Zhonghua Liao, Qi Lü
We establish a strong unique continuation property for stochastic parabolic equations. Our method is based on a suitable stochastic version of Carleman estimate. As far as we know, this is the first result for strong unique continuation property of stochastic partial differential equations.
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This work is devoted to the strong unique continuation problem for second order parabolic equations with nonsmooth coefficients. Introduction and bibliography have been revised.
In this paper, we establish space like strong unique continuation property (sucp) for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established by Ruland in [Ru] for analogous elliptic sublinear equations. Similar to that in [Ru], this is accomplished via a new $L^{2}-L^{2}$ type Carleman estimate for a class of sublinear parabolic operators.
It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is global in time and we give an example when it never becomes a classical solution again.
In this article, for a fourth-order parabolic equation which is closely related for example to the Cahn-Hilliard equation, we study an inverse source problem by interior data and the continuation of solution from lateral Cauchy data. Our method relies on a Carleman estimate and proves conditional stability for both problems.
In this paper we establish the \emph{space-like} strong unique continuation for nonlocal equations of the type $(\partial_t - Δ)^s u= Vu$, for $0<s <1$. The proof of our main result, Theorem 1.1, is achieved via a conditional elliptic type doubling property for solutions to the appropriate extension problem, followed by a blowup analysis.
In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality \[ |\operatorname{div} (A(x,t) \nabla u) - u_t| \leq \frac{M}{|x|^{2}} |u|,\ \ \ \ \] where the coefficient matrix $A$ is Lipschitz continuous in $x$ and $t$. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends a recent result of one of us with Garofalo and Manna for the heat operator.
We address the quantitative uniqueness properties of the solutions of the parabolic equation $ \partial_t u - Δu = w_j (x,t) \partial_j u + v(x,t) u $ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of vanishing is bounded by $C(\Vert v\Vert_{L^\infty}^{2/3}+\Vert w\Vert_{L^\infty}^2)$ matching the upper bound previously established in the elliptic case. in the elliptic case.
In this paper we will review the main results concerning the issue of stability for the determination unknown boundary portion of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and selfcontained proofs. We prove that such problems are severely ill-posed in the sense that under a priori regularity assumptions on the unknown boundaries, up to any finite order of differentiability, the continuous dependence of unknown boundary from the measured data is, at best, of logarithmic type.
In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lamé operator $\mathbb{H}$ and we then study the extension problem associated to such non-local operators. We also study the various regularity properties of solutions to such an extension problem via a transformation which reduces the extension problem for the parabolic Lamé operator to another system that resembles the extension problem of the fractional heat operator. Finally in the case when $s \geq 1/2$, by proving a conditional doubling property for solutions to the corresponding reduced…
This paper is devoted to a study of the unique continuation property for stochastic parabolic equations. Due to the adapted nature of solutions in the stochastic situation, classical approaches to treat the the unique continuation problem for deterministic equations do not work. Our method is based on a suitable partial Holmgren coordinate transform and a stochastic version of Carleman-type estimate.
In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-Δ)^s+Q$ with $0<s<1$ in any space dimension. We uniquely determine the unknown bounded potential $Q$ from infinitely many exterior Dirichlet-to-Neumann type measurements. This relies on Runge approximation and the dual global weak unique continuation properties of the equation under consideration. In discussing weak unique continuation of our operator, a main feature of our argument relies on a Carleman estimate for the associated fractional parabolic Caffarelli-Silvestre extension. Furthe…
This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain. This system is a simplified version of the chemotaxis model introduced by Keller and Segel. With the aid of priori L^infty-estimates (for solutions of the system) built up in this paper, we treat the semi-linear parabolic equation in the system as a linear parabolic equation, and then use the frequency function method and the localization technique to build up two unique continuation inequalities for the system. As a consequence of the above-mentioned two inequalitie…
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.
