Carleman Estimates Research Articles

Hölder stability and uniqueness for the mean field games system via Carleman estimates

AbstractWe are concerned with the mathematical study of the mean field games system (MFGS). In the conventional setup, the MFGS is a system of two coupled nonlinear parabolic partial differential equation (PDE)s of the second order in a backward–forward manner, namely, one terminal and one initial condition are prescribed, respectively, for the value function and the population density . In this paper, we show that uniqueness of solutions to the MFGS can be guaranteed if, among all four possible terminal and initial conditions, either only two terminals or only two initial conditions are given. In both cases, Hölder stability estimates are proven. This means that the accuracies of the solutions are estimated in terms of the given data. Moreover, these estimates readily imply uniqueness of corresponding problems for the MFGS. The main mathematical apparatus to establish those results is two new Carleman estimates, which may find application in other contexts associated with coupled parabolic PDEs.

Unique continuation for a fourth-order stochastic parabolic equation

In this paper, we prove a boundary unique continuation property for a fourth-order stochastic parabolic equation evolving in a domain G⊂Rn. Our result shows that the value of the solution can be determined by the observation on an arbitrary open subset of the boundary. The quantitative version of this property can be derived by the global Carleman estimate, which is deduced from a weighted identity for a fourth-order stochastic parabolic operator. The results in this paper are also new even if the fourth-order stochastic parabolic equation reduces to the corresponding fourth-order deterministic parabolic equation.

Strong unique continuation and global regularity estimates for nanoplates

In this paper, we analyze some properties of a sixth-order elliptic operator arising in the framework of the strain gradient linear elasticity theory for nanoplates in flexural deformation. We first rigorously deduce the weak formulation of the underlying Neumann problem as well as its well posedness. Under some suitable smoothness assumptions on the coefficients and on the geometry, we derive interior and boundary regularity estimates for the solution of the Neumann problem. Finally, for the case of isotropic materials, we obtain new Strong Unique Continuation results in the interior, in the form of doubling inequality and three spheres inequality by a Carleman estimates approach.

Determination of source or initial values for acoustic equations with a time-fractional attenuation

In this paper, we consider the inverse problems of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation.

Null controllability of a retarded population dynamics model with interior degeneracy

In this work, we establish a null controllability property of a dispersive age‐structured population dynamics model with finite‐time delay on the mortality rate term. We assume that the dispersion term includes an interior degeneracy satisfying some suitable assumptions. To achieve our main results, we act according to the classical procedure, which is the Carleman estimates of a suitable adjoint system, and then its associated observability inequality. It is well known that the basis of Carleman's estimates is an adjusted weight function and one of the originalities of our article is the expression of the function as a function of age and time variables.

Global Lipschitz stability for inverse problems of wave equations on Lorentzian manifolds

We prove global Lipschitz stability for an inverse source problem of a system of wave equations on a Lorentzian manifold. The method used in this paper is widely known as the Bukhgeim–Klibanov method. However, the conventional method is not sufficient for the application to the hyperbolic partial differential equation with time-dependent coefficients to obtain the Lipschitz stability, and further innovations are needed. In this paper, we present an improved global Carleman estimate and an energy estimate to obtain the Lipschitz stability.

Local null controllability for the thermistor problem

In this paper, we deal with the local null controllability of an initial boundary value problem for a thermistor equation. The control is distributed, locally in space. The main ingredients of the proof are suitable Carleman estimates for an adjoint system and Liusternik’s Inverse Mapping Theorem in Hilbert spaces. As a complement to the theoretical results, we present formulations and numerical simulations for the controllability problem, the results were obtained through the finite element method and a quasi-Newton method, simulated in the software Freefem++. Finally, large time null controllability, stabilization and others possible extensions and open problems concerning other systems are presented.

Boundary null controllability of convection-diffusion equations with constraints on the state and application to the identification of boundary pollution parameters

We study a null controllability problem for a convection-diffusion equation with the control contained in Fourier boundary condition to identify M parameters of pollution. The results are achieved by means of an observability inequality derived from new Carleman estimates for the boundary null controllability and the construction of a boundary sentinel to identify the parameters.

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.

The problem of determining multiple coefficients in an ultrahyperbolic equation

Abstract In this article, we discuss an inverse problem of determining unknown coefficients of the first-order derivatives in an ultrahyperbolic equation. By a finite set of measurements, we prove the uniqueness of solution of the problem in semi-geodesic coordinates under some conditions on the principal coefficients of the equation. Our main tool is a Carleman estimate.

A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves

In this paper, we establish a result of unique continuation for a special two-dimensional nonlinear system that models the evolution of long water waves with small amplitude in the presence of surface tension. More precisely, we will show that if (eta ,Phi ) = (eta (x,y, t),Phi (x,y, t)) is a solution of the nonlinear system, in a suitable function space, and (eta ,Phi ) vanishes on an open subset Omega of mathbb {R}^2 times [-T,T], then (eta ,Phi )equiv 0 in the horizontal component of Omega . To state such property, we use a Carleman-type estimate for a differential operator mathcal {L} related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality.

Carleman Estimates and Simultaneous Boundary Controllability of Uncoupled Wave Equations

Carleman Estimates and Simultaneous Boundary Controllability of Uncoupled Wave Equations

Space-like quantitative uniqueness for parabolic operators

We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a C1 potential V. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.

Carleman estimates for sub-Laplacians on Carnot groups

In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group mathbb G for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp vanishing order estimate for solutions to stationary Schrödinger equations.

