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Mourad Choulli

Publications and source records attributed to Mourad Choulli.

At least 73 records · Page 4Linked to original sources
arXiv2016

Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding "linearized" Dirichlet-to-Neumann map. The key ingredient in our analysis is a parabolic version of an elliptic Carleman inequality due to Bukhgeim and Uhlmann. This parabolic Carleman inequality enters in an essential way in the construction of complex geometric optics solutions that vanish at a part of the lateral boundary.

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arXiv2016

Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map

We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.

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arXiv2015

Various stability estimates for the problem of determining an initial heat distribution from a single measurement

We consider the problem of determining the initial heat distribution in the heat equation from a point measurement. We show that this inverse problem is naturally related to the one of recovering the coefficients of Dirichlet series from its sum. Taking the advantage of existing literature on Dirichlet series, in connection with M{ü}ntz 's theorem, we establish various stability estimates of H{ö}lder and logarithmic type. These stability estimates are then used to derive the corresponding ones for the original inverse problem, mainly in the case of one space dimension. In higher space dimensions, we are interested to an internal or a boundary measurement. This issue is closely related to the problem of observability arising in Control Theory. We complete and improve the existing results.

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arXiv2015

New Stability Estimates for the Inverse Medium Problem with Internal Data

A major problem in solving multi-waves inverse problems is the presence of critical points where the collected data completely vanishes. The set of these critical points depend on the choice of the boundary conditions, and can be directly determined from the data itself. To our knowledge, in the most existing stability results, the boundary conditions are assumed to be close to a set of CGO solutions where the critical points can be avoided. We establish in the present work new weighted stability estimates for an electro-acoustic inverse problem without assumptions on the presence of critical points. These results show that the Lipschitz stability far from the critical points deteriorates near these points to a logarithmic stability.

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arXiv2015

Determining the potential in a wave equation without a geometric condition. Extension to the heat equation

We prove a logarithmic stability estimate for the inverse problem of determining the potential in a wave equation from boundary measurements obtained by varying the first component of the initial condition. The novelty of the present work is that no geometric condition is imposed to the sub-boundary where the measurements are made. Our results improve those obtained by the first and second authors in [2]. We also show how the analysis for the wave equation can be adapted to an inverse coefficient problem for the heat equation

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arXiv2015

The problem of detecting corrosion by an electric measurement revisited

We establish a logarithmic stability estimate for the problem of detecting corrosion by a single electric measurement. We give a proof based on an adaptation of the method initiated in \cite{BCJ} for solving the inverse problem of recovering the surface impedance of an obstacle from the scattering amplitude. The key idea consists in estimating accurately a lower bound of the $L^2$-norm, locally at the boundary, of the solution of the boundary value problem used in modeling the problem of detection corrosion by an electric measurement

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arXiv2015

Logarithmic stability in determining a boundary coefficient in an ibvp for the wave equation

In [2] we introduced a method combining together an observability inequality and a spectral decomposition to get a logarithmic stability estimate for the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. The present work deals with an adaptation of that method to obtain a logarithmic stability estimate for the inverse problem of determining a boundary damping coefficient from boundary measurements. As in our preceding work, the different boundary measurements are generated by varying one of the initial conditions.

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arXiv2015

Logarithmic stability in determining two coefficients in a dissipative wave equation. Extensions to clamped Euler-Bernoulli beam and heat equations

We are concerned with the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. We establish stability estimates of logarithmic type when the measurements are given by the operator who maps the initial condition to Neumann boundary trace of the solution of the corresponding initial-boundary value problem. We build a method combining an observability inequality together with a spectral decomposition. We also apply this method to a clamped Euler-Bernoulli beam equation. Finally, we indicate how the present approach can be adapted to a heat equation.

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arXiv2015

Determining a boundary coefficient in a dissipative wave equation: Uniqueness and directional lipschitz stability

We are concerned with the problem of determining the damping boundary coefficient appearing in a dissipative wave equation from a single boundary measurement. We prove that the uniqueness holds at the origin provided that the initial condition is appropriately chosen. We show that the choice of the initial condition leading to uniqueness is related to a fine version of unique continuation property for elliptic operators. We also establish a Lipschitz directional stability estimate at the origin, which is obtained by a linearization process.

