On the number of singularities for the obstacle problem in two dimensions

We study the obstacle problem in two dimensions. On the one hand we improve a result of L.A. Caffarelli and N.M. Rivière: we state that every connected component of the interior of the coincidence set has at most N ₀ singular points, where N ₀ is only dependent on some geometric constants. Moreover, if the component is small enough, then this component has at most two singular points. On the other hand, we prove in a simple case a conjecture of D.G. Schaeffer on the generic regularity of the free boundary: for a family of obstacle problems in two dimensions continuously indexed by a parameter λ, the free boundary of the solution u_λ is analytic for almost every λ. Finally we present a new monotonicity formula for singular points. Dedicated to Henri Berestycki and Alexis Bonnet.     

A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.This work has been supported by funds from ACI JC 1041 “Mouvements d’interfaces avec termes non-locaux”, from ACI-JC 1025 “Dynamique des dislocations” and from ONERA, Office National d’Etudes et de Recherches. The second author was also supported by the ENPC-Région Ile de France.     

Iterative Approaches to Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

Recently, O’Hara, Pillay and Xu (Nonlinear Anal. 54, 1417–1426, 2003) considered an iterative approach to finding a nearest common fixed point of infinitely many nonexpansive mappings in a Hilbert space. Very recently, Takahashi and Takahashi (J. Math. Anal. Appl. 331, 506–515, 2007) introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In this paper, motivated by these authors’ iterative schemes, we introduce a new iterative approach to finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in a Hilbert space. The main result of this work is a strong convergence theorem which improves and extends results from the above mentioned papers.     

A shape and topology optimization technique for solving a class of linear complementarity problems in function space

A shape and topology optimization driven solution technique for a class of linear complementarity problems (LCPs) in function space is considered. The main motivating application is given by obstacle problems. Based on the LCP together with its corresponding interface conditions on the boundary between the coincidence or active set and the inactive set, the original problem is reformulated as a shape optimization problem. The topological sensitivity of the new objective functional is used to estimate the “topology” of the active set. Then, for local correction purposes near the interface, a level set based shape sensitivity technique is employed. A numerical algorithm is devised, and a report on numerical test runs ends the paper.     

Variational problem with an obstacle in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> for a class of quadratic functionals

  A variational problem with an obstacle for a certain class of quadratic functionals is considered. Admissible vector-valued functions are assumed to satisfy the Dirichlet boundary condition, and the obstacle is a given smooth (N − 1)-dimensional surface S in ℝ ^N . The surface S is not necessarily bounded. It is proved that any minimizer u of such an obstacle problem is a partially smooth function up to the boundary of a prescribed domain. It is shown that the (n − 2)-Hausdorff measure of the set of singular points is zero. Moreover, u is a weak solution of a quasilinear system with two kinds of quadratic nonlinearities in the gradient. This is proved by a local penalty method. Bibliography: 25 titles. Dedicated to V. A. Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh. Seminarov POMI, Vol. 362, 2008, pp. 15–47.     

Characterization of correlated equilibria in stochastic games

A general communication device is a device that at every stage of the game receives a private message from each player, and in return sends a private signal to each player; the signals the device sends depend on past play, past signals it sent, and past messages it received.  An autonomous correlation device is a general communication device where signals depend only on past signals the device sent, but not on past play or past messages it received.  We show that the set of all equilibrium payoffs in extended games that include a general communication device coincides with the set of all equilibrium payoffs in extended games that include an autonomous correlation device. A stronger result is obtained when the punishment level is independent of the history. Final version July 2001     

Weakly elliptic systems with obstacle constraints Part II — An N × N model problem

Existence, uniqueness (even stability), and regularity are established for a special system of variational inequalities of obstacle type. The system is only considered to be elliptic in the weakest possible sense. The system includes a version of the biharmonic and polyharmonic obstacle problems. The main aspect of this problem and the approach here is its reduction via algebraic invarients to special canonical forms. One might view this work as the beginnings of a “group analysis” of variational inequalities. This paper is dedicated to the memory of Guido Stampacchia (1922–1978).     

An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.      

On the vanishing viscosity limit in a disk

We say that a solution of the Navier–Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L ²) norm uniformly in time as the viscosity ν vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space–time energy density of the solution to the Navier–Stokes equations in a boundary layer of width proportional to ν vanish with ν, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/ν. The author was supported in part by NSF grant DMS-0705586 during the period of this work.     

