Quantitative Unique Continuation Research Articles

Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary *

Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.

On (global) unique continuation properties of the fractional discrete Laplacian

We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in general. While discretization thus counteracts the strong rigidity properties of the continuum fractional Laplacian, by discussing quantitative forms of unique continuation, we illustrate that these properties can be recovered if exponentially small (in the lattice size) correction terms are added. In particular, this allows us to deduce stability properties for a discrete, linear inverse problem for the fractional Laplacian. We complement these observations with a transference principle and the discussion of these properties on the discrete torus.

Quantitative unique continuation for parabolic equations with Neumann boundary conditions

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 by recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.

Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials

Recent (scale-free) quantitative unique continuation estimates for spectral subspaces of Schrödinger operators are extended to allow singular potentials such as certain Lp-functions. The proof is based on accordingly adapted Carleman estimates. Applications include Wegner and initial length scale estimates for random Schrödinger operators and control theory for the controlled heat equation with singular heat generation term.

A remark on quantitative unique continuation from subsets of the boundary of positive measure

The question of uniqueness of harmonic functions in a domain Ω ⊂ R d \Omega \subset \mathbb {R}^d with boundary ∂ Ω \partial \Omega , satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset of the boundary ω \omega of positive d − 1 d-1 dimensional measure has attracted a lot of attention (see e.g. V. Adolfsson and L. Escauriaza [Comm. Pure Appl. Math. 50 (1997), pp. 935–969]; F.-H. Lin [Comm. Pure Appl. Math. 44 (1991), pp. 287–308]; X. Tolsa [Comm. Pure Appl. Math. 76 (2023), pp. 305–336]). It is essentially a consequence of Carleman estimates when ω \omega contains an open subset of the boundary (uniqueness for second order elliptic operators across an hypersurface). The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, Birkhäuser/Springer, Cham, 2018, pp. 333-344; Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018, World Scientific Publishing, Hackensack, NJ, 2018, pp. 2391–2411], we consider the simpler case of W 2 , ∞ W^{2, \infty } domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2 , ∞ W^{2, \infty } domain with boundary.

Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators

We prove a scale-free quantitative unique continuation estimate for the gradient of eigenfunctions of divergence-type operators, i.e., operators of the form −divA∇, where the matrix function A is uniformly elliptic. The proof uses a unique continuation principle for elliptic second-order operators and a lower bound on the L2-norm of the gradient of eigenfunctions corresponding to strictly positive eigenvalues.As an application, we prove an eigenvalue lifting estimate that allows us to prove a Wegner estimate for random divergence-type operators. Here our approach allows us to get rid of a restrictive covering condition that was essential in previous proofs of Wegner estimates for such models.

Quantitative unique continuation for the elasticity system with application to the kinematic inverse rupture problem

We obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular, our result applies to the elasticity system and also the Maxwell system. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the stable unique continuation up to the rupture surface.

Whispering gallery modes for a transmission problem

We construct a specific family of eigenfunctions for a Laplace operator with coefficients having a jump across an interface. These eigenfunctions have an exponential concentration arbitrarily close to the interface, and therefore could be considered as whispering gallery modes. The proof is based on an appropriate Agmon estimate. We deduce as a corollary that the quantitative unique continuation result for waves propagating in singular media proved by the author in [7] is optimal.

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.

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 $n\in \mathbb{N}$.

Tunneling estimates and approximate controllability for hypoelliptic equations

This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator L \mathcal {L} on a compact manifold M \mathcal {M} assuming: ( i ) (i) the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of L \mathcal {L} , and ( i i ) (ii) the analyticity of M \mathcal {M} and the coefficients of L \mathcal {L} . The first result is the tunneling estimate ‖ φ ‖ L 2 ( ω ) ≥ C e − c λ k 2 \|\varphi \|_{L^2(\omega )} \geq Ce^{- c\lambda ^{\frac {k}{2}}} for normalized eigenfunctions φ \varphi of L \mathcal {L} from a nonempty open set ω ⊂ M \omega \subset \mathcal {M} , where k k is the hypoellipticity index of L \mathcal {L} and λ \lambda the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation ( ∂ t 2 + L ) u = 0 (\partial _t^2+\mathcal {L})u=0 : for T > 2 sup x ∈ M ( d i s t ( x , ω ) ) T>2 \sup _{x \in \mathcal {M}}(dist(x,\omega )) (here, d i s t dist is the sub-Riemannian distance), the observation of the solution on ( 0 , T ) × ω (0,T)\times \omega determines the data. The constant involved in the estimate is C e c Λ k Ce^{c\Lambda ^k} where Λ \Lambda is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation ( ∂ t + L ) v = 1 ω f (\partial _t+\mathcal {L})v=\mathbb {1}_\omega f in any time, with appropriate (exponential) cost, depending on k k . In case k = 2 k=2 (Grushin, Heisenberg...), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the analyticity assumption can be relaxed, and a boundary ∂ M \partial \mathcal {M} can be added in some situations. Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).

