We study the regularity of the “free surface” in boundary obstacle problems. We show that near a non-degenerate point the free boundary is a C (n−2)-dimensional surface in R.

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification. Particular emphasis is put on Stefan problems, and their quasi-static variants, with applications to crystal growth. New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable dis...

We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary v...

We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary v...

The aim of this work is to point out that, within group invariance theory, some classes of 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) often nonlinear problems. The theoretical content of this paper is original: results alread...

In this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like oneand two-phase Stefan problem and of obstacle type. Our approach enters the category of xed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We prove stability and convergence properties. The appr...

Free boundary problems based on mass conservation and surface tension with application in osmotic swelling are the topic of this contribution. Phase-field approximations of such models are introduced, in order to numerically investigate properties of the solutions. Formal justification of the proposed approximations is provided by matched asymptotic expansions supported by numerical tests repro...

We consider a linear-quadratic elliptic control problem (LQECP). For the problem we consider here, the control variable corresponds to the Neumann data on the boundary of a convex polygonal domain. The optimal control unknown is the one for which the harmonic extension approximates best a specified target in the interior of the domain. We propose multilevel preconditioners for the reduced Hessi...

When the option pricing problem is of several dimensions, for example, basket options, deterministic methods such as finite difference are almost intractable; because the complexity increases exponentially with the dimension and one almost inevitably needs to use Monte Carlo simulations. Moreover, many problems in finance, for example, pricing in incomplete markets and portfolio optimization, l...

We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.