arXiv subjects
Publications and source records attributed to Mourad Choulli.
We provide a simple and short proof of a multidimensional Borg-Levinson type theorem. Precisely, we prove that the spectral boundary data determine uniquely the corresponding potential appearing in the Schödinger operator on an admissible Riemannian manifold. We also sketch the proof of the case of incomplete spectral boundary data.
We discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee uniqueness of solutions of an anisotropic wave equation. In the case where the uniqueness holds, the natural problem that arise in this context is to estimate the solutions, in some appropriate space, in terms of the Cauchy data. We aim in this paper to transfer, via a reduced Fourrier-Bros-Iagolnitzer transform, the known stability estimates for the Cauchy problem for elliptic equations to that for waves equations. By proceeding in that way the main difficulty is to control the residual terms, induced by the reduced Fourrier-Bros-Iagolnitzer transform, by the Cauchy data. Also, the uniqueness of continuation from Cauchy data is obtained as byproduct of stability estimates.
The classical Hopf's lemma can be reformulated as uniqueness of continuation result. We aim in the present work to quantify this property. We show precisely that if a solution $u$ of a divergence form elliptic equation attains its maximum at a boundary point $x_0$ then both $L^1$-norms of $u-u(x_0)$ on the domain and on the boundary are bounded, up to a multiplicative constant, by the exterior normal derivative at $x_0$.
The uniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard \cite{Ha} these kind of problems are known to be ill-posed and even severely ill-posed. Until now there are only few partial results concerning the quantification of the stability for parabolic Cauchy problems. In the present article, we bring the complete answer to this issue, provided that the space domain has finite diameter with respect to the geodesic distance and assuming that solutions are sufficiently smooth.
We establish a logarithmic stability inequality for the inverse problem of determining the non linear term, appearing in a semilinear BVP, from the corresponding Dirichlet-to-Neumann map (abbreviated to DtN map in the rest of this text). Our result can be seen as a stability inequality for an earlier uniqueness result by Isakov and Sylvester [Commun. Pure Appl. Math. 47 (1994), 1403-1410].
This course is intended as an introduction to the analysis of elliptic partial differential equations. The objective is to provide a large overview of the different aspects of elliptic partial differential equations and their modern treatment. Besides variational and Schauder methods we study the unique continuation property and the stability for Cauchy problems. The derivation of the unique continuation property and the stability for Cauchy problems relies on a Carleman inequality. This inequality is efficient to establish three-ball type inequalities which are the main tool in the continuation argument. We know that historically a central role in the analysis of partial differential equations is played by their fundamental solutions. We added an appendix dealing with the construction of a fundamental solution by the so-called Levi parametrix method. We tried as much as possible to render this course self-contained. Moreover each chapter contains many exercices and problems. We have provided detailed solutions of these exercises and problems. The most parts of this course consist in an enhanced version of courses given by the author in both undergraduate and graduate levels during several years. Remarks and comments that can help to improve this course are welcome.
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown coefficients from boundary measurements by varying initial conditions. Based on observability inequalities and a special choice of initial conditions, we provide uniqueness and stability estimates for the recovery of volume and boundary lower order coefficients in wave and heat equations. Some of the results presented here are slightly improved from their original versions.
We summarize in these notes the course given at the Summer School of AIP 2019 held in Grenoble from July 1st to July 5th. This course was mainly devoted to the determination of the unbounded potential in a Schrödinger equation from the associated Dirichlet-to-Neumann map (abbreviated to DN map in this text). We establish a stability inequality for potentials belonging to $L^n$, where $n\ge 3$ is the dimension of the space. Next, we prove a uniqueness result for potentials in $L^{n/2}$, $n\ge 3$, and apply this uniqueness result to demonstrate a Borg-Levinson type theorem. We use a classical approach which is essentially based on the construction of the so-called complex geometric optic solutions (abbreviated to CGO solutions in this text).
We discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee the uniqueness of continuation of solutions of an anisotropic wave equation. In the case where the uniqueness holds, the natural problem that arise naturally in this context is to estimate the solutions, in some appropriate space, in terms of norms of the Cauchy data. We aim in this paper to convert, via a reduced Fourrier-Bros-Iagolnitzer transform, the known stability results on the Cauchy problem for elliptic equations to stability results on the Cauchy problem for waves equations. By proceeding in that way the main difficulty is to control the residual terms, induced by the reduced Fourrier-Bros-Iagolnitzer transform, by the Cauchy data. Also, the uniqueness of continuation from Cauchy data is obtained as byproduct of stability results.
