in , where ε>0, , with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1) and . The functions uε and fε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present.We obtain uniform estimates, we pass to the limit (ε→0) and we show that limit functions are solutions to the two phase free boundary problem:
where f=limfε, in a viscosity sense and in a pointwise sense at regular free boundary points.In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions.Some of the results obtained are new even in the case fε≡0.The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary.  相似文献
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The pricing equations derived from uncertain volatility modelsin finance are often cast in the form of nonlinear partial differentialequations. Implicit timestepping leads to a set of nonlinearalgebraic equations which must be solved at each timestep. Tosolve these equations, an iterative approach is employed. Inthis paper, we prove the convergence of a particular iterativescheme for one factor uncertain volatility models. We also demonstratehow non-monotone discretization schemes (such as standard Crank–Nicolsontimestepping) can converge to incorrect solutions, or lead toinstability. Numerical examples are provided.  相似文献
Regularity of viscosity solutions of a degenerate parabolic equation   总被引:3,自引:0,他引:3  
We study the Cauchy problem for the nonlinear degenerate parabolic equation of second order

and present regularity results for the viscosity solutions.


Let I: be a given bounded image function, where is an open and bounded domain which belongs to n. Let us consider n=2 for the purpose of illustration. Also, let S={xi}i be a finite set of given points. We would like to find a contour , such that is an object boundary interpolating the points from S. We combine the ideas of the geodesic active contour (cf. Caselles et al. [7,8]) and of interpolation of points (cf. Zhao et al. [40]) in a level set approach developed by Osher and Sethian [33]. We present modelling of the proposed method, both theoretical results (viscosity solution) and numerical results are given. AMS subject classification 49L25, 74G65, 68U10  相似文献
The author proves, when the noise is driven by a Brownian motion and an independent Poisson random measure, the one-dimensional reflected backward stochastic differential equation with a stopping time terminal has a unique solution. And in a Markovian framework, the solution can provide a probabilistic interpretation for the obstacle problem for the integral-partial differential equation.  相似文献
American Options Exercise Boundary When the Volatility Changes Randomly   总被引:2,自引:0,他引:2  
The American put option exercise boundary has been studied extensively as a function of time and the underlying asset price. In this paper we analyze its dependence on the volatility, since the Black and Scholes model is used in practice via the (varying) implied volatility parameter. We consider a stochastic volatility model for the underlying asset price. We provide an extension of the regularity results of the American put option price function and we prove that the optimal exercise boundary is a decreasing function of the current volatility process realization. Accepted 13 January 1998  相似文献
Optimal control problems for bilinear systems are studied and solved with a view to approximating analogous problems for general nonlinear systems. For a given bilinear optimal control problem, a sequence of linear problems is constructed, and their solutions are shown to converge to the desired solution. Also, the direct solution to the Hamilton-Jacobi equation is analyzed. A power-series approach is presented which requires offline calculations as in the linear case (Riccati equation). The methods are compared and illustrated. Relations to classical linear systems theory are discussed.  相似文献

Solutions of the optimal control and -control problems for nonlinear affine systems can be found by solving Hamilton-Jacobi equations. However, these first order nonlinear partial differential equations can, in general, not be solved analytically. This paper studies the rate of convergence of an iterative algorithm which solves these equations numerically for points near the origin. It is shown that the procedure converges to the stabilizing solution exponentially with respect to the iteration variable. Illustrative examples are presented which confirm the theoretical rate of convergence.


A Hamiltonian model is analyzed for a one-dimensional inviscid compressible fluid. The space–time evolution of the fluid is governed by the following system of the Hamilton–Jacobi and the continuity equations:
Here S and ρ designate the velocity potential and the mass density, respectively. Unless S 0 is convex, shocks form and the velocity S x becomes discontinuous in {0<ω t<π/2}. It is demonstrated that there nevertheless exists a unique viscosity–measure solution (S,ρ) when S 0 is globally Lipschitz continuous and locally semi-concave while ρ 0 is a finite Borel measure. The structure of the velocity and the density is exhibited. For initial data correlated in a certain sense, a class of classical solutions (S,ρ) is given. Negative time is also considered, and illustrating examples are given.   相似文献
A two phase elliptic singular perturbation problem with a forcing term   总被引:1,自引:0,他引:1  
We study the following two phase elliptic singular perturbation problem:
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