Narrow Results

This paper concerns a nonlocal diffusion problem with a free boundary. We first give accurate estimates on the longtime behaviors of solution by constructing suitably upper and lower solutions. In particular, for two important kinds of kernel functions, one of which is compactly supported and the other behaves like $|x{|}^{-\gamma }$ with $\gamma \in (1,2]$ near infinity, some sharp estimates on the longtime behaviors and rates of accelerated spreading are obtained. Then the limiting behaviors of the solution pair of a semi-wave problem and asymptotic dynamics of a nonlocal diffusion problem on half space are given, respectively. Finally, we investigate the limiting profiles of this free boundary problem when the expanding coefficient of free boundary converges to $0$ and $\infty$, respectively.

In this paper, we establish the existence and uniqueness of subsonic flows with a contact discontinuity in a two-dimensional finitely long slightly curved nozzle by prescribing the entropy, the Bernoulli’s quantity, and the horizontal mass flux distribution at the entrance and the flow angle at the exit. The problem is formulated as a nonlinear boundary value problem for a hyperbolic–elliptic mixed system with a free boundary. The Lagrangian transformation is employed to straighten the contact discontinuity and the Euler system is reduced to a second-order nonlinear elliptic equation for the stream function. One of the key points in the analysis is to solve the associated linearized elliptic boundary value problem with mixed boundary conditions in a weighted Hölder space. Another one is to employ the implicit function theorem to locate the contact discontinuity.

It is well acknowledged that the European options can be valued by an analytic formula, but situation is quite different for the American options. Mathematically, the Black-Scholes model for the American option pricing is a free boundary problem of partial differential equation. This model is a non-linear problem; it has no closed form solution. Although approximate solutions may be obtained by some numerical methods, but the precision and stability are hard to control since they are largely affected by the singularity at the exercise boundary near expiration date. In this paper, we propose a new numerical method, namely SDA, to solve the pricing problem of the American options. Our new method combines the advantages of the Semi-analytical Method and the Sliced-fixed Boundary Finite Difference Method while overcomes demerits of the two. Using the SDA method, we can resolve the problems resulted from the singularity near the optimal exercise boundary. Numerical experiments show that the SDA method is more accurate and more stable than other numerical methods. In this paper, we focus on the American put options, but the proposed method is also applicable to other types of options.

Let $𝔾$ be a three-dimensional Lie group with a bi-invariant metric. Consider $\mathrm{\Omega }\subset 𝔾$ a strictly convex domain in $𝔾$. We prove that if $\mathrm{\Sigma }\subset \mathrm{\Omega }$ is a stable CMC free-boundary surface in $\mathrm{\Omega }$ then $\mathrm{\Sigma }$ has genus either 0 or 1, and at most three boundary components. This result was proved by Nunes [I. Nunes, On stable constant mean curvature surfaces with free-boundary, Math. Z.287(1–2) (2017) 73–479] for the case where $𝔾={ℝ}^3$ and by R. Souam for the case where $𝔾={𝕊}^3$ and $\mathrm{\Omega }$ is a geodesic ball with radius $r<\frac{\pi }{2}$, excluding the possibility of $\mathrm{\Sigma }$ having three boundary components. Besides ${ℝ}^3$ and ${𝕊}^3$, our result also apply to the spaces ${𝕊}^1\times {𝕊}^1\times {𝕊}^1$, ${𝕊}^1\times {ℝ}^2$, ${𝕊}^1\times {𝕊}^1\times ℝ$ and $\mathrm{\text{SO}}(3)$. When $𝔾={𝕊}^3$ and $\mathrm{\Omega }$ is a geodesic ball with radius $r<\frac{\pi }{2}$, we obtain that if $\mathrm{\Sigma }$ is stable then $\mathrm{\Sigma }$ is a totally umbilical disc. In order to prove those results, we use an extended stability inequality and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces.

