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We consider the problem of maximal regularity for the semilinear non-autonomous evolution equations $$\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\, t \text {-a.e.}, \, u(0)=u_0. \end{aligned}$$ u ′ ( t ) + A ( t ) u ( t ) = F ( t , u ) , t -a.e. , u ( 0 ) = u 0 . Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space $$\mathcal {H}.$$ H . We prove the maximal regularity result in temporally weighted $$L^2$$ L 2 -spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value $$u_0$$ u 0 and the inhomogeneous term F. Our results are motivated by boundary value problems.

International audience; We consider the inverse problem of determining a general semilinear term appearing in nonlinear parabolic equations. For this purpose, we derive a new criterion that allows to prove global recovery of some general class of semilinear terms from lateral boundary measurements of solutions of the equation with initial condition fixed at zero. More precisely, we prove, for what seems to be the first time, the unique and stable recovery of general semilinear terms depending on time and space variables independently of the solution of the nonlinear equation from the knowledge of the parabolic Dirichlet-to-Neumann map associated with the solution of the equation with initial condition fixed at zero. Our approach is based on the second linearization of the inverse problem under consideration.

International audience; This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space dependent variable-order of fractional time derivative of the governing equation deviates from a known homogeneous background one. The observation data is given by the Neumann data of the solution of the governing equation for a specially designed Dirichlet data. Under a suitable jump condition on the deviation, it is shown that the most recent version of the time domain enclosure method enables one to extract information about the geometry of the obstacle and a qualitative nature of the jump, from the observation data.

We study the inverse problem of recovering a semilinear diffusion term $a(t,\lambda)$ as well as a quasilinear convection term $\mathcal B(t,x,\lambda,\xi)$ in a nonlinear parabolic equation $$\partial_tu-\textrm{div}(a(t,u) \nabla u)+\mathcal B(t,x,u,\nabla u)\cdot\nabla u=0, \quad \mbox{in}\ (0,T)\times\Omega,$$ given the knowledge of the flux of the moving quantity associated with different sources applied at the boundary of the domain. This inverse problem that is modeled by the solution dependent parameters $a$ and $\mathcal B$ has many physical applications related to various classes of cooperative interactions or complex mixing in diffusion processes. Our main result states that, under suitable assumptions, it is possible to fully recover the nonlinear diffusion term $a$ as well as the nonlinear convection term $\mathcal B$. The recovery of the diffusion term is based on the idea of solutions to the linearized equation with singularities near the boundary $\partial \Omega$. Our proof of the recovery of the convection term is based on the idea of higher order linearization to reduce the inverse problem to a density property for certain anisotropic products of solutions to the linearized equation. We show this density property by constructing sufficiently smooth geometric optic solutions concentrating on rays in $\Omega$.

Combining the methods of Cuenin [2019] and Borichev-Golinskii-Kupin [2009, 2018], we obtain the so-called Lieb-Thirring inequalities for non-selfadjoint perturbations of an effective Hamiltonian for bilayer graphene., 22 pages; few typos corrected

We consider the inverse problem of determining an electromagnetic potential appearing in an infinite cylindrical domain from boundary measurements. More precisely, we prove the stable recovery of some general class of magnetic field and electric potential from boundary measurements. Assuming some knowledge of the unknown coefficients close to the boundary, we obtain also some results of stable recovery with measurements restricted to some portion of the boundary. Our approach combines construction of complex geometric optics solutions and Carleman estimates suitably designed for our stability results stated in an unbounded domain.

The aim of this paper is the study of the signal received by an antenna (RX) when a transmit antenna (TX) sends a short pulse in a large scale space, for instance, in an urban environment. Integral equations are established which link the densities of charges and currents inside the environment objects with the incident field created by the TX antenna. From this equations we define an integral operator K. The densities can be obtained inverting 1 − K. The introduction of Daubechies wavelets allows us to obtain sparse matrices for KK * which is computationally convenient to get (1 − K) −1 .