Unique continuation inequalities for the parabolic-elliptic chemotaxis system

This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain Ω. This system is a simplified version of the chemotaxis model introduced by Keller and Segel in [14]. With the aid of priori L∞-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 inequalities, we have the following qualitative unique continuation property: if one component of a solution vanishes in a nonempty open subset ω⊂Ω at some time T>0, then the solution is identically zero.

Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

<p style='text-indent:20px;'>In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.</p>

Quantitative unique continuation of solutions to the bi‐Laplace equations

In this paper, we prove a three‐ball inequality for y satisfying an equation of the form in some open, connected set Ω of with and . The derivation of such estimate relies on a delicate Carleman estimate for the bi‐Laplace equation and some Caccioppoli inequalities to estimate the lower‐terms. Based on three‐ball inequality, we then derive the vanishing order of y is less than , where | · |∞ means the L∞ norm, which is a quantitative version of the strong unique continuation property for y. Furthermore, under some priori assumptions on Vj and y, we prove that the nontrivial solution y satisfies the decay property around the point at infinity. In particular, if , this decaying rate can be improved to .

Large‐Scale Analyticity and Unique Continuation for Periodic Elliptic Equations

AbstractWe prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large‐scale Ck, 1 estimate scale exponentially in k, just as for the classical estimate for harmonic functions, and the minimal scale grows at most linearly in k. As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations that exhibit growth like O(exp(δ| x| )) for small δ > 0. The large‐scale analyticity also implies quantitative unique continuation results, namely a three‐ball theorem with an optimal error term as well as a proof of the nonexistence of L2 eigenfunctions at the bottom of the spectrum. © 2020 Wiley Periodicals LLC.

On two methods for quantitative unique continuation results for some nonlocal operators

In this article, we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the logarithmic stability of the moment problem. In our second method we rely on the presence of branch-cut singularities for certain Fourier multipliers. As an application we present quantitative Runge approximation results for the operator with and acting on functions on

Quantitative unique continuation for hyperbolic and hypoelliptic equations

We review recent results of the authors concerning quantitative unique continuation estimates for operators with coefficients that are analytic in some (or all the) variables. We describe several applications for wave-like equations, but also equations based on hypoelliptic operators. These proceedings are a survey of the general results in [LL19] together with applications to wave equations [LL16] and to hypoelliptic equations [LL17, LL20b].

Band Edge Localization Beyond Regular Floquet Eigenvalues

We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in L^2({mathbb {R}}^d) is universal. By this, we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.

Quantitative unique continuation for Schrödinger 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+Vu=0 in some open, connected subset of Rn, we estimate the vanishing order of solutions in terms of the Lt-norm of V. Our results apply to all t>n2 and n≥3. With these maximal order of vanishing estimates, we employ a scaling argument to produce quantitative unique continuation at infinity estimates for global solutions to Δu+Vu=0. To handle V∈Lt for every t∈(n2,∞], we prove a novel Lp−Lq Carleman estimate by interpolating a known Lp−L2 estimate with a new endpoint Carleman estimate. This new Carleman estimate may also be used to establish improved order of vanishing estimates for equations with a first order term, those of the form Δu+W⋅∇u+Vu=0.

Quantitative Unique Continuation for Second Order Elliptic Operators with Singular Coefficients

We establish some quantitative strong unique continuation properties for solution of |Pu|≤ C1|x|− 1|∇u| + C0|x|− 2|u| where P is a second order elliptic operator. As in (Rev. Mat. Iberoam. 27: 475–491, 9), our result is of quantitative nature but requires weaker conditions on the coefficients of P.