Free boundary problems arising in the freezing of soils in a bounded region. III. Hydrodynamics of the unfrozen phase. IV. Thermal stresses in the frozen phase

In this paper, the method of characteristics is utilized to construct unique solutions to Problems II and III stated in Part I, and we demonstrate the continuous dependence of these solutions on the respective initial and boundary data. Moreover, asymptotic estimates for the critical time of breakdown in these solutions are also obtained.     

On a nonlinear elliptic equation arising in a free boundary problem

 Let p ^* =n/(n−2) and n≥3. In this paper, we first classify all non-constant solutions of

On a free boundary problem in ground freezing

We consider a boundary value problem describing the stationary flow of a non‐Newtonian fluid through the frozen ground, with a free interface between the liquid and the solid phases. We prove the existence of at least one weak solution of the problem. Copyright © 2000 John Wiley & Sons, Ltd.     

Nonexistence for a boundary value problem arising in parabolic theory

The boundary value problemc _t=c _xx−c _yy+q(t,x)c with {fx349-1} was solved by Colton [1] forq analytic int. The solution may be used for mapping solutions of the heat equation into solutions ofu _t=u _xx+q(t,x)u. Solutions (of the boundary value problem) no longer exist ifq is not analytic int. Erica and Ludwig Jesselson Professor of Theoretical Mathematics, The Weizmann Institute of Science. This research was partially supported by the Minerva Foundation.     

On a parabolic free boundary problem arising from a Bingham-like flow model with a visco-elastic core

In this paper we study a two-phase one-dimensional free boundary problem for parabolic equation, arising from a mathematical model for Bingham-like fluids with visco-elastic core presented in [L. Fusi, A. Farina, A mathematical model for Bingham-like fluids with visco-elastic core, Z. Angew. Math. Phys. 55 (2004) 826-847]. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Local existence is proved using a fixed point argument based on Schauder's theorem. Uniqueness is proved using a non-standard technique based on a weak formulation of the problem.     

The numerical solution of a periodic parabolic problem subject to a nonlinear boundary condition. II

Approximating the differential equation by the standard fully implicit finite difference approximation reduces the problem to that of solving a set of non-linear algebraic equations. In the first part of this paper it was shown that Newton's method provided a satisfactory technique for the solution of this set of equations. In this paper the implementation of Newton's method is discussed, and it is shown that a device due toBickley andMacNamee can be used very successfully. A more accurate difference formula is also considered and a summary of numerical results presented.     

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.     

A general class of free boundary problems for fully nonlinear parabolic equations

Annali di Matematica Pura ed Applicata (1923 -) - In this paper, we consider the fully nonlinear parabolic free boundary problem $$\begin{aligned} \left\{ \begin{array}{ll} F(D^2u) -\partial _{t}...     

Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain

This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in $[0,M]$ and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation

$v_t?\underset{0\le a\le 1}{\max }?(\frac{1}{2}{a}^2{\sigma }^2{v}_{xx}+a\mu v_x)+cv?\underset{0\le l\le M}{\max }?[(1?v_x)l]=0$

under certain boundary conditions on an angular domain

$Q_T=\{(x,t)|0<x<Mt,0<t\le T\}.$

The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company.     

Convergence of bounded solutions of nonlinear parabolic problems on a bounded interval: the singular case

In this article we prove that for every ${1 < p \le 2}$ and for every continuous function ${f: [0,1]\times\mathbb{R}\to\mathbb{R}}$ , which is Lipschitz continuous in the second variable, uniformly with respect to the first one, each bounded solution of the one-dimensional heat equation $$\begin{array}{ll}u_{t}-\{|u_{x}|^{p-2}u_{x} \}_{x}+f(x,u)=0\qquad{\rm in} \quad (0,1)\times (0,+\infty) \end{array}$$ with homogeneous Dirichlet boundary conditions converges as ${t\to+\infty}$ to a stationary solution. The proof follows an idea of Matano which is based on a comparison principle. Thus, a key step is to prove a comparison principle on non-cylindrical open sets.     

Solution of a boundary control problem for a nonlinear parabolic equation

An optimal boundary control problem for a nonlinear parabolic equation of second order is considered. An existence theorem is proved and necessary optimality conditions are obtained in the form of point and integral maximum principles.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 30–37, 1986.     

Smooth solutions to a class of free boundary parabolic problems

We establish existence, uniqueness, and regularity results for solutions to a class of free boundary parabolic problems, including the free boundary heat equation which arises in the so-called ``focusing problem' in the mathematical theory of combustion. Such solutions are proved to be smooth with respect to time for positive , if the data are smooth.

     

Viscosity solution theory of a class of nonlinear degenerate parabolic equations II. Lipschitz continuity of free boundary

In [1] we construct a unique bounded Hölder continuous viscosity solution for the nonlinear PDEs with the evolutionp-Laplacian equation and its anisotropic version as typical examples. In this part, we investigate the Lipschitz continuity of the free boundary of viscosity solution and its asymptotic spherical symmetricity, however, this result does not include the anisotropic case.This research is supported by the National Natural Sciences Foundation of China.     

Higher regularity of the free boundary in the parabolic Signorini problem

We show that the quotient of two caloric functions which vanish on a portion of an \(H^{k+ \alpha }\) regular slit is \(H^{k+ \alpha }\) at the slit, for \(k \ge 2\). In the case \(k=1\), we show that the quotient is in \(H^{1+\alpha }\) if the slit is assumed to be space-time \(C^{1, \alpha }\) regular. This can be thought of as a parabolic analogue of a recent important result in De Silva and Savin (Boundary Harnack estimates in slit domains and applications to thin free boundary problems, 2014), whose ideas inspired us. As an application, we show that the free boundary near a regular point of the parabolic thin obstacle problem studied in Danielli et al. (Optimal regularity and the free boundary in the parabolic Signorini problem. Mem. Am. Math. Soc., 2013) with zero obstacle is \(C^{\infty }\) regular in space and time.