A uniqueness theorem for a free boundary problem

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

     

A free boundary problem of diffusion equation with integral condition

We consider a free boundary problem of heat equation with integral condition on the unknown free boundary. Results of solution regularity and problem well-posedness are presented.     

An existence result for a free boundary shallow water model using a Lagrangian scheme

We propose a free boundary shallow water model for which we give an existence theorem. The proof uses an original Lagrangian discrete scheme in order to build a sequence of approximate solutions. The properties of this scheme allow to treat the difficulties linked to the boundary motion. These approximate solutions verify some compactness results which allow us to pass to the limit in the discrete problem.     

The regularity of the free boundary in a multidimensional filtration problem

We prove the regularity of the free boundary for a filtration problem with capillarity in more than one space dimension. The free boundary is the interface between the saturated region (in which the governing equation is elliptic) and the unsaturated region (where a degenerate parabolic equation is to be solved).This work was partially supported by National Project Equazioni di Evoluzione e Applicazioni Fisico Matematiche (M.U.R.S.T.).     

A free boundary problem arising from DCIS mathematical model

Ductal carcinoma in situ – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma in situ model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.     

A critical case of stability in a free boundary problem

We study the stability of the planar travelling wave solution to a free boundary problem for the heat equation in the whole . We turn the problem into a fully nonlinear parabolic system and establish a stability result which is the proper generalization of the one-dimensional case. The curvature terms contribute a gradient squared corresponding to critical growth. The latter is eliminated by means of the Hopf-Cole transformation. Received August 18, 2000, accepted September 27, 2000.     

A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon

In this paper we consider chemical vapor deposition of pyrolytic carbon from methane in hot wall reactors. Especially, we deal with the interaction of homogeneous gas-phase and heterogeneous surface reactions. The resulting mathematical model is composed of a system of reaction-diffusion equations in a corner domain supplied with the Gibbs-Thomson law, which describes the movement of the free boundary, arising from the carbon deposition. We prove a short time existence and uniqueness result in Hölder spaces. We achieve this by contraction arguments and transforming the Gibbs-Thomson law to local coordinates to obtain a nonlinear parabolic equation on a manifold.     

Lagrangian approach to the study of level sets II: A quasilinear equation in climatology

We study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equation

     

Global classical solution of Muskat free boundary problem

In this paper the Muskat problem which describes a two-phase flow of two fluids, for example, oil and water, in porous media is discussed. The problem involves in seeking two time-dependent harmonic functions u₁(x,y,t) and u₂(x,y,t) in oil and water regions, respectively, and the interface between oil and water, i.e., the free boundary Γ:y=ρ(x,t), such that on the free boundary

     

Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ρ<R, there exists a radially symmetric stationary solution with tumor boundary r=R and necrotic core boundary r=ρ. The system depends on a positive parameter μ, which describes the tumor aggressiveness. There also exists a sequence of values μ₂<μ₃<? for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch.     

Regularity of solutions to a free boundary problem modeling tumor growth by Stokes equation

In this paper we investigate regularity of solutions to a free boundary problem modeling tumor growth in fluid-like tissues. The model equations include a quasi-stationary diffusion equation for the nutrient concentration, and a Stokes equation with a source representing the proliferation density of the tumor cells, subject to a boundary condition with stress tensor effected by surface tension. This problem is a fully nonlinear problem involving nonlocal terms. Based on the employment of the functional analytic method and the theory of maximal regularity, we prove that the free boundary of this problem is real analytic in temporal and spatial variables for initial data of less regularity.     

Linearization of a free boundary problem in corrosion detection

We consider a boundary identification problem arising in nondestructive testing of materials. The problem is to recover a part ΓI⊂∂Ω of the boundary of a bounded, planar domain Ω from one Cauchy data pair (u,∂u/∂ν) of a harmonic potential u in Ω collected on an accessible boundary subset ΓA⊂∂Ω. We prove Fréchet differentiability of a suitably defined forward map, and discuss local uniqueness and Lipschitz stability results for the linearized problem.     

Global stability of solutions to a free boundary problem of ductal carcinoma in situ

In the paper we present some remarks on the global stability of steady state solutions to a free boundary model studied by Xu (2004) and also prove some new results of global stability of steady state solutions to the model.     

A BOUNDARY ELEMENT METHOD FOR A NONLINEAR BOUNDARY VALUE PROBLEM IN STEADY-STATE HEAT TRANSFER IN DIMENSION THREE

In this paper, a new method of boundary reduction is proposed, which reduces thesteady-state heat transfer equation with radiation. Moreover, a boundary element method is pre-sented for its solution and the error estimates of the numerical approximations are given.     

One-phase parabolic free boundary problem in a convex ring

This paper studies a free boundary problem for the heat equation in a convex ring. It is proved that the considered problem has unique solution under some conditions on the initial data.