Dual-Role Based Pricing in a Dynamic and Competitive Environment

The dual role of price, as a product attribute signaling quality and as a measure of sacrifice, serving as a benchmark for comparing the utility gains from superior product quality, is now well established in the marketing and economic literature. However, knowledge about their long-run impact and the influence of competition on these effects still remains very sparse. In the current paper, with reference to a dynamic and competitive framework, an analytical model is proposed to help determining optimal decision rules for price incorporating both roles. The main results are as follows: (i) The optimal pricing policy is determined as a Nash equilibrium strategy. (ii) The resulting equilibrium price is higher than an equilibrium that disregards the carryover price effects. (iii) For a symmetric competition, we provide normative rules on how firms should set prices as a function of the perceived quality; particularly, how the price should be set initially, when there is little product familiarity and the perceived quality is low, and how this price should vary as the perceived quality increases. (iv) At steady state, we find that the level of equilibrium margin, in percentage terms, decreases with the elasticity of demand with respect to the brand price, but this decrease is moderated by the elasticity of demand with respect to the brand perceived quality, the cross elasticity of demand with respect to the competitor’s perceived quality and the effect of the competitor’s current price on the firm’s perceived quality deterioration. The author thanks Konstantin Kogan for helpful discussions and comments.     

On equilibrium in pure strategies in games with many players

We demonstrate that, if there are sufficiently many players, any Bayesian equilibrium of an incomplete information game can be “ε-purified” . That is, close to any Bayesian equilibrium there is an approximate Bayesian equilibrium in pure strategies. Our main contribution is obtaining this result for games with a countable set of pure strategies. In order to do so we derive a mathematical result, in the spirit of the Shapley–Folkman Theorem, permitting countable strategy sets. Our main assumption is a “large game property,” dictating that the actions of relatively small subsets of players cannot have large affects on the payoffs of other players. E. Cartwright and M. Wooders are indebted to Phillip Reny, Frank Page and two anonymous referees for helpful comments.     

Crowding games are sequentially solvable

A sequential-move version of a given normal-form game Γ is an extensive-form game of perfect information in which each player chooses his action after observing the actions of all players who precede him and the payoffs are determined according to the payoff functions in Γ. A normal-form game Γ is sequentially solvable if each of its sequential-move versions has a subgame-perfect equilibrium in pure strategies such that the players' actions on the equilibrium path constitute an equilibrium of Γ.  A crowding game is a normal-form game in which the players share a common set of actions and the payoff a particular player receives for choosing a particular action is a nonincreasing function of the total number of players choosing that action. It is shown that every crowding game is sequentially solvable. However, not every pure-strategy equilibrium of a crowding game can be obtained in the manner described above. A sufficient, but not necessary, condition for the existence of a sequential-move version of the game that yields a given equilibrium is that there is no other equilibrium that Pareto dominates it. Received July 1997/Final version May 1998     

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set. Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren’s frequency formula, first established by Caffarelli, Salsa, and Silvestre. N. Garofalo is supported in part by NSF grant DMS-0701001. A. Petrosyan is supported in part by NSF grant DMS-0701015.     

Confinement of Brownian motion among Poissonian obstacles in ℝ d , d≥ 3

Consider Brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of ℝ ^d , d≥ 1, at the points of a Poisson cloud of constant intensity v <: 0. Assume that Brownian motion is absorbed instantaneously upon entering the obstacle set. In SZN-conf Sznitman has shown that in d = 2, conditionally on the event that the process does not enter the obstacle set up to time t, the probability that Brownian motion remains within distance ∼t ^1/4 from its starting point is going to 1 as t goes to infinity. We show that the same result holds true for d≥ 3, with t ^1/4 replaced by t ^{1/( d +2)}. The proof is based on Sznitmans refined method of enlargement of obstacles [10] as well as on a quantitative isoperimetric inequality due to Hall [4]. Received: 6 July 1998     

