This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in ${ℝ}^n$ (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc:= ${ℝ}^n\setminus \overline{\Omega }$. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD)...

This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in ${ℝ}^n$ (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := ${ℝ}^n\setminus \overline{\Omega }$. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping...

The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace’s equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem...

The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace's equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem...

In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in ${𝐑}^3$. The singular set of such a new minimizer belongs to a three parameters family of sets $(0<{\delta }_1,{\delta }_2,{\delta }_3<\pi )$. We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of ${𝐒}^2$ with three reentrant corners. The necessary conditions are...