Related papers: Probe method and a Carleman function

We consider an inverse source problem for the Helmholtz equation in a bounded domain. The problem is to reconstruct the shape of the support of a source term from the Cauchy data on the boundary of the solution of the governing equation. We…

A general method for solving nonlinear ill-posed problems is developed. The method consists of solving a Cauchy problem with a regularized operator and proving that the solution of this problem tends, as time grows, to a solution of the…

A Carleman Weight Function (CWF) is used to construct a new cost functional for a Coefficient Inverse Problems for a hyperbolic PDE. Given a bounded set of an arbitrary size in a certain Sobolev space, one can choose the parameter of the…

We consider the inverse Calder\'on problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually…

In this work we develop a new numerical approach for recovering a spatially dependent source component in a standard parabolic equation from partial interior measurements. We establish novel conditional Lipschitz stability and H\"{o}lder…

We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are…

A steady state plane problem of an inhomogeneous half-plane subjected to a load running along the boundary at subsonic speed is analyzed. The Lame coefficients and the density of the half-plane are assumed to be power functions of depth.…

In his deep and prolific investigations of heat diffusion, Lam\'e was led to the investigation of the eigenvalues and eigenfunctions of the Laplace operator in an equilateral triangle. In particular he derived explicit results for the…

It is proved the existence of nonclassical solutions of the Neumann problem for the harmonic functions in the Jordan rectifiable domains with arbitrary measurable boundary distributions of normal derivatives. The same is stated for the…

We consider a Neumann problem for the Laplace equation in a periodic domain. We prove that the solution depends real analytically on the shape of the domain, on the periodicity parameters, on the Neumann datum, and on its boundary integral.

Employing a limiting case of a conjecture for constructing piecewise separable-variables functions, the elements of the Pseudoanalytic Function Theory are used for numerically approaching solutions of the forward Dirichlet boundary value…

It has been recently discovered that a convex function can be determined by its slopes and its infimum value, provided this latter is finite. The result was extended to nonconvex functions by replacing the infimum value by the set of all…

We propose a global convergent numerical method to reconstruct the initial condition of a nonlinear parabolic equation from the measurement of both Dirichlet and Neumann data on the boundary of a bounded domain. The first step in our method…

We solve a weakly singular integral equation by Laplace transformation over a finite interval of R. The equation is transformed into a Cauchy integral equation, whose resolution amounts to solving two Fredholm integral equations of the…

We are concerned with inverse boundary problems for first order perturbations of the Laplacian, which arise as model operators in the acoustic tomography of a moving fluid. We show that the knowledge of the Dirichlet--to--Neumann map on the…

We propose a new numerical method for the solution of the problem of the reconstruction of the initial condition of a quasilinear parabolic equation from the measurements of both Dirichlet and Neumann data on the boundary of a bounded…

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain…

A new version of the convexification method is developed analytically and tested numerically for a 1-D coefficient inverse problem in the frequency domain. Unlike the previous version, this one does not use the so-called "tail function",…

The study of the Dirichlet-to-Neumann map and the associated Steklov problem for the Laplace equation has been a central topic in spectral geometry over the past decade. In this survey, we consider a more general framework in which the…

In this paper we develop a general theoretical tool for the establishment of the boundedness of notoriously difficult operators (such as potentials) on certain specific types of rearrangement-invariant function spaces from analogous…