We consider the problem of maximal regularity for the semilinear non-autonomous fractional equations B α u(t) + A(t)u(t) = F (t, u), t-a.e. Here, B α denotes the Riemann-Liouville fractional derivative of order α ∈ (0, 1) w.r.t. time and the time dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space H. We prove maximal L p-regularity results and other regularity properties for the solutions of the above equation under minimal regularity assumptions on the forms and the inhomogeneous term F.

We investigate normal forms of germs of glancing Lagrangian intersections, and their semiclassical counterparts. This is motivated by singularities of Bessel beams, or diffraction problems by an obstacle with a non convex boundary.

International audience; Abstract We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

We make a spectral analysis of the massive Dirac operator in a tubular neighborhood of an unbounded planar curve,subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip limit. We also investigate the existence of bound states in the non-relativistic limit and give a geometric quantitative condition for the bound states to exist., Some technical results have been moved to the Appendix. To appear on Ann. Henri Poincar\'e

We consider the inverse problem of determining some class of nonlinear terms appearing in an elliptic equation from boundary measurements. More precisely, we study the stability issue for this class of inverse problems. Under suitable assumptions, we prove a Lipschitz and a Hölder stability estimate associated with the determination of quasilinear and semilinear terms appearing in this class of elliptic equations from measurements restricted to an arbitrary part of the boundary of the domain. Besides their mathematical interest, our stability estimates can be useful for improving numerical reconstruction of this class of nonlinear terms. Our approach combines the linearization technique with applications of suitable class of singular solutions.

International audience

International audience; We study a class of dynamical semigroups $(\mathbb{L}^n)_{n\in\mathbb{N}}$ that emerge, by a Feynman--Kac type formalism, from a random quantum dynamical system $(\mathcal{L}_{\omega_n}\circ\cdots\circ\mathcal{L}_{\omega_1}(\rho_{\omega_0}))_{n\in\mathbb{N}}$ driven by a Markov chain $(\omega_n)_{n\in\mathbb{N}}$. We show that the almost sure large time behavior of the system can be extracted from the large $n$ asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator $\mathbb{L}$. As a physical application, we consider the case where the $\mathcal{L}_\omega$'s are the reduced dynamical maps describing the repeated interactions of a system $\mathcal{S}$ with thermal probes $\mathcal{C}_\omega$. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.

We show that the knowledge of the Dirichlet-to-Neumann maps given on an arbitrary open non-empty portion of the boundary of a smooth domain in $\mathbb{R}^n$, $n\ge 2$, for classes of semilinear and quasilinear conductivity equations, determines the nonlinear conductivities uniquely. The main ingredient in the proof is a certain $L^1$-density result involving sums of products of gradients of harmonic functions which vanish on a closed proper subset of the boundary.

In this paper we study the inverse problem of identifying a source or an initial state in a time-fractional diffusion equation from the knowledge of a single boundary measurement. We derive logarithmic stability estimates for both inversions. These results show that the ill-posedness increases exponentially when the fractional derivative order tends to zero, while it exponentially decreases when the regularity of the source or the initial state becomes larger. The stability estimate concerning the problem of recovering the initial state can be considered as a weak observability inequality in control theory. The analysis is mainly based on Laplace inversion techniques and a precise quantification of the unique continuation property for the resolvent of the time-fractional diffusion operator as a function of the frequency in the complex plane. We also determine a global time regularity for the time-fractional diffusion equation which is of interest itself.

We prove a global uniqueness result for the Calderon inverse problem for a general quasilinear isotropic conductivity equation on a bounded open set with smooth boundary in dimension n ≥ 3 . Performing higher order linearizations of the nonlinear Dirichlet–to–Neumann map, we reduce the problem of the recovery of the differentials of the quasilinear conductivity, which are symmetric tensors, to a completeness property for certain anisotropic products of solutions to the linearized equation. The completeness property is established using complex geometric optics solutions to the linearized conductivity equation, whose amplitudes concentrate near suitable two dimensional planes.