This article is concerned with the unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $Ω$ prescribed with some regularity and growth conditions. Our result shows that the value of the solutions can be determined uniquely by its value on an arbitrary open subset $ω$ in $Ω$ at any given positive time $T$. We also derive the quantitative nature of this unique continuation, that is, the estimate of a $L^2(Ω)$ norm of the initial data on $Ω$, which is majorized by that of solution on the bounded open subset $ω$ at termin…
We study the strong unique continuation property backwards in time for the nonlocal equation in $\mathbb{R}^{n} \times \mathbb{R}$ \begin{equation}\label{one} (\partial_t - Δ)^{s} u = V(x,t)u \end{equation} for $s \in (0,1)$. Our main result Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained by Poon for the case when $s=1$. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation above. With such theory in hands we establish: i) a basic monotonicity result for an adjusted frequency function which plays a central role…
Parabolic Frequency on Gaussian Spaces and Unique Continuation
本文在高斯空间中研究抛物线频率，通过加权 $L^2$ 框架推导强唯一持续性原理，扩展了 Poon 的结果并补充了 Colding 和 Minicozzi 的几何推广。
Consider the higher order parabolic operator $\partial_t+(-Δ_x)^m$ and the higher order Schrödinger operator $i^{-1}\partial_t+(-Δ_x)^m$ in $X=\{(t,x)\in\mathbb{R}^{1+n};~|t|<A,|x_n|<B\}$, where $m$ and $n$ are any positive integers. Under certain lower order and regularity assumptions, we prove that if solutions to the linear problems vanish when $x_n>0$, then the solutions vanish in $X$. Such results are global if $n>1$, and we also prove some relevant local results.
We consider a class of parabolic equations with critical electromagnetic potentials, for which we obtain a classification of local asymptotics, unique continuation results, and an integral representation formula for solutions.
In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an opti…
We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of an isolated invariant set. As the frequency of the oscillations tends to infinity, we prove that every isolated invariant set of the averaged autonomous equation can be continued to an isolated invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index …
In this paper, we establish a Hölder-type quantitative estimate of unique continuation for solutions to the heat equation with Coulomb potentials in either a bounded convex domain or a $C^2$-smooth bounded domain. The approach is based on the frequency function method, as well as some parabolic-type Hardy inequalities.
Unique continuation and approximate controllability for a degenerate parabolic equation
针对一维弱退化抛物方程，通过建立局部Carleman估计证明解在退化集上消失且共法向导数为零时必恒为零，并推导Dirichlet边界条件下的近似可控性。
A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren-Poon monotonicity formula combined with a blow-up analysis.
We obtain unique continuation results for Schrodinger equations with time dependent gradient vector potentials. This result with an appropriate modification also yields unique continuation properties for solutions of certain nonlinear Schrodinger equations.
The Calderón problem for space-time fractional parabolic operators with variable coefficients
本文研究了变系数时空分数阶抛物算子的反问题，利用全局弱唯一延拓性质与Runge型逼近论证，结合新建立的Carleman估计，实现了从外部测量数据对势函数的唯一恢复，并扩展至带漂移算子的情形。
In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality \[ |Δu - u_t| \leq \frac{M}{|x|^2} |u|, \] with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.
We quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product we obtain a new stability estimate in the linear case. We also show the so-called strong uniqueness of continuation and the uniqueness of continuation from a set of positive measure. These results are derived by using a linearization procedure.
In this paper, Hardy's uncertainty principle and unique continuation properties of abstract Schrödinger equations in vector-valued classes are obtained
In this paper, we derive a local unique continuation property for stochastic hyperbolic equations without boundary conditions. This result is proved by a global Carleman estimate.
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 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.
This paper deals with the unique continuation of solutions for a one-dimensional anomalous diffusion equation with Caputo derivative of order $α\in(0,1)$. Firstly, the uniqueness of solutions to a lateral Cauchy problem for the anomalous diffusion equation is given via the Theta function method, from which we further verify the unique continuation principle.
We obtain a unique continuation result for fractional Schrödinger operators with potential in Morrey spaces. This is based on Carleman inequalities for fractional Laplacians.
Quantitative strong unique continuation for the Schrödinger operator with unbounded potential
本文通过重访JK定理6.3，建立了具有无界势的薛定谔算子的定量强唯一延拓，并结合全局定量唯一延拓得到全局定量强唯一延拓。
We consider a time-dependent structured population model equation and establish a Carleman estimate. We apply the Carleman estimate to prove the unique continuation which means that Cauchy data on any lateral boundary determine the solution uniquely.
In this paper we describe some recent works on quantitative unique continuation for elliptic, parabolic and dispersive equations. The elliptic results are joint work with J.Bourgain, while the remainder of the works discussed are joint works with L.Escauriaza, G.Ponce and L.Vega.
In this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth order equations and higher order fractional problems.
Comments on the quantitative uniqueness of continuation for evolution equations
本文通过结合可观测性不等式与能量估计，为侧边界消失的演化方程解建立了接近最优的延拓定量唯一性结果。
We establish the strong unique continuation property of fractional orders of linear elliptic equations with Lipschitz coefficients by establishing monotonicity of some Almgren-type frequency functional via an extension procedure.
We prove that solutions to elliptic equations in two variables in divergence form, possibly non-selfadjoint and with lower order terms, satisfy the strong unique continuation property.
