalities. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution to the Stokes system from those three-ball inequal…
In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the Lebesgue norms of the lower order terms for all admissible exponents. Then we show that a scaling argument allows us to pass from these vanishing order estimates to estimates for the rate of decay of solutions at infinity. Our proofs rely on a new $L^p - L^q$ Carleman estimate for the Laplacian in $\mathbb{R}^2$.
In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schrödinger equations) on a compact, smooth Riemannian manifold, $(M,g)$, without boundary. Moreover, with only slight modifications these results generalize to equations with $C^1$ potentials. Here Carleman estimates are a key tool. These yield a quantitative three balls inequality which implies quantitative bulk and boundary doubling estimates and hence leads to the control of the maximal order of vanishing. Using the boundary…
In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated from a recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.
In this article we prove quantitative unique continuation results for wave operators of the form $\partial$ 2 t -- div(c(x)$\nabla$$\bullet$) where the scalar coefficient c is discontinuous across an interface of codimension one in a bounded domain or on a compact Riemannian manifold. We do not make any assumptions on the geometry of the interface or on the sign of the jumps of the coefficient c. The key ingredient is a local Carleman estimate for a wave operator with discontinuous coefficients. We then combine this estimate with the recent techniques of Laurent-L{é}autaud [LL19] to propagate …
We revisit the regularity theory of Escauriaza, Seregin, and Šverák for solutions to the three-dimensional Navier-Stokes equations which are uniformly bounded in the critical $L^3_x(\mathbf{R}^3)$ norm. By replacing all invocations of compactness methods in these arguments with quantitative substitutes, and similarly replacing unique continuation and backwards uniqueness estimates by their corresponding Carleman inequalities, we obtain quantitative bounds for higher regularity norms of these solutions in terms of the critical $L^3_x$ bound (with a dependence that is triple exponential in natur…
In this article, we study the quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms. We quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when $u$ is a non-trivial solution to $\triangle u + W \cdot \nabla u + V u = 0$ in some open, connected subset of $\mathbb R^n$, where $n \geq 3$, we characterize the vanishing order of solutions in terms of the norms of $V$ and $W$ in their respective Lebesgue spaces. Using these maximal order of vanishing estimates, we also establish quantitat…
We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty L_x^q}<\infty$ where $r=\sqrt{x_1^2+x_2^2}$ and either $q\in(3,\infty)$, or $u$ is axisymmetric and $q\in(2,3]$. Using the strategy of Tao (2019), we obtain improved subcritical estimates for such solutions depending only on the double exponential of the critical norm. One consequence is a double logarithmic lower bound on the blowup rate. We make use of som…
The question of unique continuation of harmonic functions in a domain $Ω$ $\subset$ R d with boundary $\partial$$Ω$, satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset $ω$ of the boundary is a classical problem. When $ω$ contains an open subset of the boundary it is a consequence of Carleman estimates (uniqueness for second order elliptic operators across an hypersurface). The case where $ω$ is a set of positive (d -- 1) dimensional measure has attracted a lot of attention, see e.g. [10, 3, 15], where qualitative results have been obtained in various sit…
In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequality, for the second order elliptic equation with jump discontinuous coefficients. The derivation of the inequality relies on the Carleman estimate proved in our previous work. We then apply the three-region inequality to study the size estimate problem with one boundary measurement.
In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru, Robbiano-Zuily and Hörmander. We provide local stability estimates that can be propagated, leading to global ones. Then, we specify the previous results to the wave operator on a Riemannian manifold $\mathcal{M}$ with boundary. For this operator, we also prove Carleman estimates and local quantitative unique continuation from and up to the boundary $\partial \mathcal{M}$. This allows us to obtain a global stability estimate from any…
This paper is concerned with quantitative estimates for the Navier-Stokes equations. First we investigate the relation of quantitative bounds to the behaviour of critical norms near a potential singularity with Type I bound $\|u\|_{L^{\infty}_{t}L^{3,\infty}_{x}}\leq M$. Namely, we show that if $T^*$ is a first blow-up time and $(0,T^*)$ is a singular point then $$\|u(\cdot,t)\|_{L^{3}(B_{0}(R))}\geq C(M)\log\Big(\frac{1}{T^*-t}\Big),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}).$$ We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretel…
The aim of the paper is twofold. Firstly, we would like to derive quantitative uniqueness estimates for solutions of the general complex conductivity equation. It is still unknown whether the \emph{strong} unique continuation property holds for such equations. Nonetheless, in this paper, we show that the unique continuation property in the form of three-ball inequalities is satisfied for the complex conductivity equation under only Lipschitz assumption on the leading coefficients. The derivation of such estimates relies on a delicate Carleman estimate. Secondly, we study the problem of estimat…
We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequ…
Quantitative Landis-type result for Dirac operators
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；Ikerbasque, Basque Foundation for Science（伊克拉巴斯科，巴斯克科学基金会）；Universidad del País Vasco / Euskal Herriko Unibertsitatea（巴斯克大学）
Approximate null-controllability of discrete heat equations with potentials on lattices
Facultad de Ciencia y Tecnología, Universidad del País Vasco /Euskal Herriko Unibertsitatea (UPV/EHU)（巴斯克大学）；Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学）；Louisiana State University（路易斯安那州立大学）