International audience; This text deals with multidimensional Borg-Levinson inverse theory. Its main purpose is to establish that the Dirichlet eigenvalues and Neumann boundary data of the operator −∆+q, acting in a bounded domain of R d with d 2, uniquely determine the realvalued bounded potential q. We first address the case of incomplete spectral data, where finitely many boundary spectral eigen-pairs remain unknown. Under suitable summability condition on the Neumann data, we also consider the case where only the asymptotic behavior of the eigenvalues is known. Finally, we use the multidimensional Borg-Levinson theory for solving parabolic inverse coefficient problems.

This paper is a continuation of a previous work about the study of the survival probability modelizing the molecular predissociation in the Born-Oppenheimer framework. Here we consider the critical case where the reference energy corresponds to the value of a crossing of two electronic levels, one of these two levels being confining while the second dissociates. We show that the survival probability associated to a certain initial state is a sum of the usual time-dependent exponential contribution, and a reminder term that is jointly polynomially small with respect to the time and the semiclassical parameter. We also compute explicitly the main contribution of the remainder., 37 pages, 1 figure

In this paper, we prove Lipschitz stable determination of the complex-valued electric potential and the direction of the magnetic field appearing in the dynamic Schrodinger equation with static coefficients, by finitely many partial boundary measurements of the solution. This is by means of the Bukhgeim–Klibanov method, based on an appropriate Carleman estimate. Since the time symmetrization of the static magnetic Schrodinger equation around t = 0 is not possible, we preliminarily establish a Carleman inequality specifically designed for this problem.

In this work, we study the spectrum of the Dirichlet Laplacian in a sheared 2D quantum waveguide. First, we establish Hardy-type inegualities for the Dirichlet Laplacian. Then, we locate the essential spectrum of the operator under a certain assumption on the geometry of the waveguide. We establish the stability of the essential spectrum with respect to a perturabtion in the case where the shear deformation admits a limit (possibly infinite) at infinity. Finally, we establish sufficient conditions which guarantee the existence of discrete eigenvalues below the threshold of the essential spectrum.; Dans ce travail, nous étudions le spectre de l’opérateur Laplacien dans un guide d’ondes quantique 2D cisaillé, avec une condition au bord de Dirichlet. Dans un premier temps, nous établissons des inégalités de type Hardy pour l’opérateur Laplacien. Ensuite, nous localisons le spectre essentiel de l’opérateur sous une certaine hypothèse sur la géométrie du guide d’ondes. Nous montrons la stabilité du spectre essentiel vis-à-vis d’une perturbation dans le cas où la déformation de cisaillement admet une limite (éventuellement infinie) à l’infini. Enfin, nous établissons des conditions suffisantes qui garantissent l’existence des valeurs propres discrètes au-dessous du seuil du spectre essentiel.

We prove some local estimates on the trace of spectral projectors for random Schrodinger operators restricted to cubes \(\Lambda \subset \mathbb {R}^d\). We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrodinger operators on cubes \(\Lambda \subset \mathbb {R}^d\), for \(d \geqslant 1\). The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g., diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Further, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear least-squares fitting with either fractional polynomials or rational approximations as the model function, and provide numerical experiments to illustrate the approach for small time $t$., 20 pages, 2 figures, to appear at Proceedings of the Royal Society A

In this report we present preliminary results about the tunneling problem for a magnetic Schr\"odinger operator. As a motivation we consider the 3-D time-dependent Schr\"odinger operator $H(t)=-h^2\Delta+V+E(t)\cdot x$ where $V$ is a radial potential and $E(t)$ a circularly polarized field with uniform frequency $\omega$. The quantum monodromy operator (QMO) that takes the system through a complete period $T=2\pi/\omega$, turns out to be unitarily equivalent to $e^{iTP_A(x,hD_x)/h}$, where $P_A(x,hD_x))$ identifies with a magnetic Schr\"odinger operator. When $V$ is sufficiently confining, $P_A(x,hD_x))$ presents a double magnetic well. Then we construct its semi-classical ground state and examine the splitting between its two first eigenvalues.

