On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

趋化模型中的一个自由边界问题

本文就一个常见的生物趋化问题建立了相应的自由边界模型,对上述模型得到了其局部弱解的存在性.     

Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors

In this paper we study a free boundary problem modelling the growth of nonnecrotic tumors. The main trait of this free boundary problem is that it is essentially multidimensional, so that its well-posedness is hard to establish by using the usual methods in the classical theory of free boundary problems. In this paper we use the functional analysis method based on the theory of analytic semigroups to prove that this problem has a unique local solution in suitable function spaces. Continuous dependence of the solution on the initial data and regularities of the solution can also be easily obtained by using the argument of this paper.     

On a multiple credit rating migration model with stochastic interest rate

In this paper, we explore a pricing model for corporate bond accompanied with multiple credit rating migration risk and stochastic interest rate. The bond price volatility strongly depends on potentially multiple credit rating migration and stochastic change of interest rate. A free boundary problem of partial differential equation is presented, which is the equivalent transformation of the pricing model. The existence, uniqueness, and regularity for the free boundary problem are established to guarantee the rationality of the pricing model. Due to the stochastic change of interest rate, the discontinuous coefficient in the free boundary problem depends explicitly on the time variable but is convergent as time tends to infinity. Accordingly, an auxiliary free boundary problem is constructed, whose coefficient is the convergent limit of the coefficient in the original free boundary problem. With some constraint on the risk discount rate satisfied, we prove that a unique traveling wave exists in the auxiliary free boundary problem. The inductive method is adopted to fit the multiplicity of credit rating. Then we show that the solution of the original free boundary problem converges to the traveling wave in the auxiliary free boundary problem. Returning to the pricing model with multiple credit rating migration and stochastic interest rate, we conclude that the bond price profile can be captured by a traveling wave pattern coupling with a guaranteed bond price with face value equal to one at the maturity.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

Regularity of flat free boundaries in two-phase problems for the p-Laplace operator

In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.     

Lagrangian surfaces with Legendrian boundary

Science China Mathematics - In this paper, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces....     

Bifurcation for a free boundary problem modeling the growth of multi-layer tumors

In this paper we study bifurcations for a free boundary problem modeling the growth of multi-layer tumors under the action of inhibitors. An important feature of this problem is that the surface tension effect of the free boundary is taken into account. By reducing this problem into an abstract bifurcation equation in a Banach space, overcoming some technical difficulties and finally using the Crandall–Rabinowitz bifurcation theorem, we prove that this problem has infinitely many branches of bifurcation solutions bifurcating from the flat solution.     

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.     

Error estimates on homogenization of free boundary velocities in periodic media

In this paper we consider a free boundary problem which describes contact angle dynamics on inhomogeneous surface. We obtain an estimate on convergence rate of the free boundaries to the homogenization limit in periodic media. The method presented here also applies to more general class of free boundary problems with oscillating boundary velocities.     

On the geometry and regularity of largest subsolutions for a free boundary problem in \mathbf{R }^2: elliptic case

We consider a one-phase exterior Bernoulli type free boundary problem in $\mathbb{R }^2$ with a non constant, only continuous, free boundary condition. In this article, we present geometry and regularity properties of the largest viscosity subsolution of this problem. Moreover, we provide density bounds for the positivity set and its complement near the free boundary.     

A free boundary problem for the stokes system with contact lines

In this paper we study an evolution free boundary problem for the two-dimensional Stoltes system in the case in which the free boundary intersects the solid boundary of a container. We prove existence and uniqueness of solutionsfor this problem in suitable classes of functions and under suitable smallness conditions for the initial data. For these solutions the contact point moves with an uniform velocity with respect to the container.     

具有两种流体的轴对称渗流的一个自由边界问题

本文处理带有两种流体的轴对称的一个自由边界问题，其中在渗流区域的上部都是油，下部是水，这是同时取油注水的一个数学模型。下面，我们将用复分析方法求出此自由边界问题的一个解，并证明其解的唯一性。     

Long time regularity of solutions of the Hele–Shaw problem

In this paper we study the long-time behavior of solutions of the one phase Hele–Shaw problem without surface tension. We show that after a finite time solutions of the Hele–Shaw problem become starshaped and Lipschitz continuous in space. Based on this observation we then prove that the free boundary become smooth in space and time with nondegenerate free boundary speed.     

Mixed Elastico-Plasticity Problems with Partially Unknown Boundaries

In this paper,we study mixed elastico-plasticity problems in which part of the boundary is known,while the other part of the boundary is unknown and is a free boundary.Under certain conditions,this problemcan be transformed into a Riemann-Hilbert boundary value problem for analytic functions and a mixed boundaryvalue problem for complex equations.Using the theory of generalized analytic functions,the solvability of theproblem is discussed.     

Porosity of the Free Boundary for p-Parabolic Type Equations in Non-Divergence Form

Potential Analysis - In this article we establish the exact growth of the solution to the singular quasilinear p-parabolic free boundary problem in non-divergence form near the free boundary from...     

SOLVABILITY OF A PARABOLIC-HYPERBOLIC TYPE CHEMOTAXIS SYSTEM IN 1-DIMENSIONAL DOMAIN

In this paper, we use contraction mapping principle, operator-theoretic approach and some uniform estimates to establish local solvability of the parabolic-hyperbolic type chemotaxis system with fixed boundary in 1-dimensional domain. In addition, local solvability of the free boundary problem is considered by straightening the free boundary.     

Smooth zero-contact-angle solutions to a thin-film equation around the steady state

In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the “complete wetting” regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide “minimal” conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary.     

轴对称渗流中自由边界问题的近似解法

１．问题的提法 本文讨论各向异性非均匀介质的轴对称稳定渗流问题,我们延用［７］的记号。设有一水井（或油井）,其截面如图１所示,ｚ轴为对称轴,Ｄ为渗流区域,其边界为ＡＢＣＥＦＡ,Ｋ＝｛ｋｉｊ（ｒ,ｚ,ｈ,ｑ）｝为对称渗流张量,它依赖于柱坐标中的ｒ,ｚ,压头ｈ＝ｚ＋ｐ／ρｇ及渗流速率其中ｐ为点（ｒ,ｚ）处的压力,ρ为流体密度,ｇ为重力加速度．ｒ０为井的半径,Ｈ１为液面的高度,同时假设当ｒ≥Ｒ时,其渗流速度Ｖ＝０． 由渗流理论,有引入热函数,流函数 Ｄ中一点）,则满足下列一阶非线性方程组其中,若（ｉ＝ １…