On an obstacle problem arising in large exponent asymptotics for one dimensional fully nonlinear diffusions of power type

2020 ◽  
Author(s):  
Qing Liu
Keyword(s):  
Obstacle Problem ◽  
Power Type ◽  
Fully Nonlinear ◽  
One Dimensional ◽  
Large Exponent ◽  
Nonlinear Diffusions

scholarly journals Asymptotics for a parabolic double obstacle problem

1994 ◽  
Vol 444 (1922) ◽  
pp. 429-445 ◽  
Keyword(s):  
Asymptotic Behaviour ◽  
Obstacle Problem ◽  
Thin Strip ◽  
Normal Velocity ◽  
One Dimensional ◽  
Cahn Equation ◽  
Small Constant ◽  
The One ◽  
Short Time ◽  
Double Well Potential

We consider a parabolic double obstacle problem which is a version of the Allen-Cahn equation u t = Δ u — ϵ -2 ψ '( u ) in Ω x (0, ∞), where Ω is a bounded domain, ϵ is a small constant, and ψ is a double well potential; here we take ψ such that ψ ( u ) = (1 — u 2 ) when | u | ≤ 1 and ψ ( u ) = ∞ when | u | > 1. We study the asymptotic behaviour, as ϵ → 0, of the solution of the double obstacle problem. Under some natural restrictions on the initial data, we show that after a short time (of order ϵ 2 |ln ϵ |), the solution takes value 1 in a region Ω + t and value — 1 in Ω - t , where the region Ω ( Ω + t U Ω - t ) is a thin strip and is contained in either a O ( ϵ |ln ϵ |) or O ( ϵ ) neighbourhood of a hypersurface Γ t which moves with normal velocity equal to its mean curvature. We also study the asymptotic behaviour, as t → ∞, of the solution in the one-dimensional case. In particular, we prove that the ω -limit set consists of a singleton.

scholarly journals Regularity for the fully nonlinear parabolic thin obstacle problem

2021 ◽  
pp. 2150011
Author(s):  
Georgiana Chatzigeorgiou
Keyword(s):  
Parabolic Equation ◽  
Viscosity Solution ◽  
Nonlinear Parabolic Equation ◽  
Obstacle Problem ◽  
Elliptic Case ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Nonlinear Parabolic ◽  
Parabolic Case ◽  
Thin Obstacle

We prove [Formula: see text] regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.

scholarly journals Regularity of the free boundary in the biharmonic obstacle problem

2019 ◽  
Vol 58 (6) ◽  
Author(s):  
Gohar Aleksanyan
Keyword(s):  
Free Boundary ◽  
Obstacle Problem ◽  
Small Ball ◽  
One Dimensional ◽  
Coincidence Set ◽  
Accessible Domain

Abstract In this article we use a flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible domain, we derive the $$C^{1,\alpha }$$C1,α-regularity of the free boundary in a small ball centred at the origin. From the $$C^{1,\alpha }$$C1,α-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $$ C^{3,\alpha }$$C3,α up to the free boundary, and therefore $$C^{2,1}$$C2,1. In the end we study an example, showing that in general $$ C^{2,\frac{1}{2}}$$C2,12 is the best regularity that a solution may achieve in dimension $$n \ge 2$$n≥2.

Sensitivity analysis of the optimal exercise boundary of the American put option

2016 ◽  
Vol 23 (3) ◽  
pp. 429-433
Author(s):  
Nasir Rehman ◽  
Sultan Hussain ◽  
Wasim Ul-Haq
Keyword(s):  
Stock Price ◽  
Obstacle Problem ◽  
One Dimensional ◽  
American Put Option ◽  
Optimal Exercise Boundary ◽  
Put Option ◽  
Dimensional Diffusion ◽  
Exercise Boundary ◽  
American Put ◽  
Continuity Estimate

AbstractWe consider the American put problem in a general one-dimensional diffusion model. The risk-free interest rate is constant, and volatility is assumed to be a function of time and stock price. We use the well-known parabolic obstacle problem and establish the continuity estimate of the optional exercise boundaries of the American put option with respect to the local volatilities, which may be considered as a generalization of the Achdou results [1].

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