International audience; In this paper, we investigate inverse source problems for a wide range of PDEs of parabolic and hyperbolic types as well as time-fractional evolution equations by partial interior observation. Restricting the source terms to the form of separated variables, we establish uniqueness results for simultaneously determining both temporal and spatial components without non-vanishing assumptions at [Formula: see text], which seems novel to the best of our knowledge. Remarkably, mostly we allow a rather flexible choice of the observation time not necessarily starting from [Formula: see text], which fits into various situations in practice. Our main approach is based on the combination of the Titchmarsh convolution theorem with unique continuation properties and time-analyticity of the PDEs under consideration.

We study semi-classical asymptotics for problems with localized right-hand sides by considering a Hamiltonian H(x, p) positively homogeneous of degree m ≥ 1 on T * R^2 \ 0. The energy shell is E = 1, and the right-hand side f_h is microlocalized: (1) on the vertical plane Λ = {x = x_0 }; (2) on the "cylinder" Λ = {(X, P) = ϕω(ψ), ω(ψ) ; ϕ ∈ R, ω(ψ) = (cos ψ, sin ψ)}. We restrict essentially to the isotropic case H(x, p) = |p|^m/ ρ(x) , with ρ a smooth positive function. In case (2), Λ is the frequency set of Bessel function J_0 (|x|/h), and the solution u_h of (H(x, hD_x) − 1)u_h = f _h , which is called a "Bessel beam", arises in the theory of optical fibers.

In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion type equations, including both classical and time-fractional diffusion equations, from the flux measurement observed at one point on the boundary. The mathematical model for time-fractional diffusion equations involves a Djrbashian-Caputo fractional derivative in time. We prove a uniqueness result in an unknown medium (e.g., diffusion coefficients, obstacle, initial condition and source), i.e., the recovery of the order of derivation in a diffusion process having several pieces of unknown information. The proof relies on the analyticity of the solution at large time, asymptotic decay behavior, strong maximum principle of the elliptic problem and suitable application of the Hopf lemma. Further we provide an easy-to-implement reconstruction algorithm based on a nonlinear least-squares formulation, and several numerical experiments are presented to complement the theoretical analysis., Comment: 22 pages, 3 figures, to appear at SIAM Journal on Applied Mathematics

In this paper, we study the opening of a spectral gap for a class of 2-dimensional periodic Hamiltonians which include those modelling multilayer graphene. The kinetic part of the Hamiltonian is given by $\sigma \cdot F(-i\nabla)$, where $\sigma$ denotes the Pauli matrices and $F$ is a sufficiently regular vector-valued function which equals 0 at the origin and grows at infinity. Its spectrum is the whole real line. We prove that a gap appears for perturbations in a certain class of periodic matrix-valued potentials depending on $F$, and we study how this gap depends on different parameters.

We study the stability issue for the inverse problem of determining a coefficient appearing in a Schr\"odinger equation defined on an infinite cylindrical waveguide. More precisely, we prove the stable recovery of some general class of non-compactly and non periodic coefficients appearing in an unbounded cylindrical domain. We consider both results of stability from full and partial boundary measurements associated with the so called Dirichlet-to-Neumann map. To the best of our knowledge, our results are the first results of stable recovery of such class of coefficients for an elliptic equation in an unbounded domain.

In this article we study two inverse problems of recovering a space-time-dependent source component from the lateral boundary observation in a subdiffusion model. The mathematical model involves a ...

This article is concerned with an inverse problem on simultaneously determining some unknown coefficients and/or an order of derivative in a multidimensional time-fractional evolution equation either in a Euclidean domain or on a Riemannian manifold. Based on a special choice of the Dirichlet boundary input, we prove the unique recovery of at most two out of four $$\varvec{x}$$ -dependent coefficients (possibly with an extra unknown fractional order) by a single measurement of the partial Neumann boundary output. Especially, both a vector-valued velocity field of a convection term and a density can also be uniquely determined. The key ingredient turns out to be the time-analyticity of the decomposed solution, which enables the construction of Dirichlet-to-Neumann maps in the frequency domain and thus the application of inverse spectral results.

