The boundary inverse problem for the Laplace equation in two dimensions

Abstract

We have studied the problem of determining part of the boundary of a domain where a potential satisfies the Laplace equation. The potential and its normal derivative have prescribed values on the known part of the boundary that encloses while its normal derivative must vanish on the remaining part. We establish a sufficient condition for the potential to be monotonic along the unknown boundary. This allows us to use the potential to parametrize the boundary. Two methods are presented that solve the problem under this assumption. The first one solves the problem in a closed form and it can be used to define a parameter that will describe the ill-posedness of the problem. The effect of this parameter on the second method presented has been determined for a particular numerical example.

Optimal Stability for the Inverse Problemof Multiple Cavities

“…Let us also mention that a companion problem, in a different topological setting, when s … W intersects "W on a nontrivial arc in such a way that W 0 s remains simply connected, has already been treated by various authors, [5], [9], [10], [12,14], [17] and [20,21].…”

Shape Identification in Nonlinear Boundary Problems Solved bby Pies Method

“…is the number of collocation points on the segment) are square matrices of elements expressed by integrals over the boundary from (2), and are vectors which contain the coefficients of approximation series (Zieniuk, 2007), whilst is the vector of elements expressed by the integral over the domain from equation (2).…”

Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $\mathcal {R}^3$

“…Related results can be found in Andrieux, Abda & Jaoua [2], Aparicio & Pidcock [3] and Kaup, Santosa & Vogelius [14]. In all of these papers, the results were only given for the two dimensional case.…”

“…Consider a static field in Ω with a suitable function u = u(x, y, z), (x, y, z) ∈ Ω ⊂ R 3 . Throughout this paper, we assume that u takes a given constant value on γ.…”