We prove that a free boundary problem for an incompressible Euler equation with surface tension is uniquely solvable, locally in time, in a class of functions of finite smoothness. Moreover, it is shown that the solution of this problem converges to the solution of the problem without surface tension as the coefficient of the surface tension tends to zero.

An installment option is a derivative financial security where the price is paid in installments instead of as a lump sum at the time of purchase. The valuation of these options involves a free boundary problem in that at each installment date, the holder of the derivative has the option of continuing to pay the premiums or allowing the contract to lapse, and the decision will depend upon whether the present value of the expected pay-off is greater or less than the present value of the remaining premiums. Using a model installment option where the premiums are paid continuously rather than on discrete dates, an integral equation is derived for the position of this free boundary by applying a partial Laplace transform to the underlying partial differential equation for the value of the security. Asymptotic analysis of this integral equation allows us to deduce the behavior of the free boundary close to expiry.

We study the Bénard problem in the Boussinesq approximation with upper free surface in the presence of surface tension. Both buoyancy and Marangoni effect are taken into account. Defining a suitable energy functional, we obtain sufficient conditions for nonlinear exponential stability of the rest state.

We consider the compressible Navier–Stokes system for 1D flows of a viscous heat-conducting gas. The zero velocity and stress together with the (positive) temperature or zero heat flux are prescribed on the boundaries. The large-time behavior of the solution is studied for the arbitrarily large initial data. Namely, the stabilization with the power rate of the specific volume divided by 1 + t and the temperature to constants as well as of the velocity to a linear function is proved in L² and H¹-norm by means of the Lyapunov functional method. The study is accomplished in the Lagrangian mass coordinates.

In this paper we present a new approach to describe the behaviour of a pollutant slick at the sea surface. To this end, we consider that the pollutant and the water are immiscible and we propose a two-layer model where the lower layer corresponds to the water and the upper layer represents the pollutant. Since the dimension of the pollutant slick is generally much smaller than the domain occupied by the sea, we propose to compute the motion of the pollutant with a shallow water model with free boundary only in the domain occupied by the pollutant. To discretize in time the problem with free boundary, we use an ALE formulation coupled with the characteristic method. Then, to solve the space discretized problem, we approximate the pollutant velocity by using a Galerkin method with a special basis which verifies the boundary conditions and simplifies significantly the resolution. Finally we test this work in a real situation: the dam of Calacuccia (Corsica).

We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered as functions of the flux intensity. We prove that if one approximates the porosity with a constant then the solution of the hyperbolic problem converges to the classical continuous Green–Ampt solution, also in the presence of shocks. In general, however, the shocks remain present in any approximating solution.

We investigate the derivation and the mathematical properties of a Saint-Venant model with an energy equation and with temperature-dependent transport coefficients. These equations model shallow water flows as well as thin viscous sheets over fluid substrates like oil slicks, atlantic waters in the Strait of Gilbraltar or float glasses. We exhibit an entropy function for the system of partial differential equations and by using the corresponding entropic variable, we derive a symmetric conservative formulation of the system. The symmetrized Saint-Venant quasilinear system of partial differential equations is then shown to satisfy the nullspace invariance property and is recast into a normal form. Upon establishing the local dissipative structure of the linearized normal form, global existence results and asymptotic stability of equilibrium states are obtained. We finally derive the Saint-Venant equations with an energy equation taking into account the temperature-dependence of transport coefficients from an asymptotic limit of a three-dimensional model.