We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold., Comment: 31 pages

This PhD is devoted to the determination of unknown coefficients appearing in the linear Boltzmann equation or the parabolic equation. Its main objective is to establish results of uniqueness and stability in the identification of these coefficients from ad-hoc data. The first part of the thesis is devoted to the study of the inverse problems associated with the linear Boltzmann equation (transport equation). The inverse problem considered concerns the identification of the diffusion and absorption coefficients from knowledge of the albedo operator, which models all possible boundary measurements. The second part of the thesis is devoted to treat inverse problems that concern parabolic equations. Precisely, we specify our study on determining a coefficient, arising in the parabolic equation, from full and partial measurement of the solution on the boundary in an arbitrary time interval.; Cette thèse est consacrée à la détermination de coefficients inconnus apparaissant dans l’équation linéaire de Boltzmann ou l’équation de la chaleur. Son objectif principal est d’établir des résultats d’unicité et de stabilité dans l’identification de ces coefficients à partir de données ad-hoc. La première partie de la thèse est consacrée à l’étude des problèmes inverses associés à l’équation linéaire de Boltzmann (équation de transport). Le problème inverse considéré concerne l’identification des coefficients de diffusion et d’absorption à partir de la mesure latérale du flux sortant de la solution. La deuxième partie de la thèse est consacrée à la résolution d’un problème inverse pour l’équation de la chaleur. Nous étudions la stabilité de la détermination du coefficient d’ordre zéro à partir de données de type Neumann complètes ou partielles, mesurées sur un intervalle de temps arbitraire.

For a parabolic equation in the spatial variable x = ( x 1 , … , x n ) {x=(x_{1},\ldots,x_{n})} and time t, we consider an inverse problem of determining a coefficient which is independent of one spatial component x n {x_{n}} by lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also, we prove similar results for the corresponding inverse source problem.

This thesis deals with various aspects of spectral theory of operators used to model graphene. It is made of twoparts.The first parts deals with the periodic case. I begin by presenting a general theory of periodic systems. Iintroduce then different models of graphene and compare them. Finally, I look at various ways to makegraphene a semiconductor. In particular, I study different types of nanoribbons and I give a result of gapopening for a pseudodifferential operator.The second part deals with the disordered case. I begin by presenting a general theory of random operators.Then, I briefly explain multiscale analysis, which is the method used to prove the main result of this theory,which is called Anderson localization. Finally, I give a proof of this localization for a model of graphene and aresult on the integrated density of states.; Cette thèse traite de différents aspects de la théorie spectrale d’opérateurs utilisés pour modéliser le graphène.Elle est constituée de deux parties.La première traite du cas périodique. Je commence par présenter la théorie générale des systèmes périodiques.J’introduis ensuite les différents modèles de graphène en les comparant. Enfin, je m’intéresse à différentesfaçons de rendre le graphène semi-conducteur. Je fais en particulier une étude de nanorubans de divers types etprésente un résultat d’ouverture d’une lacune spectrale pour un opérateur pseudo-différentiel.La deuxième partie traite du cas désordonné. Je commence par présenter la théorie générale des opérateursaléatoires. J’explique ensuite succinctement l’analyse multi-échelles qui est la méthode permettant de montrerle résultat essentiel de cette théorie, appelé localisation d’Anderson. Enfin, je donne la preuve de cettelocalisation pour un modèle de graphène ainsi qu’un résultat sur la densité d’états intégrée.

This thesis deals with various aspects of spectral theory of operators used to model graphene. It is made of two parts.The first parts deals with the periodic case. I begin by presenting a general theory of periodic systems. I introduce then different models of graphene and compare them. Finally, I look at various ways to make graphene a semiconductor. In particular, I study different types of nanoribbons and I give a result of gap opening for a pseudodifferential operator. The second part deals with the disordered case. I begin by presenting a general theory of random operators. Then, I briefly explain multiscale analysis, which is the method used to prove the main result of this theory, which is called Anderson localization. Finally, I give a proof of this localization for a model of graphene and a result on the integrated density of states., Cette thèse traite de différents aspects de la théorie spectrale d’opérateurs utilisés pour modéliser le graphène. Elle est constituée de deux parties. La première traite du cas périodique. Je commence par présenter la théorie générale des systèmes périodiques. J’introduis ensuite les différents modèles de graphène en les comparant. Enfin, je m’intéresse à différentes façons de rendre le graphène semi-conducteur. Je fais en particulier une étude de nanorubans de divers types et présente un résultat d’ouverture d’une lacune spectrale pour un opérateur pseudo-différentiel. La deuxième partie traite du cas désordonné. Je commence par présenter la théorie générale des opérateurs aléatoires. J’explique ensuite succinctement l’analyse multi-échelles qui est la méthode permettant de montrer le résultat essentiel de cette théorie, appelé localisation d’Anderson. Enfin, je donne la preuve de cette localisation pour un modèle de graphène ainsi qu’un résultat sur la densité d’états intégrée.

