In this short note we prove the logarithmic stability of the single measurement uniqueness result for the fractional Calderón problem which had been derived in \cite{GRSU18}. To this end, we use the quantitative uniqueness results established in \cite{RS20a} and complement these bounds with a boundary doubling estimate. The latter yields control of the order of vanishing of solutions to the fractional Schrödinger equation. Then, following a scheme introduced in \cite{S10,ASV13} in the context of the determination of a surface impedance from far field measurements, this allows us to deduce logarithmic stability of the potential $q$.
In this note we prove the exponential instability of the fractional Calderón problem and thus prove the optimality of the logarithmic stability estimate from \cite{RS17}. In order to infer this result, we follow the strategy introduced by Mandache in \cite{M01} for the standard Calderón problem. Here we exploit a close relation between the fractional Calderón problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in \cite{RS1…
We determine both the magnetic potential and the electric potential from the exterior partial measurements of the Dirichlet-to-Neumann map in the fractional linear magnetic Calderón problem by using an integral identity. We also determine both the magnetic potential and the nonlinearity in the fractional semilinear magnetic Calderón problem by using a first order linearization.
We review recent progress in the fractional Calderón problem, where one tries to determine an unknown coefficient in a fractional Schrödinger equation from exterior measurements of solutions. This equation enjoys remarkable uniqueness and approximation properties, which turn out to yield strong results in related inverse problems.
We prove fine higher regularity results of Calderón-Zygmund-type for equations involving nonlocal operators modelled on the fractional $p$-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.
We introduce the fractional magnetic operator involving a magnetic potential and an electric potential. We formulate an inverse problem for the fractional magnetic operator. We determine the electric potential from the exterior partial measurements of the associated Dirichlet-to-Neumann map by using Runge approximation property.
In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $q \in L^{\infty}(Ω) $ in the equation $((-Δ)^s+ q)u = 0 \mbox{ in } Ω\subset \mathbb{R}^n$ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $L^{\infty}(Ω)$. Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on $q$. We allow for the …
We obtain Musielak Orlicz bumps conditions on a pair of weights for the boundedness of Calderón Zygmund operators and their commutators between variable Lebesgue spaces with different weights. The symbols of the commutators belong to a wider class of functions. We also give Bloom type estimates for commutators of Calderón Zygmund and fractional integral operators in the variable Lebesgue context. The techniques involved in both type of results are related with the theory of sparse domination.
We study the fractional Schrödinger equation with quasilocal perturbations. These are a family of nonlocal perturbations vanishing at infinity, which include e.g. convolutions against Schwartz functions. We show that the qualitative unique continuation and Runge approximation properties hold in the assumption of sufficient decay. Quantitative versions of both results are also obtained via a propagation of smallness analysis for the Caffarelli-Silvestre extension. The results are then used to show uniqueness in the inverse problem of retrieving a quasilocal perturbation from DN data under suita…
We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically d…
We study an inverse problem for the fractional Schrödinger equation (FSE) with a local perturbation by a linear partial differential operator (PDO) of order smaller than the order of the fractional Laplacian. We show that one can uniquely recover the coefficients of the PDO from the Dirichlet-to-Neumann (DN) map associated to the perturbed FSE. This is proved for two classes of coefficients: coefficients which belong to certain spaces of Sobolev multipliers and coefficients which belong to fractional Sobolev spaces with bounded derivatives. Our study generalizes recent results for the zeroth a…
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…
The Calderón problem for the fractional Schrödinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant $L^p$ or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful…
We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many \emph{generic} measurements is discussed. Here the genericity is obtained through…
We show global uniqueness in the fractional Calderón problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work \cite{GhoshSaloUhlmann} considered the case of infinitely many measurements. The method is again based on the strong uniqueness properties for the fractional equation, this time combined with a unique continuation principle from sets of measure zero. We also give a constructive procedure for determining an unknown potential from a single exterior measurement, based on constructive versions of the unique continuation resu…
In this paper we prove log log type stability estimates for inverse boundary value problems on admissible Riemannian manifolds of dimension $n \geq 3$. The stability estimates correspond to a couple of uniqueness results by Dos Santos Ferrera, Kenig, Salo and Uhlmann. These inverse problems arise naturally when studying the anisotropic Calderón problem.
