On a Class of Geometry-Driven Free Boundary Problems

On a Class of Geometry-Driven Free Boundary Problems

An abstract class of two-dimensional geometry-driven free boundary problems is considered in which the evolution of the Cauchy transform of the time-evolving domain satisfies a partial differential equation of conservation type. This idea generalizes an association between free boundary problems and the inverse gravitational problem. It is shown that this class of free boundary problems admits ...

An Existence Result for a Class of Free Boundary Problems an Existence Result for a Class of Free Boundary Problems

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...

On the existence of nonnegative solutions for a class of fractional boundary value problems

‎In this paper‎, ‎we provide sufficient conditions for the existence of nonnegative solutions of a boundary value problem for a fractional order differential equation‎. ‎By applying Kranoselskii`s fixed--point theorem in a cone‎, ‎first we prove the existence of solutions of an auxiliary BVP formulated by truncating the response function‎. ‎Then the Arzela--Ascoli theorem is used to take $C^1$ ...

A Class of Codimension-Two Free Boundary Problems

This review collates a wide variety of free boundary problems which are characterized by the uniform proximity of the free boundary to a prescribed surface. Such situations can often be approximated by mixed boundary value problems in which the boundary data switches across a “codimension-two” free boundary, namely, the edge of the region obtained by projecting the free boundary normally onto t...

On Free Boundary Problems