International audience; We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.

We consider a mathematical model for the weak decay of the intermediate boson $Z^0$ into neutrinos and antineutrinos. We prove that the total Hamiltonian has a unique ground state in Fock space and we establish a limiting absorption principle, local decay and a property of relaxation to the ground state for initial states and observables suitably localized in energy and position. Our proofs rest, in particular, on Mourre's theory and a low-energy decomposition., Comment: 42 pages

We are concerned with the dependence of the lowest positive eigenvalue of the Dirac operator on the geometry of rectangles, subject to infinite-mass boundary conditions. We conjecture that the square is a global minimiser both under the area or perimeter constraints. Contrary to well-known non-relativistic analogues, we show that the present spectral problem does not admit explicit solutions. We prove partial optimisation results based on a variational reformulation and newly established lower and upper bounds to the Dirac eigenvalue. We also propose an alternative approach based on symmetries of rectangles and a non-convex minimisation problem; this implies a sufficient condition formulated in terms of a symmetry of the minimiser which guarantees the conjectured results., Comment: 11 pages; due to a gap in the proof in our previous version (see Remark 1), we obtain just partial results, by an alternative approach; version accepted for publication in Journal of Mathematical Physics

Given ( M , g ) , a compact connected Riemannian manifold of dimension d ⩾ 2 , with boundary ∂M, we consider an initial boundary value problem for a fractional diffusion equation on ( 0 , T ) × M , T > 0 , with time-fractional Caputo derivative of order α ∈ ( 0 , 1 ) ∪ ( 1 , 2 ) . We prove uniqueness in the inverse problem of determining the smooth manifold ( M , g ) (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solutions on a subset of ∂M at fixed time. In the “flat” case where M is a compact subset of R d , two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation ρ ∂ t α u − div ( a ∇ u ) + q u = 0 on ( 0 , T ) × M are recovered simultaneously.

International audience; Prices of European call options in a regime-switching local-volatility model can be computed by solving a parabolic system which generalizes the classical Black and Scholes equation, giving these prices as functionals of the local-volatilities. We prove Lipschitz stability for the inverse problem of determining the local-volatilities from quoted call option prices for a range of strikes, if the calls are indexed by the different states of the continuous Markov chain which governs the regime switches.

We study the inverse problem of determining a magnetic Schrodinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even prove the unique determination of the magnetic field and the electric potential from measurements restricted to a bounded subset of the infinite boundary. Finally, in the case of a waveguide taking the form of an infinite cylindrical domain, we prove the recovery of the magnetic field and the electric potential from partial data corresponding to restriction of Neumann boundary measurements to slightly more than half of the boundary. We establish all these results by mean of a new class of complex geometric optics solutions and of Carleman estimates suitably designed for our problem stated in an unbounded domain and with bounded electromagnetic potentials.

International audience; This article is devoted to the simultaneous resolution of three inverse problems, among the most important formulation of inverse problems for partial differential equations, stated for some class of diffusion equations from a single boundary measurement. Namely, we consider the simultaneous unique determination of several class of coefficients, some internal sources (a source term and an initial condition) and an obstacle appearing in a diffusion equation from a single boundary measurement. Our problem can be formulated as the simultaneous determination of information about a diffusion process (velocity field, density of the medium), an obstacle and of the source of diffusion. We consider this problems in the context of a classical diffusion process described by a convection-diffusion equation as well as an anomalous diffusion phenomena described by a time fractional diffusion equation.

