Regularity of the Solutions for Parabolic Two-Phase Free Boundary Problems

In this paper we start to develop the regularity theory of general two-phase free boundary problems for parabolic equations. In particular we consider uniformly parabolic operators in nondivergence form and we are mainly concerned with the optimal regularity of the viscosity solutions. We prove that under suitable nondegenerate conditions the solution is Lipschitz across the free boundary.     

Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

Let u and solve the problem

Boundary Regularity and Nontransversal Intersection for the Fully Nonlinear Obstacle Problem

In this paper a classification is given of blowup solutions at the intersection of the free and fixed boundary corresponding to obstacle problems generated by fully nonlinear, uniformly elliptic operators. As a consequence, nontransversal intersection is shown to hold in any dimension. Several regularity results are also obtained. © 2019 Wiley Periodicals, Inc.     

The Well Posedness in a Parabolic Multiple Free Boundary Problem

We consider a free boundary problem in a parabolic partial differential equation with multiple interfacial curves which is reduced to a reaction-diffusion equation. The forcing term of this problem is not continuously differentiable and thus we use Green's function to make a regular one. The existence, uniqueness and dependence. on initial conditions will be shown in this paper.     

A Two-Phase Parabolic Free Boundary Problem with Coefficients Below the Lipschitz Threshold

We study the regularity of a parabolic free boundary problem of two-phase type with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the optimal ${C_x^{1,1}\cap C_t^{0,1}}$ -regularity of the solution and that the free boundary is, near the so-called branching points, the union of two graphs that are Lipschitz in time and C ¹ in space. In our case, the same monotonicity formula does not apply in the same way. Instead we use scaling arguments similar to the ones used for the elliptic case in Edquist et al. (Ann Inst Henri Poincareé, Anal Non Linéaire 26(6):2359?C2372, 2009) to prove the optimal regularity. However, whenever the spatial gradient does not vanish on the free boundary, we are in the parabolic setting faced with some extra difficulties, that forces us to strain our assumptions slightly.     

On the Curvature of the Free Boundary for the Obstacle Problem in Two Dimensions

We give a new proof of the fact that the free boundary for the obstacle problem in two dimensions satisfies a natural and sharp inner ball condition.     

On the Curvature of the Free Boundary for the Obstacle Problem in Two Dimensions

We give a new proof of the fact that the free boundary for the obstacle problem in two dimensions satisfies a natural and sharp inner ball condition.     

Numerical Solution of a Parabolic Free Boundary Problem via Newton's Method

We develop a Newton-like numerical method to solve a free boundaryproblem of parabolic type. From some numerical tests we concludethat the speed of convergence of this procedure is very high,probably quadratic in the neighbourhood of the solution.     

A One-Dimensional Parabolic Problem Arising in Studies of Some Free Boundary Problems

The paper is concerned with a one-dimensional parabolic problem in a domain bounded by two lines x = 0 and x = kt, k > 0, (x, t) ², with the Neumann boundary condition on the line x = 0 and with dynamic boundary condition on the line x = kt. For the solution of this problem, a coercive estimate in a weighted Hölder norm is obtained. It is shown that this estimate can be useful for the analysis of parabolic free boundary problems. Bibliography: 7 titles.     

由退化抛物型方程产生的常微分方程一种自由边界问题

Assume B∈C[0,+∞)∩C¹(0,+∞) and B(υ) is strictly increasingand concave. That the free boundary Problem for ODE     

The Hopf Bifurcation in a Parabolic Free Boundary Problem with Double Layers

We consider a parabolic free boundary problem which has a bifurcation parameter and double interfaces. We investigate the sign change in a real part of eigenvalues and the transversality condition as a bifurcation parameter cross the critical value in order to examine the stability of the stationary solutions. The occurence of a Hopf bifurcation will be shown at a critical value.     

Properties of Solutions to the Parabolic Obstacle Problem in a Neighborhood of the Boundary of a Domain

For the parabolic obstacle problem with homogeneous Dirichlet boundary condition the regularity of the free boundary is studied in a neighborhood of the boundary of a domain. Bibliography: 6 titles.     

A Free Boundary Problem Governed by Nonlinear Degenerate Equations of Parabolic Type

We consider a free boundary problem connected with non-Newtonian fluid motion, i.e. the flow of power law fluids with the yield stress. We obtain the solution of the relevant approximation problem by means of a parabolic quasi-variational inequality, and then obtain the weak solution of the original problem after a passage to the limit. Finally, we study the regularity of the weak solution.     

On a Discrete Fractional Boundary Value Problem with Nonlocal Fractional Boundary Conditions

In this paper, we investigate the nonlinear fractional difference equation with nonlocal fractional boundary conditions. We derive the Green＇s function for this problem and show that it satisfies certain properties. Some existence results are obtained by means of nonlinear alternative of Leray-Schauder type theorem and Krasnosel-skii＇s fixed point theorem.     

On Regularity and Singularity of Free Boundaries in Obstacle Problems

The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems. This approach is based on the monotonicity of several variational integrals, the Federer-Almgren dimension reduction and stratification theorems, and some simple PDE arguments.     

分数阶微分方程多点分数阶边值问题

研究分数阶微分方程多点分数阶边值问题解的存在性与唯一性,利用不动点定理,得到了边值问题存在唯一解和至少存在1个解的充分条件.     

Optimal Regularity for the No‐Sign Obstacle Problem

In this paper we prove the optimal $C^{1,1}(B_{1/2})$ ‐regularity for a general obstacle‐type problem under the assumption that $f*N$ is $C^{1,1}(B_1)$ , where N is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$ ‐regularity. As a by‐product of the $C^{1,1}$ ‐regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$ , the free boundary is locally a $C^1$ ‐graph close to $x^0$ provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades. © 2012 Wiley Periodicals, Inc.