Stability for the determination of unknown boundary and impedance with a Robin boundary condition

We consider an inverse problem arising in corrosion detection. We prove a stability result of logarithmic type for the determination of the corroded portion of the boundary and impedance by two measurements on the accessible portion of the boundary

Smoothness dependent stability in corrosion detection

We consider the stability issue for the determination of a linear corrosion in a conductor by a single electrostatic measurement. We established a global log-log type stability when the corroded boundary is simply Lipschitz. We also improve such a result obtaining a global log stability by assuming that the damaged boundary is $C^{1,1}$-smooth

Wave equation with Robin condition, quantitative estimates of strong unique continuation at the boundary

The main result of the present paper consists in a quantitative estimate of unique continuation at the boundary for solutions to the wave equation. Such estimate is the sharp quantitative counterpart of the following strong unique continuation property: let $u$ be a solution to the wave equation that satisfies an homogeneous Robin condition on a portion $S$ of the boundary and the restriction of $u_{\mid S}$ on $S$ is flat on a segment $\{0\}\times J$ with $0\in S$ then $u_{\mid S}$ vanishes in a neighborhood of $\{0\}\times J$

Cracks with impedance, stable determination from boundary data

We discuss the inverse problem of determining the possible presence of an (n-1)-dimensional crack \Sigma in an n-dimensional body \Omega with n > 2 when the so-called Dirichlet-to-Neumann map is given on the boundary of \Omega. In combination with quantitative unique continuation techniques, an optimal single-logarithm stability estimate is proven by using the singular solutions method. Our arguments also apply when the Neumann-to-Dirichlet map or the local versions of the D-N and the N-D map are available.Comment: 40 pages, submitte

Simultaneous numerical determination of a corroded boundary and its admittance

In this paper, an inverse geometric problem for Laplace’s equation arising in boundary corrosion detection is considered. This problem, which consists of determining an unknown corroded portion of the boundary of a bounded domain and its admittance Robin coefficient from two pairs of boundary Cauchy data (boundary temperature and heat flux), is solved numerically using the meshless method of fundamental solutions. A non-linear minimization of the objective function is regularized, and the stability of the numerical results is investigated with respect to noise in the input data and various values of the regularization parameters involved