A remark on quantitative unique continuation from subsets of the boundary of positive measure

The question of uniqueness of harmonic functions in a domain Ω ⊂ R d \Omega \subset \mathbb {R}^d with boundary ∂ Ω \partial \Omega , satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset of the boundary ω \omega of positive d − 1 d-1 dimensional measure has attracted a lot of attention (see e.g. V. Adolfsson and L. Escauriaza [Comm. Pure Appl. Math. 50 (1997), pp. 935–969]; F.-H. Lin [Comm. Pure Appl. Math. 44 (1991), pp. 287–308]; X. Tolsa [Comm. Pure Appl. Math. 76 (2023), pp. 305–336]). It is essentially a consequence of Carleman estimates when ω \omega contains an open subset of the boundary (uniqueness for second order elliptic operators across an hypersurface). The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, Birkhäuser/Springer, Cham, 2018, pp. 333-344; Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018, World Scientific Publishing, Hackensack, NJ, 2018, pp. 2391–2411], we consider the simpler case of W 2 , ∞ W^{2, \infty } domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2 , ∞ W^{2, \infty } domain with boundary.

The Bulk-Boundary Correspondence for the Einstein Equations in Asymptotically Anti-de Sitter Spacetimes

In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes ( mathscr {M}, g ) with conformal boundary ( mathscr {I}, mathfrak {g}). We establish a correspondence, near mathscr {I}, between such spacetimes and their conformal boundary data on mathscr {I}. More specifically, given a domain mathscr {D} subset mathscr {I}, we prove that the coefficients mathfrak {g}^{scriptscriptstyle (0)} = mathfrak {g} and mathfrak {g}^{scriptscriptstyle (n)} (the undetermined term, or stress energy tensor) in a Fefferman–Graham expansion of the metric g from the boundary uniquely determine g near mathscr {D}, provided mathscr {D} satisfies a generalised null convexity condition (GNCC). The GNCC is a conformally invariant criterion on mathscr {D}, first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in mathscr {M} near mathscr {D}, and with the pseudoconvexity degenerating in the limit at mathscr {D}. As a corollary of this result, we deduce that conformal symmetries of ( mathfrak {g}^{scriptscriptstyle (0)}, mathfrak {g}^{scriptscriptstyle (n)} ) on domains mathscr {D} subset mathscr {I} satisfying the GNCC extend to spacetime symmetries near mathscr {D}. The proof, which does not require any analyticity assumptions, relies on three key ingredients: (1) a calculus of vertical tensor-fields developed for this setting; (2) a novel system of transport and wave equations for differences of metric and curvature quantities; and (3) recently established Carleman estimates for tensorial wave equations near the conformal boundary.

Discrete Carleman estimates and application to controllability for a fully-discrete parabolic operator with dynamic boundary conditions

We consider a fully-discrete approximation of a 1-D heat equation with dynamic boundary conditions for which we provide a controllability result. The proof of this result is based on a relaxed observability inequality for the corresponding adjoint system. This is conducted using a suitable Carleman estimate for such models where the discrete parameters h and Δt are connected to one of the large Carleman parameters.

Upper Bound of Critical Sets of Solutions of Elliptic Equations in the Plane

In this note, we investigate the measure of singular sets and critical sets of real-valued solutions of elliptic equations in two dimensions. These singular sets and critical sets are finitely many points in the plane. Adapting the Carleman estimates involving polynomial functions at singularities by Donnelly and Fefferman (J. Amer. Math. Soc. 3, 333–353, 1990), we obtain the upper bounds of singular points and critical points.

The mean field games system: Carleman estimates, Lipschitz stability and uniqueness

Abstract An overdetermination is introduced in an initial condition for the second order mean field games system (MFGS). This makes the resulting problem close to the classical ill-posed Cauchy problems for PDEs. Indeed, in such a problem an overdetermination in boundary conditions usually takes place. A Lipschitz stability estimate is obtained. This estimate implies uniqueness. A new Carleman estimate is derived. This latter estimate is called “quasi-Carleman estimate”, since it contains two test functions rather than a single one in conventional Carleman estimates. These two estimates play the key role. Carleman estimates were not applied to the MFGS prior to the recent work of Klibanov and Averboukh in [M. V. Klibanov and Y. Averboukh, Lipschitz stability estimate and uniqueness in the retrospective analysis for the mean field games system via two Carleman estimates, preprint 2023, https://arxiv.org/abs/2302.10709].

Inverse problems of damped wave equations with Robin boundary conditions: an application to blood perfusion

Knowledge of the properties of biological tissues is essential in monitoring any abnormalities that may be forming and have a major impact on organs malfunctioning. Therefore, these disorders must be detected and treated early to save lives and improve the general health. Within the framework of thermal therapies, e.g. hyperthermia or cryoablation, the knowledge of the tissue temperature and of the blood perfusion rate are of utmost importance. Therefore, motivated by such a significant biomedical application, this paper investigates, for the first time, the uniqueness and stable reconstruction of the space-dependent (heterogeneous) perfusion coefficient in the thermal-wave hyperbolic model of bio-heat transfer from Cauchy boundary data using the powerful technique of Carleman estimates. Additional novelties consist in the consideration of Robin boundary conditions, as well as developing a mathematical analysis that leads to stronger stability estimates valid over a shorter time interval than usually reported in the literature of coefficient identification problems for hyperbolic partial differential equations. Numerically, the inverse coefficient problem is recast as a nonlinear least-squares minimization that is solved using the conjugate gradient method (CGM). Both exact and noisy data are inverted. To achieve stability, the CGM is stopped according to the discrepancy principle. Numerical results for a physical example are presented and discussed, showing the convergence, accuracy and stability of the inversion procedure.