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arXiv2015

Observations on gaussian upper bounds for Neumann heat kernels

Given a domain $Ω$ of a complete Riemannian manifold $\mathcal{M}$ and define $\mathcal{A}$ to be the Laplacian with Neumann boundary condition on $Ω$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound $$ h(t,x,y)\leq \frac{C}{\left[V\_Ω(x,\sqrt{t})V\_Ω(y,\sqrt{t})\right]^{1/2}}\left( 1+\frac{d^2(x,y)}{4t}\right)^δe^{-\frac{d^2(x,y)}{4t}},\;\; t\textgreater{}0,\; x,y\in Ω. $$ Here $d$ is the geodesic distance on $\mathcal{M}$, $V\_Ω(x,r)$ is the Riemannian volume of $B(x,r)\cap Ω$, where $B(x,r)$ is the geodesic ball of center $x$ and radius $r$, and $δ$ is a constant related to the doubling property of $Ω$. As a consequence we obtain analyticity of the semigroup $e^{-t {\mathcal A}}$ on $L^p(Ω)$ for all $p \in [1, \infty)$ as well as a spectral multiplier result.

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arXiv2015

An inverse anisotropic conductivity problem induced bytwisting a homogeneous cylindrical domain

We consider the inverse problem of determining the unknown function $α: \mathbb{R} \rightarrow \mathbb{R}$ from the DN map associated to the operator $\mbox{div}(A(x',α(x\_3))\nabla \cdot)$ acting in the infinite straight cylindrical waveguide $Ω=ω\times \mathbb{R}$, where $ω$ is a bounded domain of $\mathbb{R}^2$. Here $A=(A\_{ij}(x))$, $x=(x',x\_3) \in Ω$, is a matrix-valued metric on $Ω$ obtained by straightening a twisted waveguide. This inverse anisotropic conductivity problem remains generally open, unless the unknown function $α$ is assumed to be constant. In this case we prove Lipschitz stability in the determination of $α$ from the corresponding DN map. The same result remains valid upon substituting a suitable approximation of the DN map, provided the function $α$ is sufficiently close to some {\it a priori} fixed constant.

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arXiv2015

Gaussian lower bound for the Neumann Green function of ageneral parabolic operator

Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish gaussian two-sided bounds for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for the gaussian two-sided bounds for the fundamental solution. We also indicate how our method can be adapted to get a gaussian lower bound for the Neumann heat kernel of a compact Riemannian manifold with boundary having non negative Ricci curvature.

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arXiv2015

Local boundedness property for parabolic BVP's and the gaussian upper bound for their Green functions

In the present note, we give a concise proof for the equivalence between the local boundedness property for parabolic Dirichlet BVP's and the gaussian upper bound for their Green functions. The parabolic equations we consider are of general divergence form and our proof is essentially based on the gaussian upper bound by Daners \cite{Da} and a Caccioppoli's type inequality. We also show how the same analysis enables us to get a weaker version of the local boundedness property for parabolic Neumann BVP's assuming that the corresponding Green functions satisfy a gaussian upper bound.

math.AP↗
arXiv2015

Double logarithmic stability estimate in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map

We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr{ö}dinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H minus 1-norm) of bounded (resp. L2) potentials whose difference is lying in any Sobolev space of order positive order.

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arXiv2013

Heat trace asymptotics and compactness of isospectral potentials for the Dirichlet Laplacian

Let $Ω$ be a $C^\infty$-smooth bounded domain of $\mathbb{R}^n$, $n \geq 1$, and let the matrix ${\bf a} \in C^\infty (\overlineΩ;\R^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(Ω)$-realization $A$ of the operator $-\mydiv ( {\bf a} \nabla \cdot)$ with Dirichlet boundary conditions. We perturb $A$ by some real valued potential $V \in C_0^\infty (Ω)$ and note $A_V=A+V$. We compute the asymptotic expansion of $\mbox{tr}\left( e^{-t A_V}-e^{-t A}\right)$ as $t \downarrow 0$ for any matrix ${\bf a}$ whose coefficients are homogeneous of degree $0$. In the particular case where $A$ is the Dirichlet Laplacian in $Ω$, that is when ${\bf a}$ is the identity of $\R^{n^2}$, we make the four main terms appearing in the asymptotic expansion formula explicit and prove that $L^\infty$-bounded sets of isospectral potentials of $A$ are $H^s$-compact for $s <2$.

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