Computing a closest bifurcation instability in multidimensional parameter space

Summary Engineering and physical systems are often modeled as nonlinear differential equations with a vector λ of parameters and operated at a stable equilibrium. However, as the parameters λ vary from some nominal value λ₀, the stability of the equilibrium can be lost in a saddle-node or Hopf bifurcation. The spatial relation in parameter space of λ₀ to the critical set of parameters at which the stable equilibrium bifurcates determines the robustness of the system stability to parameter variations and is important in applications. We propose computing a parameter vector λ_* at which the stable equilibrium bifurcates which is locally closest in parameter space to the nominal parameters λ₀. Iterative and direct methods for computing these locally closest bifurcations are described. The methods are extensions of standard, one-parameter methods of computing bifurcations and are based on formulas for the normal vector to hypersurfaces of the bifurcation set. Conditions on the hypersurface curvature are given to ensure the local convergence of the iterative method and the regularity of solutions of the direct method. Formulas are derived for the curvature of the saddle node bifurcation set. The methods are extended to transcritical and pitchfork bifurcations and parametrized maps, and the sensitivity to λ₀ of the distance to a closest bifurcation is derived. The application of the methods is illustrated by computing the proximity to the closest voltage collapse instability of a simple electric power system.     

The Obstacle Version of the Geometric Dynamic Programming Principle: Application to the Pricing of American Options Under Constraints

We provide an obstacle version of the Geometric Dynamic Programming Principle of Soner and Touzi (J. Eur. Math. Soc. 4:201–236, 2002) for stochastic target problems. This opens the doors to a wide range of applications, particularly in risk control in finance and insurance, in which a controlled stochastic process has to be maintained in a given set on a time interval [0,T]. As an example of application, we show how it can be used to provide a viscosity characterization of the super-hedging cost of American options under portfolio constraints, without appealing to the standard dual formulation from mathematical finance. In particular, we allow for a degenerate volatility, a case which does not seem to have been studied so far in this context.     

Credible deviations from signaling equilibria

In games with costly signaling, some equilibria are vulnerable to deviations which could be “unambiguously” interpreted as coming from a unique set of Sender-types. This occurs when these types are precisely the ones who gain from deviating for any beliefs the Receiver could form over that set. We show that this idea characterizes a unique equilibrium outcome in two classes of games. First, in monotonic signaling games, only the Riley outcome is immune to this sort of deviation. Our result therefore provides a plausible story behind the selection made by Cho and Kreps’s (Q J Econ 102:179–221, 1987) D1 criterion on this class of games. Second, we examine a version of Crawford and Sobel (Econometrica 50:1431–1451, 1982) model with costly signaling, where standard refinements have no effect. We show that only a Riley-like separating equilibrium is immune to these deviations.     

On balking from an empty queue

The intuition while observing the economy of queueing systems, is that one’s motivation to join the system, decreases with its level of congestion. Here we present a queueing model where sometimes the opposite is the case. The point of departure is the standard first-come first-served single server queue with Poisson arrivals. Customers commence service immediately if upon their arrival the server is idle. Otherwise, they are informed if the queue is empty or not. Then, they have to decide whether to join or not. We assume that the customers are homogeneous and when they consider whether to join or not, they assess their queueing costs against their reward due to service completion. As the whereabouts of customers interact, we look for the (possibly mixed) join/do not join Nash equilibrium strategy, a strategy that if adopted by all, then under the resulting steady-state conditions, no one has any incentive not to follow it oneself. We show that when the queue is empty then depending on the service distribution, both ‘avoid the crowd’ (ATC) and ‘follow the crowd’ (FTC) scenarios (as well as none-of-the-above) are possible. When the queue is not empty, the situation is always that of ATC. Also, we show that under Nash equilibrium it is possible (depending on the service distribution) that the joining probability when the queue is empty is smaller than it is when the queue is not empty. This research was supported by The Israel Science Foundation Grant No. 237/02.     

Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers

In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type “admissible strategy” against “admissible control”, and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman–Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in Hamadène, Hassani (Probab Theory Relat Fields 132:237–264, 2005). Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier. Partially supported by the NSF of P.R.China (No. 10701050; 10671112), Shandong Province (No. Q2007A04), and National Basic Research Program of China (973 Program) (No. 2007CB814904).     

Invariant measures of minimal post-critical sets of logistic maps

We construct logistic maps whose restriction to the ω-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the ω-limit set of its critical point. Furthermore, we show that such a logistic map f can be taken so that each such invariant measure has zero Lyapunov exponent and is an equilibrium state of f for the potential −ln |f′|.