In these notes we prove log-type stability for the Calderón problem with conductivities in $ C^{1,\varepsilon}(\barΩ) $. We follow the lines of a recent work by Haberman and Tataru in which they prove uniqueness for $ C^1(\barΩ) $.
In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension $n\geq2$. Our results generalize the recent initiative [16] of introducing and solving inverse problem for fractional Schrödinger operator $((-Δ)^{s}+q)$ for $0<s<1$. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differ…
This is a follow-up of a previous article where we proved local stability estimates for a potential in a Schrödinger equation on an open bounded set in dimension $n=3$ from the Dirichlet-to-Neumann map with partial data. The region under control was the penumbra delimited by a source of light outside of the convex hull of the open set. These local estimates provided stability of log-log type corresponding to the uniqueness results in Calderón's inverse problem with partial data proved by Kenig, Sjöstrand and Uhlmann. In this article, we prove the corresponding global estimates in all dimension…
The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems $$ \left\{\begin{array}{rcll} (-Δ)^s u-λ\dfrac{u}{|x|^{2s}}&=&f(x,u) &\hbox{ in } Ω,\\ u&=&0 &\hbox{ in } \mathbb{R}^N\setminusΩ,\\ u&>&0 &\hbox{ in }Ω, \end{array}\right. $$ where $(-Δ)^s$, $s\in(0,1)$, is the fractional laplacian operator, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary such that $0\inΩ$ and $N>2s$. We will mainly consider the solvability in two cases: 1) The linear problem, that is, $f(x,t)=f(x)$, whe…
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 \cite{AU04, KU19}. 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.
We prove an optimal stability estimate for Electrical Impedance Tomography with local data, in the case when the conductivity is precisely known on a neighborhood of the boundary. The main novelty here is that we provide a rather general method which enables to obtain the Hölder dependence of a global Dirichlet to Neumann map from a local one on a larger domain when, in the layer between the two boundaries, the coefficient is known.
We deal with a global Calderón-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1<p \le 2-\frac{1}{n}$. In addition, the emphasis of this paper is in obtaining the Lorentz bounds for the gradient of solutions with the use of fractional maximal operators.
We prove a local Lipschitz stability estimate for Gel'fand-Calderón's inverse problem for the Schrödinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of $L^\infty$. A similar result for Calderón's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based …
It is proved that, in two dimensions, the Calderón inverse conductivity problem in Lipschitz domains is stable in the $L^p$ sense when the conductivities are uniformly bounded in any fractional Sobolev space $W^{α,p}$ $α>0, 1<p<\infty$.
We study the inverse problem of determining the coefficients of the fractional power of a general second order elliptic operator given in the exterior of an open subset of the Euclidean space. We show the problem can be reduced into determining the coefficients from the boundary Cauchy data of the elliptic operator on the open set, the Calderón problem. As a corollary we establish several new results for nonlocal inverse problems by using the corresponding results for the local inverse problems. In particular the isotropic nonlocal Calderón problem can be resolved completely, assuming some reg…
We are concerned with the Calderón problem of determining an unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of a convex polygonal inclusion by a single partial boundary measurement. We also derive the uniqueness result in a more general scenario where the conductivities are piecewise constants supported in a nested polygonal geometry. Our methods in establishing the stability and uniqueness results have a significant technical initiative and a strong potent…
We consider the Calderòn problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution.
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds
University of Toronto（多伦多大学）；University of California, Irvine（加州大学尔湾分校）；University of Washington（华盛顿大学）；Institute for Advanced Study of the Hong Kong University of Science and Technology（香港科技大学高等研究院）
We prove new global stability estimates for the Gel'fand-Calderon inverse problem in 3D. For sufficiently regular potentials this result of the present work is a principal improvement of the result of [G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal. 27 (1988), 153-172].
We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey-Stewartson II equation. We then use it to prove global well-posedness and scattering in $L^2$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension $2$, for conductivities $σ>0$ with $\log σ\in \dot H^1$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $L^2$-boundedness of pseudo-dif…
This note is devoted to the study of Hytönen's extrapolation theorem of compactness on weighted Lebesgue spaces. Two criteria of compactness of linear operators in the two-weight setting are obtained. As applications, we obtain two-weight compactness of commutators of Calderón--Zygmund operators, fractional integrals and bilinear Calderón--Zygmund operators.
