A Schwarz function on an open domain Ω is a holomorphic satisfying S(ζ)=ζ ¯ Γ, which part of the boundary Ω. Sakai in 1991 gave complete characterization admitting function. In fact, if simply connected and Γ=∂Ω∩D(ζ 0 ,r), then Γ has to be regular real analytic (with possible cusps). Sakai’s result natural applications 1) quadrature domains, 2) free problem for Δu=1 equation. our scenarios can ...

The aim of this work is to point out that within a similarity approach some classes of free boundary value problems governed by ordinary differential equations can be transformed to initial value problems. The interest in the numerical solution of free boundary problems arises because these are always nonlinear problems. Furthermore we show that free boundary problems arise also via a similarit...

In this work we consider diierent formulations for a free boundary problem and analyze the related numerical methods. In particular, we formulate various shape optimization problems that are weak forms of a free boundary problem and analyze their properties using the tools of 'shape calculus'. The main result is that it is possible to innuence the conditioning of the shape cost functional and t...

In this paper we study the existence of C 2;; domain solutions for a class of free boundary problems which are critical domains for a shape functional depending on a general term (in most PDE applications this term represents the energy associated) and the perimeter, under the constraint that the measure of is given. As an application there are considered two problems. They involve respectively...

A new domain decomposition technique on non-matching grids for free boundary problems is considered. An iterative DD scheme is used to reduce the original free boundary problem to a sequence of problems on two sub-domains, one of which includes the free boundary and is described by a variational inequality and the other includes the remainder of the problem and is described by a second order pa...

In free-boundary problems, the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation. We develop duality-based a posteriori error estimates for functional outputs of solutions of free-boundary problems that include both sources of error. The derivation of an appropriate dual problem (linearized adjoint) is, ...