A stability result concerning the obstacle problem for a plate

It is proposed here to study the free boundary of the obstacle problem in the case of an elastic plate. Under a nondegeneracy assumption, we prove a stability theorem which relates the variations of the contact zone to the variations the external forces. The statement of this result obtained and the steps in the proof are very close to those given by D.G. Schaeffer in 1975, except for the very important fact that the present study deals with the biharmonic operator.     

Random homogenization of an obstacle problem

We study the homogenization of an obstacle problem in a perforated domain, when the holes are periodically distributed and have random shape and size. The main assumption concerns the capacity of the holes which is assumed to be stationary ergodic.     

The Keldys-Fichera obstacle problem

In this paper we consider a quasilinear obstacle problem with coefficients of higher growth orders and the Keldys-Fichera boundary condition. Research supported by the Fund of IMAS     

An unconditional existence result for the quasi-variational elastohydrodynamic free boundary value problem

In this paper a two-dimensional quasi-variational inequality arising in elastohydrodynamic lubrication is studied for non-constant viscosity. So far, existence results for such piezo-viscous problems require an L^∞ property for an auxiliary problem. For the usual pressure-viscosity relations, this property needs small data assumptions which are not observed in experimental conditions. In the present work, such small data assumptions are proved unnecessary for existence results. Besides well-established monotonicity behavior for the viscosity-pressure relation, the only condition used here is on the asymptotic behavior for this law as the pressure tends to infinity. If the procedure used here, namely the introduction of a reduced pressure by Grubin transform followed by a regularization procedure, appears somewhat classical, the way in which an upper bound is obtained is completely new.     

A uniqueness theorem for a free boundary problem

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

     

The Caffarelli Alternative in Measure for the Nondivergence Form Elliptic Obstacle Problem with Principal Coefficients in VMO

We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in [3 Caffarelli , L.A. ( 1977 ). The regularity of free boundaries in higher dimensions . Acta Math. 139 : 155 – 184 .[Crossref], [Web of Science ®] , [Google Scholar]].     

The virtual element method for a nonhomogeneous double obstacle problem of Kirchhoff plate

This paper is intended to propose and analyze the lowest order conforming and nonconforming virtual element methods (VEMs) for a nonhomogeneous double obstacle problem in fourth-order variational inequality. We first present an abstract framework for the error analysis of an abstract discrete method. Then, we develop the conforming and fully nonconforming VEMs for the previous problem, with the optimal error estimates obtained by using the previous abstract framework combined with two modified interpolation operators and the related estimates. The discrete problem is solved by the primal–dual active set algorithm and the numerical results are in good agreement with theoretical findings.     

The behavior of the free boundary near the fixed boundary for a minimization problem

We show that the free boundary ∂{u > 0}, arising from the minimizer(s) u, of the functional

Symmetry and rigidity for the hinged composite plate problem

The composite plate problem is an eigenvalue optimization problem related to the fourth order operator ${(?\mathrm{\Delta })}^2$. In this paper we continue the study started in [10], focusing on symmetry and rigidity issues in the case of the hinged composite plate problem, a specific situation that allows us to exploit classical techniques like the moving plane method.     

Free boundary regularity close to initial state for parabolic obstacle problem

In this paper we study the behavior of the free boundary , arising in the following complementary problem:

Here denotes the parabolic boundary, is a parabolic operator with certain properties, is the upper half of the unit cylinder in , and the equation is satisfied in the viscosity sense. The obstacle is assumed to be continuous (with a certain smoothness at , ), and coincides with the boundary data at time zero. We also discuss applications in financial markets.

     

Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R2 of the form 0 〈 y 〈p（x,t）, where p（x,t） is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little H61der spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient y is larger than a threshold value y〉0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.     

A third-order mixed finite-element method for the numerical solution of the biharmonic problem

This paper is devoted to the introduction of a mixed finite element for the solution of the biharmonic problem. We prove optimal rate of convergence for the element. The mixed approach allows the simultaneous approximation of both displacement and tensor of its second derivatives.     

分期付款购房模型及其抛物障碍问题

考虑常数利率条件下的分期付款购房模型,若风险资产市场价格服从几何布朗运动,则合约可以运用美式期权定价方法给出合理的价格.该文给出了分期付款购房合约定价的偏微分方程方法,得到一个一维抛物障碍问题,同时给出了贷款合约价格函数所联系的自由边界,并讨论了自由边界的单调性与有界性.     

On the asymptotic free boundary for the American put option problem

In practical work with American put options, it is important to be able to know when to exercise the option, and when not to do so. In computer simulation based on the standard theory of geometric Brownian motion for simulating stock price movements, this problem is fairly easy to handle for options with a short lifespan, by analyzing binomial trees. It is considerably more challenging to make the decision for American put options with long lifespan. In order to provide a satisfactory analysis, we look at the corresponding free boundary problem, and show that the free boundary—which is the curve that separates the two decisions, to exercise or not to—has an asymptotic expansion, where the coefficient of the main term is expressed as an integral in terms of the free boundary. This raises the perspective that one could use numerical simulation to approximate the integral and thus get an effective way to make correct decisions for long life options.     

On the boundary value problem for the spectrally loaded heat conduction operator

We consider the boundary value problems in a quarter-plane for a loaded heat conduction operator (one-dimensional in the space variable). A peculiarity of the operator in question is as follows: first, the spectral parameter is the coefficient of the loaded summand; second, the order of the derivative in the loaded summand is equal to that of the differential part of the operator, and third, the load point moves with a variable velocity. We demonstrate that the boundary value problem under study is Noetherian.     

Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem

This paper concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon. From the angle of stochastic control, it is a singular control problem, whose value function is governed by a time-dependent HJB equation with gradient constraints. We reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. This enables us to make use of the well-developed theory of obstacle problem to attack the problem. The C2,1 regularity of the value function is proven and the behaviors of the free boundaries are completely characterized.     

Inequalities for eigenvalues of a clamped plate problem

Let be a connected bounded domain in an -dimensional Euclidean space . Assume that

are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator:

Then, we give an upper bound of the -th eigenvalue in terms of the first eigenvalues, which is independent of the domain , that is, we prove the following:

Further, a more explicit inequality of eigenvalues is also obtained.

     

Linearized stability of traveling cell solutions arising from a moving boundary problem

In 2003, Mogilner and Verzi proposed a one-dimensional model on the crawling movement of a nematode sperm cell. Under certain conditions, the model can be reduced to a moving boundary problem for a single equation involving the length density of the bundled filaments inside the cell. It follows from the results of Choi, Lee and Lui (2004) that this simpler model possesses traveling cell solutions. In this paper, we show that the spectrum of the linear operator, obtained from linearizing the evolution equation about the traveling cell solution, consists only of eigenvalues and there exists such that if is a real eigenvalue, then . We also provide strong numerical evidence that this operator has no complex eigenvalue.