The axially symmetric free surface problem of an ideal incompressible jet issuing from a nozzle and impinging on an uneven wall is investigated in this paper. More precisely, we show that given a semi-infinitely long axially symmetric nozzle, a mass flux $M_0>0$ in the inlet and a constant atmospheric pressure, there exists a unique incompressible impinging jet whose free surface goes to infinity and is close to the impermeable wall at far field. Moreover, the free surface of the impinging jet initiates at the edge of the semi-infinitely long nozzle and the pressure remains the constant atmospheric pressure on the free surface. The main ingredient to show the existence and the uniqueness of the impinging jet is based on the variational method developed in a series of the celebrated works [Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math.325 (1981) 105–144; Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics (John Wiley & Sons, 1982)] by Alt, Caffarelli and Friedman. Furthermore, some important properties of the axially symmetric impinging jet, such as positivity of the radial velocity, asymptotic behavior of the impinging jet, and the optimal decay rate of the free surface and the impinging jet, are obtained. Moreover, the problem of the axially symmetric jet impinging on a hemispherical cup is also considered. Finally, we establish the well-posedness theory on the incompressible impinging jet in two dimensions.

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Choné (Econometrica, 1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet–Choné to a novel free boundary problem, giving the first analytical description of an overlooked market segment.

We consider an optimal stopping problem arising in connection with the exercise of an executive stock option by an agent with inside information. The agent is assumed to have noisy information on the terminal value of the stock, does not trade the stock or outside securities, and maximises the expected discounted payoff over all stopping times with regard to an enlarged filtration which includes the inside information. This leads to a stopping problem governed by a time-inhomogeneous diffusion and a call-type reward. We establish conditions under which the option value exhibits time decay, and derive the smooth fit condition for the solution to the free boundary problem governing the maximum expected reward, and derive the early exercise decomposition of the value function. The resulting integral equation for the unknown exercise boundary is solved numerically and this shows that the insider may exercise the option before maturity, in situations when an agent without the privileged information may not. Hence we show that early exercise may arise due to the agent having inside information on the future stock price.

We study the long-time behavior of the solutions of the Porous Medium Equation u_t = Δ u^m, m > 1, posed in a tube Ω = ℝ × D, where D is a bounded domain in ℝⁿ, and with homogeneous Dirichlet conditions on the lateral boundary. We show that the asymptotic behavior of general nonnegative solutions follows the KPP pattern in suitable rescaled variables. We proceed as follows: we pass to the renormalized problem and show that this problem admits a wave solution that travels along the tube with constant speed with respect to the new time (which is logarithmic in the old time scale). This solution has a bounded free boundary as a forward front.

We also show that the universal asymptotic pattern for solutions with compactly supported data is described in the renormalized form by two of these traveling waves going out in different directions, joined by a stationary profile in the middle region.

We compare this situation with the heat equation case u_t = Δu that behaves quite differently.

In this paper, we consider a parabolic variational inequality arising from the valuation of European installment put option. We prove the existence and uniqueness of the solution to the problem. Moreover, we obtain C^∞ regularity and the bounds of the free boundary. Eventually, we show its numerical result by the binomial method.

We construct three families of singular critical points for a variational free boundary problem. These critical points are homogeneous solutions of degree one to some overdetermined boundary value problem. The intersections of the level sets of these solutions with the unit sphere are isoparametric hypersurfaces and their focal submanifolds.

We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.-T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, J. Functional Anal. 277 (2019) 2772–2814; C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound. 21 (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form $J_{𝜖}(x)=\frac{1}{𝜖}J(\frac{x}{𝜖})$ for small $𝜖>0$, where $J(x)$ has compact support. We also give an estimate of the error term of the approximation by some positive power of $𝜖$.

This work deals with the variation of the solution to an obstacle problem with respect to the variation of its parameters. More precisely, a mechanical structure is pushed by some external forces against an obstacle in such a way that the equilibrium solution involves a part of the domain in which the structure is strictly in contact with the obstacle. It is known from the theory of variational inequalities that studying the variation of the solution as the external forces vary amounts to studying the variation of the boundary of this contact zone. This problem has been studied in previous works in the scalar case, and it was open in the general case where the unknown is a vector field, due to the coupling between the components. As a first step, the present work considers the case of a linearly elastic shallow membrane shell where the coupling between the in-plane and normal components of the displacement arises from the curvature.