We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schr{ö}dinger equation to the magnetic Schr{ö}dinger equation. That is in presence of a magnetic potential. We establish observability inequalities and exponential stabilization by extending the usual multiplier method, under the same geometric condition to that needed for the Schr{ö}dinger equation. We also prove, with the help of elliptic Carleman inequalities, logarithmic stabilization results through a resolvent estimate. Although the approach is classical, these results on logarithmic stabilization seem to be new even for the Schr{ö}dinger equation.
We establish two-sided Gaussian bounds for fundamental solutions of general non-divergence form parabolic operators with Hölder continuous coefficients. The result we obtain is essentially based on parametrix method.
We deal with the problem of determining the shape of an inclusion embedded in a homogenous background medium. The multifre-quency electrical impedance tomography is used to image the inclusion. For different frequencies, a current is injected at the boundary and the resulting potential is measured. It turns out that the potential solves an elliptic equation in divergence form with discontinuous leading coefficient. For this inverse problem we aim to establish a logarithmic type stability estimate. The key point in our analysis consists in reducing the original problem to that of determining an unknown part of the inner boundary from a single boundary measurment. The stability estimate is then used to prove uniqueness results. We also provide an expansion of the solution of the BVP under consideration in the eigenfunction basis of the Neumann-Poincar{é} operator associated to the Neumann-Green function.
In this short paper we prove a global logarithmic stability of the Cauchy problem for H 2-solutions of an anisotropic elliptic equation in a Lip-schitz domain. The result we obtained is based on tools borrowed from the existing technics to establish stability estimate for the Cauchy problem [4] (see also [1]) combined with tools we already used in [6] to study an inverse medium problem.
This article is concerned with uniqueness and stability issues for the inverse spectral problem of recovering the magnetic field and the electric potential in a Riemannian manifold from some asymptotic knowledge of the boundary spectral data of the corresponding Schrödinger operator under Dirichlet boundary conditions. The spectral data consist of some asymptotic knowledge of a subset of eigenvalues and Neumann traces of the associated eigenfunctions of the magnetic Laplacian. We also address the same question for Schrödinger operators under Neumann boundary conditions, in which case we measure the Dirichlet traces of eigenfunctions. In our results we characterize the uniqueness of the magnetic field from a rate of growth of the eigenvalues, combined with suitable asymptotic properties of boundary observation of eigenfunctions, of the associated magnetic Schrödinger operator. To our best knowledge this is the first result proving uniqueness from such general asymptotic behavior of boundary spectral data.
We are interested in an inverse medium problem with internal data. This problem is originated from multi-waves imaging. We aim in the present work to study the well-posedness of the inversion in terms of the boundary conditions. We precisely show that we have actually a stability estimate of Hölder type. For sake of simplicity, we limited our study to the class of Helmholtz equations $Δ$+V with bounded potential V.
Corrigendum : An inverse problem in corrosion detection:stability estimates, J. Inv. Ill-posed Problems 12 (4) (2004), 349-367.
We analyze the inverse problem of identifying the diffusivity coefficient of a scalar elliptic equation as a function of the resolvent operator. We prove that, within the class of measurable coefficients, bounded above and below by positive constants, the resolvent determines the diffusivity in an unique manner. Furthermore we prove that the inverse mapping from resolvent to the coefficient is Lipschitz in suitable topologies. This result plays a key role when applying greedy algorithms to the approximation of parameter-dependent elliptic problems in an uniform and robust manner, independent of the given source terms. In one space dimension the results can be improved using the explicit expression of solutions, which allows to link distances between one resolvent and a linear combination of finitely many others and the corresponding distances on coefficients. These results are also extended to multi-dimensional elliptic equations with variable density coefficients. We also point out towards some possible extensions and open problems.
We improve the preceding results obtained by the first and the second authors in [3]. They concern the stability issue of the inverse problem that consists in determining the potential and the damping coefficient in a wave equation from an initial-to-boundary operator. We partially modify the arguments in [3] to show that actually we have H{ö}lder stability instead of logarithmic stability.