International audience

International audience; This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form $f(x)g(t)$ and $f(x_1,x_2,t) h(x_3)$, where $g(t)$ and $h(x_3)$ are given and $x=(x_1, x_2, x_3)$ is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variables. For moving point sources radiating periodic signals, the data recorded at four receivers are proven sufficient to uniquely recover the orbit function. Simultaneous determination of embedded obstacles and source terms was verified in an inhomogeneous background medium using the observation data of infinite time period. Our approach depends heavily on the Laplace transform.

We are motivated by tripartite entanglement for fermions. While GHZ or W states involve 3-fold intrication, we consider here piecewise intrication of 3 fermions in ${\bf C}^2$, namely of type $ab+bc+ca$. Before interaction with Stern-Gerlach apparatus, qu-bits are distinguishable; at the output however they turn into un-distinguishable particles, whose anti-symmetric wave function is of the form $\det(b-a,c-a)$ (affine determinant). More generally, $d+1$ intricated fermions in ${\bf C}^d$ can be represented by the anti-symmetric wave function $\det(a_1-a_0,a_2-a_0,\cdots,a_d-a_0)$. We investigate also properties of affine Slater determinants, as expectation values or reduced density matrices., Conference ISQS26, Prague, July 2019

International audience; We consider the inverse source problem of determining a source term depending on both time and space variable for fractional and classical diffusion equations in a cylindrical domain from boundary measurements. With suitable boundary conditions we prove that some class of source terms which are independent of one space direction, can be reconstructed from boundary measurements. Actually , we prove that this inverse problem is well-posed. We establish also some results of Lipschitz stability for the recovery of source terms which we apply to the stable recovery of time-dependent coefficients.

International audience; We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.

We consider a restricted Dirichlet-to-Neumann map Λ S , R T associated with the operator ∂ t 2 − Δ g + A + q where Δ g is the Laplace-Beltrami operator of a Riemannian manifold ( M , g ) , and A and q are a vector field and a function on M. The restriction Λ S , R T corresponds to the case where the Dirichlet traces are supported on ( 0 , T ) × S and the Neumann traces are restricted on ( 0 , T ) × R . Here S and R are open sets, which may be disjoint, on the boundary of M. We show that Λ S , R T determines uniquely, up the natural gauge invariance, the lower order terms A and q in a neighborhood of the set R assuming that R is strictly convex and that the wave equation is exactly controllable from S in time T / 2 . We give also a global result under a convex foliation condition. The main novelty is the recovery of A and q when the sets R and S are disjoint. We allow A and q to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.

We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions.Particles interact via Coulomb potential and electrons have relativistic kinetic energy given by (p2+m2)1/2-m.We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance |D| between the atoms. We rigorously compute all terms in the binding energy up to order |D|-9 with error term of order O(|D|-10). As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schrödinger operators with permutation symmetry imposed by the Pauli principle and new estimates of the localization error. In addition we prove the van der Waals-London law for the projected Dirac operator, known as the Brown-Ravenhall operator. In this case we do not calculate the coefficients explicitly and we obtain an error term of order O(|D|-7).; On considère un système constitué de plusieurs atomes où les noyaux sont supposés fixes et ponctuels. Les particules interagissent via le potentiel de Coulomb et les électrons ont une énergie cinétique pseudo-relativiste donnée par (p2+m2)1/2-m. On démontre la loi de van der Waals-London qui dit que l'énergie d'interaction entre atomes neutres décroit comme |D|-6 où |D| est la distance entre atomes. Nous calculons rigoureusement tous les termes de l'énergie de liaison jusqu'à l'ordre |D|-9 avec un terme d'erreur en O(|D|-10). Comme étape intermédiaire, nous établissons la décroissance exponentielle des fonctions propres de l'opérateur de Schrödinger à plusieurs particules avec les symétries imposées par le principe de Pauli, et des estimations sur le terme d'erreur de localisation. Nous montrons de plus la loi de van der Waals-London pour l'opérateur de Dirac projeté connu sous le nom d'opérateur de Brown et Ravenhall. Dans ce dernier cas on obtient un terme d'erreur en O(|D|-7).