In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong unique continuation property. Moreover, we also discuss the unique continuation property from measurable sets, which shows that nodal domains to these equations must have vanishing Lebesgue measure. Our methods rely on suitable Carleman estimates, for which we include the sublinear potential into the main part of the operator.
Unique continuation for a mean field game system
该研究针对平均场博弈系统，证明了其唯一延拓性，即任意侧向子边界上柯西数据为零时解恒为零，丰富了相关数学理论。
Asymptotics of solutions to relativistic fractional elliptic equations with Hardy type potentials is established in this paper. As a consequence, unique continuation properties are obtained.
Unique continuation for the momentum ray transform
本文利用分数阶拉普拉斯算子的唯一延拓性质，证明了动量射线变换作用于张量的保守性以及加权射线与锥变换的反局部性。
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to the weight function is equivalent to the unique continuation property of eigenfunctions. In addition, we discuss some unique continuation results for the special case of the fractional Laplacian.
In this paper we study unique continuation theorems for magnetic Schrödinger equation via Carleman estimates. We use integration by parts techniques in order to show these estimates. We consider electric and magnetic potentials with strong singularities at the origin and some decay at infinity.
We prove the unique continuation property for the differential inequality $|(-Δ)^{α/2}u|\leq|V(x)u|$, where $0<α<n$ and $V\in L_{\textrm{loc}}^{n/α,\infty}(\mathbb{R}^n)$, $n\geq3$.
In this survey we discuss the frequency function method so as to study the problem of unique continuation for elliptic partial differential equations. The methods used in the note were mainly introduced by Garofalo and Lin.
In this paper, we prove the unique continuation property for the weak solution of the plate equation with non-smooth coefficients. Then, we apply this result to study the global attractor for the semilinear plate equation with a localized damping.
We obtain a global unique continuation result for the differential inequality $|(i\partial_t+Δ)u|\leq|V(x)u|$ in $\mathbb{R}^{n+1}$. This is the first result on global unique continuation for the Schrödinger equation with time-independent potentials $V(x)$ in $\mathbb{R}^{n}$. Our method is based on a new type of Carleman estimates for the operator $i\partial_t+Δ$ on $\mathbb{R}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.
A logarithmic convexity approach to quantitative unique continuation for the complex Ginzburg-Landau operator
研究复金兹堡 - 朗道方程解的定量唯一延拓估计，用对数凸性方法证明该性质，导出明确常数并刻画其参数依赖性。
New quantitative unique continuation result for elliptic equations
本文基于Carleman不等式，通过简单直接的证明方法，为椭圆方程提供了新的定量化唯一持续性结果，并展示了斯托克斯方程的类似结果。
We prove the following unique continuation result: if a solution to the level set equation for mean curvature flow in a mean-convex domain agrees to infinite order at the point where it attains its maximum with the solution for a ball, then it agrees everywhere and the domain is a ball.
Assuming that the Lamé moduli $μ$, $λ$ are $C^{\tiny{1}}$ and $n\geq2$, we prove quantitative estimates of a weak sense of strong unique continuation for thesolutions of the n-dimensional Lamé system of the form of three spheres inequalities.
We establish a unique continuation property for stochastic heat equations evolving in a bounded domain $G$. Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of $G$ at any given positive time constant. Further, when $G$ is convex and bounded, we also give a quantitative version of the unique continuation property. As applications, we get an observability estimate for stochastic heat equations, an approximate result and a null controllability result for a backward stochastic heat equation.
This article deals with the weak and strong unique continuation principle for fractional Schrödinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to variable coefficient versions of fractional Schrödinger equations.
In this note we obtain a unique continuation result for the differential inequality $|\bar{\partial}u|\leq|Vu|$, where $\bar{\partial}=(i\partial_y+\partial_x)/2$ denotes the Cauchy-Riemann operator and $V(x,y)$ is a function in $L^2(\mathbb{R}^2)$.
We consider the problem of the strong unique continuation for an elasticity system with general residual stress. Due to the known counterexamples, we assume the coefficients of the elasticity system are in the Gevrey class of appropriate indices. The main tools are Carleman estimates for product of two second order elliptic operators.
In this paper, we give the strong unique continuation property for a general two dimensional anisotropic elliptic system with real coefficients in a Gevrey class under the assumption that the principal symbol of the system has simple characteristics. The strong unique continuation property is derived by obtaining some Carleman estimate. The derivation of the Carleman estimate is based on transforming the system to a larger second order elliptic system with diagonal principal part which has complex coefficients.
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