Estimates Of Unique Continuation Research Articles

Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schrödinger type equation via local Cauchy data

We consider the inverse problem of the simultaneous identification of the coefficients σ and q of the equation div(σ∇u)+qu=0 from the knowledge of the Cauchy data set. We assume that σ=γA, where A is a given matrix function and γ and q are unknown piecewise affine scalar functions. No sign, nor spectrum condition on q is assumed. We derive a result of global Lipschitz stability in dimension n≥3. The proof relies on the method of singular solutions and on the quantitative estimates of unique continuation.

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.

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).

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 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].

Size estimates of an obstacle in a stationary Stokes fluid

In this work we are interested in estimating the size of a cavity D immersed in a bounded domain d = 2, 3, filled with a viscous fluid governed by the Stokes system, by means of velocity and Cauchy forces on the external boundary . More precisely, we establish some lower and upper bounds in terms of the difference between the external measurements when the obstacle is present and without the object. The proof of the result is based on interior regularity results and quantitative estimates of unique continuation for the solution of the Stokes system.

Size estimates for the EIT problem with one measurement: the complex case

In this paper we estimate the size of a measurable inclusion in terms of power measurements for a single applied boundary current. This problem arises in medical imaging for the screening of organs. For this kind of problem one has to deal mathematically with the complex conductivity (admittivity) equation. In this case we are able to establish, for certain classes of admittivities, lower and upper bounds of the measure of the inclusion in terms of the power measurements. A novelty of our result is that we are able to estimate also the volume of inclusions having part of its boundary in common with the reference body. Our analysis is based on the derivation of energy bounds and of fine quantitative estimates of unique continuation for solutions to elliptic equations.

Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators

We prove a unique continuation principle for spectral projections of Schrodinger operators. We consider a Schrodinger operator H = −Δ + V on \({{\rm L}^2(\mathbb{R}^d)}\) , and let HΛ denote its restriction to a finite box Λ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type χI (HΛ) WχI (HΛ) ≥ κ χI (HΛ) with κ > 0 for appropriate potentials W ≥ 0 and intervals I. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schrodinger operators with alloy-type random potentials (‘crooked’ Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.

Quantitative uniqueness for elliptic equations with singular lower order terms

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.

Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation

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 inequality is derived, is proved in an elementary and direct way based on standard integration by parts.

Doubling properties of caloric functions

We obtain quantitative estimates of unique continuation for solutions to parabolic equations – doubling properties and two-sphere one-cylinder inequalities.

Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds

This article concerns some quantitative versions of unique continuation known as observability inequalities. One of them is a lower bound on the spectral projectors of the Dirichlet Laplacian which generalizes the unique continuation of an eigenfunction from any open set Omega. Another one is equivalent to the interior null-controllability in time T of the heat equation with Dirichlet condition (the input function is a source in (0,T) x Omega). On a compact Riemannian manifolds, these inequalities are known to hold for arbitrary T and Omega. This article states and links these observability inequalities on a complete non-compact Riemannian manifold, and tackles the quite open problem of finding which Omega and T ensure their validity. It proves that it is sufficient for Omega to be the exterior of a compact set (for arbitrary T), but also illustrates that this is not necessary. It provides a necessary condition saying that there is no sequence of balls going infinitely far away from Omega without shrinking in a generalized sense (depending on T) which also applies when the distance to Omega is bounded.

Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries

In this paper we obtain quantitative estimates of strong unique continuation for solutions to parabolic equations. We apply these results to prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain Ω in R n , from the knowledge of overdetermined boundary data for parabolic boundary value problems.

STRONG UNIQUE CONTINUATION FOR THE LAMÉ SYSTEM OF ELASTICITY1*

We prove quantitative estimates of unique continuation for the solutions of the Lamé system of the form of three–spheres–inequalities and of doubling inequalities. *Work supported in part by MURST and by CNR.