We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results.

in an infinite strip (global version) or a finite parabolic cylinder (localized version), where L is a uniformly parabolic operatorwith double Dini continuous A and uniformly bounded b and c. We also prove the elliptic counterpart of this estimate.This closes the gap between the known conditions in the literature (both in the elliptic and parabolic case) imposed on u ± in order to obtain an almost monotonicity estimate.At the end of the paper, we demonstrate how to use this new almost monotonicity formula to prove the optimal C 1,1 regularity in a fairly general class of quasilinear obstacle-type free boundary problems.

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ballWe prove that the free boundary touches the fixed boundary (uniformly) tangentially if the boundary data f and its first and second derivatives vanish at the touch-point.

In this paper we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape.Using the shape sensitivities we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem.Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments.

Abstract. The aim of this paper is to study a free boundary problem for a uniformly elliptic fully non-linear operator. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points.