Unique Continuation Research Articles

Localisation for Delone operators via Bernoulli randomisation

Delone operators are Schrödinger operators in multi-dimensional Euclidean space with a potential given by the sum of all translates of a given “single-site potential” centred at the points of a Delone set. In this paper, we use randomisation to study dynamical localisation for families of Delone operators. We do this by suitably adding more points to a Delone set and by introducing i.i.d. Bernoulli random variables as coupling constants at the additional points. The resulting non-ergodic continuum Anderson model with Bernoulli disorder is accessible to the latest version of the multiscale analysis. The novel ingredient here is the initial length-scale estimate whose proof is hampered due to the non-periodic background potential. It is obtained by the use of a quantitative unique continuation principle. As applications we obtain both probabilistic and topological statements about dynamical localisation. Among others, we show that Delone sets for which the associated Delone operators exhibit dynamical localisation at the bottom of the spectrum are dense in the space of Delone sets.

Stabilization of the critical nonlinear Klein-Gordon equation with variable coefficients on R^3

We prove the exponential stability of the defocusing critical semilinear wave equation with variable coefficients and locally distributed damping on \(\mathbb{R}^3\). The construction of the variable coefficients is almost equivalent to the geometric control condition. We develop the traditional Morawetz estimates and the compactness-uniqueness arguments for the semilinear wave equation to prove the unique continuation result. The observability inequality and the exponential stability are obtained subsequently.

Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields

Let \(I_{m}\) denote the Euclidean ray transform acting on compactly supported symmetric m-tensor field distributions f, and \(I_{m}^{*}\) be its formal \(L^2\) adjoint. We study a unique continuation result for the operator \(N_{m}=I_{m}^{*}I_{m}\). More precisely, we show that if \(N_{m}f\) vanishes to infinite order at a point \(x_0\) and if the Saint-Venant operator W acting on f vanishes on an open set containing \(x_0\), then f is a potential tensor field. This generalizes two recent works of Ilmavirta and Mönkkönen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying the Saint-Venant operator acting on higher-order tensor fields as the right generalization of the exterior derivative operator acting on 1-forms, which makes unique continuation results for ray transforms of higher-order tensor fields possible. In the second half of the paper, we prove analogous unique continuation results for momentum ray and transverse ray transforms.

A Gauge-Invariant Unique Continuation Criterion for Waves in Asymptotically Anti-de Sitter Spacetimes

We reconsider the unique continuation property for a general class of tensorial Klein–Gordon equations of the form □gϕ+σϕ=G(ϕ,∇ϕ),σ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Box _{g} \\phi + \\sigma \\phi = {\\mathcal {G}}(\\phi ,\ abla \\phi ) \ ext {,} \\qquad \\sigma \\in {\\mathbb {R}} \\end{aligned}$$\\end{document}on a large class of asymptotically anti-de-Sitter spacetimes. In particular, we aim to generalize the previous results of Holzegel, McGill, and the second author (Holzegel and Shao in Commun Math Phys 347(3):723–775, 2016; Commun Partial Differ Equ 42(12):1871–1922, 2017; McGill and Shao in Class Quantum Gravity 38(5):054001, 2021) (which established the above-mentioned unique continuation property through novel Carleman estimates near the conformal boundary) in the following ways: We replace the so-called null convexity criterion—the key geometric assumption on the conformal boundary needed in McGill and Shao (2021) to establish the unique continuation properties—by a more general criterion that is also gauge invariant.Our new unique continuation property can be applied from a larger, more general class of domains on the conformal boundary.Similar to McGill and Shao (2021), we connect the failure of our generalized criterion to the existence of certain null geodesics near the conformal boundary. These geodesics are closely related to the classical Alinhac-Baouendi counterexamples to unique continuation (Alinhac and Baouendi in Math Z 220(4):561–568, 1995). Finally, our gauge-invariant criterion and Carleman estimate will constitute a key ingredient in proving unique continuation results for the full nonlinear Einstein-vacuum equations, which will be addressed in a forthcoming paper of Holzegel and the second author (Holzegel and Shao in Unique continuation for the Einstein equations in asymptotically anti-de sitter spacetimes (in preparation), 2022).

Structural and qualitative properties of a geometrically integrable equation

Lie symmetries of a Novikov geometrically integrable equation are found and group-invariant solutions are obtained. Local conservation laws up to second order are established as well as their corresponding conserved quantities. Sufficient conditions for the L1 norm of the solutions to be invariant are presented, as well as conditions for the existence of positive solutions. Two demonstrations for unique continuation of solutions are given: one of them is just based on the invariance of the L1 norm of the solutions, whereas the other is based on well-posedness of Cauchy problems. Finally, pseudo-spherical surfaces determined by the solutions of the equation are studied: all invariant solutions that do not lead to pseudo-spherical surfaces are classified and the existence of an analytic metric for a pseudo-spherical surface is proved using conservation of solutions and well-posedness results.

Well-posedness and blow-up results for a class of nonlinear fractional Rayleigh-Stokes problem

Abstract In this article, we consider the fractional Rayleigh-Stokes problem with the nonlinearity term satisfies certain critical conditions. The local existence, uniqueness and continuous dependence upon the initial data of ε \varepsilon -regular mild solutions are obtained. Furthermore, a unique continuation result and a blow-up alternative result of ε \varepsilon -regular mild solutions are given in the end.

