Abstract
We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.
AbstractThe integro-differential system of viscoelasticity equations with a source of explosive type is considered.
It is assumed that the coefficients of the equations depend only on one spatial variable.
The problem of determining the kernel included in the integral terms of the equations is studied. The solution of the problem is reduced to one inverse problem for scalar hyperbolic equations.
This inverse problem is replaced by an equivalent system of integral equations for unknown functions.
The principle of constricted mapping in the space of continuous functions with weighted norms to the latter is applied.
The theorem of global unique solvability is proved and the stability estimate of solution to the inverse problem is obtained.
AbstractWe consider the inverse boundary value problem for the dynamical steady-state convection-diffusion equation.
We prove that the first-order coefficient and the scalar potential are uniquely determined by the Dirichlet-to-Neumann map.
More precisely, we show in dimension {n\geq 3} a log-type stability estimate for the inverse problem under consideration.
The method is based on reducing our problem to an auxiliary inverse problem and the construction of complex geometrical optics solutions of this problem.
We prove new global Hölder-logarithmic stability estimates for the near-field inverse scattering problem in dimensiond≥3. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. In addition, a global logarithmic stability estimate for this inverse problem in dimensiond=2is also given.
AbstractThe problem of the recovery of a real-valued potential in the two-dimensional Schrödinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient
International audience
Dans ce papier, on a prouvé une estimation de stabilité pour le problème inverse de dé-termination du champ magnétique dans l'équation des ondes donné sur un domaine non borné à partir de l'opérateur de Dirichlet-to-Neumann. On a montré un résultat de stabilité pour ce problème inverse, dont la démonstration est basée sur la construction de solutions optique géométrique pour l'équation des ondes avec un potentiel magnétique 1-périodique. ABSTRACT. We consider the boundary inverse problem of determining the aligned magnetic field appearing in the magnetic wave equation in a periodic quantum cylindrical waveguide from boundary observations. The observation is given by the Dirichlet to Neumann map associated to the wave equation. We prove by means of the geometrical optics solutions of the magnetic wave equation that the knowledge of the Dirichlet-to-Neumann map determines uniquely the aligned magnetic field induced by a time independent and 1-periodic magnetic potential. We establish a Hölder-type stability estimate in the inverse problem.
On an inverse problem for a fractional semilinear elliptic equation involving a magnetic potential