The conditional stability for unique continuation on the sphere for Helmholtz equations

Unique continuation is one of the most important properties for the solution of partial differential equation, which means the local information of the solution can determine the global one. In this paper, we discuss a special unique continuation for Helmholtz equations on a sphere in , which is different with the classical unique continuation. The unique continuation holds only on the sphere and may not be extended to the domain in . A Hölder type conditional stability of unique continuation is proved by complex extension method. Then a Tikhonov regularized scheme is proposed and the convergence rate is obtained by using the conditional stability estimate. Numerical examples are presented to show the performance of the scheme. It should be remarked here that our results may be applied to the problem of recovering a far field pattern with its local information.

Unique continuation from a crack’s tip under Neumann boundary conditions

We derive local asymptotics of solutions to second order elliptic equations at the edge of a (N−1)-dimensional crack, with homogeneous Neumann boundary conditions prescribed on both sides of the crack. A combination of blow-up analysis and monotonicity arguments provides a classification of all possible asymptotic homogeneities of solutions at the crack’s tip, together with a strong unique continuation principle.

The Calderón Problem for the Fractional Wave Equation: Uniqueness and Optimal Stability

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension $n\in \mathbb{N}$.

Green’s Functions for First-Order Systems of Ordinary Differential Equations without the Unique Continuation Property

This paper is a contribution to the spectral theory associated with the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with $Ju'+qu=wf$ and show the existence of Green's functions for self-adjoint relations even if unique continuation of solutions fails.

The fractional Schrödinger equation on compact manifolds: global controllability results

The goal of this work is to prove global controllability and stabilization properties for the fractional Schrödinger equation on d-dimensional compact Riemannian manifolds without boundary (M, g). To prove our main results we use techniques of pseudo-differential calculus on manifolds. More precisely, by using microlocal analysis, we are able to prove propagation of regularity which together with the so-called Geometric Control Condition and Unique Continuation Property help us to prove global control results for the system under consideration. As a main novelty this manuscript presents the relation between the geometric control condition and the controllability for the fractional Schrödinger equation.

Correction to: Boundary Unique Continuation on $$C^1$$-Dini Domains and the Size of the Singular Set

Correction to: Boundary Unique Continuation on $$C^1$$-Dini Domains and the Size of the Singular Set

Boundary controllability of two coupled wave equations with space-time first-order coupling in 1-D

This paper is devoted to study exact controllability of two one-dimensional coupled wave equations with first-order coupling terms with coefficients depending on space and time. We give a necessary and sufficient condition for both exact controllability in high frequency in the general case and the unique continuation in the cascade case.

Reconstruction and stability in Gelfand’s inverse interior spectral problem

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $\Delta_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit $\log\log$-type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gel'fand's inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric Boundary Control method.

Boundary Unique Continuation on $$C^1$$-Dini Domains and the Size of the Singular Set

Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in \overline{D}: u(X) = |\nabla u(X)| = 0\} $ and give an estimate of its $(d-2)$-dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of $u$ centered at $0$. Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.

Exponential decay for the KdV equation on ℝ with new localized dampings

In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.

Uniqueness of inverse source problems for general evolution equations

In this paper, we investigate inverse source problems for a wide range of PDEs of parabolic and hyperbolic types as well as time-fractional evolution equations by partial interior observation. Restricting the source terms to the form of separated variables, we establish uniqueness results for simultaneously determining both temporal and spatial components without non-vanishing assumptions at [Formula: see text], which seems novel to the best of our knowledge. Remarkably, mostly we allow a rather flexible choice of the observation time not necessarily starting from [Formula: see text], which fits into various situations in practice. Our main approach is based on the combination of the Titchmarsh convolution theorem with unique continuation properties and time-analyticity of the PDEs under consideration.

The Cauchy problem and continuation of periodic solutions for a generalized Camassa–Holm equation

We consider a three-parameter family of non-linear equations with -order non-linearities. Such a family includes as a particular member the well-known b-equation, which encloses the famous Camassa–Holm equation. For certain choices of the parameters, we establish a global existence result and show a scenario that prevents the wave breaking of solutions. Also, we explore unique continuation properties for some values of the parameters.

Partial Data Problems and Unique Continuation in Scalar and Vector Field Tomography

We prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.

The nodal set of solutions to some nonlocal sublinear problems

We study the nodal set of stationary solutions to equations of the form (-Delta )^s u = lambda _+ (u_+)^{q-1} - lambda _- (u_-)^{q-1}quad text {in }B_1, where lambda _+,lambda _->0, q in [1,2), and u_+ and u_- are respectively the positive and negative part of u. This collection of nonlinearities includes the unstable two-phase membrane problem q=1 as well as sublinear equations for 1<q<2. We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case s=1, we prove that the admissible vanishing orders can not exceed the critical value k_q= 2s/(2- q). Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that k_q< 1, we prove a remarkable difference with the local case: solutions can only vanish with order k_q and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.