Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing lagrangians,eikonal equations,and shape-from-shading

We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique bounded-from-below viscosity solution of the Hamilton-Jacobi equation that is null on the target. The result applies to problems with the property that all trajectories satisfying a certain integral condition must stay in a bounded set. We allow problems for which the Lagrangian is not uniformly bounded below by positive constants, in which the hypotheses of the known uniqueness results for Hamilton-Jacobi equations are not satisfied. We apply our theorems to eikonal equations from geometric optics, shape-from-shading equations from image processing, and variants of the Fuller Problem.     

Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces Lφ(Rd). We give a necessary condition for the continuity of M on Lφ(Rd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces Lp(⋅)(Rd) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also Mq is continuous for some q>1, where .     

Asymptotic analysis for the eikonal equation with the dynamical boundary conditions

We study the dynamical boundary value problem for Hamilton‐Jacobi equations of the eikonal type with a small parameter. We establish two results concerning the asymptotic behavior of solutions of the Hamilton‐Jacobi equations: one concerns with the convergence of solutions as the parameter goes to zero and the other with the large‐time asymptotics of solutions of the limit equation.     

Some homogenization results for non-coercive Hamilton–Jacobi equations

Recently, C. Imbert and R. Monneau study the homogenization of coercive Hamilton–Jacobi Equations with a u/ε-dependence: this unusual dependence leads to a non-standard cell problem and, in order to solve it, they introduce new ideas to obtain the estimates on the oscillations of the solutions. In this article, we use their ideas to provide new homogenization results for “standard” Hamilton–Jacobi Equations (i.e. without a u/ε-dependence) but in the case of non-coercive Hamiltonians. As a by-product, we obtain a simpler and more natural proof of the results of C. Imbert and R. Monneau, but under slightly more restrictive assumptions on the Hamiltonians.     

A counterexample to uniqueness of generalized characteristics in Hamilton-Jacobi theory

The notion of generalized characteristics plays a pivotal role in the study of propagation of singularities for Hamilton-Jacobi equations. This note gives an example of nonuniqueness of forward generalized characteristics emanating from a given point.     

Homogenization and penalization of functional first-order PDE

In this paper we study the asymptotic behavior of viscosity solutions for a functional partial differential equation with a small parameter as the parameter tends to zero. We study simultaneous effects of homogenization and penalization in functional first-order PDE. We establish a convergence theorem in which the limit equation is identified with some first order PDE.     

Retractability of set theoretic solutions of the Yang-Baxter equation

It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-Z groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted union is constructed. This answers in the negative another problem of Gateva-Ivanova. The constructed solution is of multipermutation level 3. Retractability of solutions is also proved in the case where the natural generators of the IYB group are cyclic permutations. Moreover, it is shown that such solutions are generalized twisted unions.     

The Bellman equation for time-optimal control of noncontrollable,nonlinear systems

For a general nonlinear system and closed target set we study the value functions and of the control problems of reaching and, respectively, its interior, in minimum time. Under no controllability assumptions on the system, we characterize them as, respectively, the minimal viscosity supersolution and the maximal viscosity subsolution of the Bellman equation with appropriate boundary conditions. Then we prove that is the unique upper semicontinuous complete solution of such a boundary value problem, which means in particular that the (completed) graph of contains the graph of any solution, as well as all the limits of reasonable approximating sequences. We give some applications to verifications theorems and to the stability of the minimum time function with respect to general perturbations.The authors are partially supported by the Italian National Projects Equazioni di evoluzione e applicazioni fisico-matematiche and Equazioni differenziali e calcolo delle variazioni, respectively.     

Cauchy problem of the generalized double dispersion equation

In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the generalized double dispersion equation are proved. The blow-up of the solution for the Cauchy problem of the generalized double dispersion equation is discussed by the concavity method under some conditions.     

SOME ASPECTS OF THE THEORY OF SYMMETRIC OPERATOR SPACES

Abstract

We discuss several aspects of the theory of symmetric Banach spaces of measurable operators, including their construction and certain topological and geometric properties. Particular emphasis is given to the role played by rearrangement inequalities.     

Transmission conditions on interfaces for Hamilton–Jacobi–Bellman equations

We establish a comparison principle for a Hamilton–Jacobi–Bellman equation, more appropriately a system, related to an infinite horizon problem in presence of an interface. Namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place. Since corresponding Hamiltonians, at least for the subsolution part, do not enjoy any semicontinuity property, the comparison argument is rather based on a separation principle of the controlled dynamics across the interface. For this, we essentially use the notion of ε-partition and minimal ε-partition for intervals of definition of an integral trajectory.     

Real analytic generalized functions

Real analytic generalized functions are introduced and investigated. The analytic singular support and analytic wave front of a generalized function in are introduced and described. Authors’ addresses: S. Pilipović, Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovića 4, 21000 Novi Sad, Serbia; D. Scarpalezos, U.F.R. de Mathématiques, Université Paris 7, 2 Place Jussieu, Paris 5^e, 75005, France; V. Valmorin, Université des Antilles et de la Guyane, Département Math-Info, Campus de Fouillole, 97159 Pointe á Pitre Cedex, France     

Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary

In this paper the asymptotic behaviour of a second-order linear evolution problem is studied in a domain, a part of wich has an oscillating boundary. An homogeneous Neumann condition is given on the whole boundary of the domain. Moreover the behaviour of associated optimal control problem is analyzed.      

A bound for minimal graphs with a normal at infinity

It is shown that a minimal graph with a normal at infinity is in a-priori bounded vertical distance from its approximating halfcatenoid. This is used to show that the exterior contact angle problem is wellposed under natural geometric conditions on the domain, while the exterior Dirichlet problem can be solvable only for data which satisfy an oscillation bound.This paper was written under the support of the Deutsche Forschungsgemeinschaft while the author was visiting the department of mathematics at Stanford University.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

Phragmén-Lindelöf and continuous dependence type results in generalized heat conduction

Using an energy method we investigate the decay of end effects for a generalized heat conduction problem defined on a semi-infinite cylindrical region. With homogeneous Dirichlet conditions on the lateral surface of the cylinder it is shown that solutions either grow exponentially or decay exponentially in the distance from the finite end of the cylinder. The effect of perturbing the equation parameters is also investigated.     

On viscosity solutions of irregular Hamilton-Jacobi equations

It is proved that the initial-value problem for admits a unique continuous viscosity solution under certain conditions which do not exclude that H(x, p) is discontinuous in x. Particular attention is devoted to the linear transport equation , where a may be discontinuous. Received: 21 October 2002     

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton–Jacobi–Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles–Souganidis (2000) for first-order Hamilton–Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.     

Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation

In this paper, we prove the existence and the uniqueness of global solution for the Cauchy problem for the generalized Boussinesq equation. Under some assumptions, we also show that the _L∞$L_{\infty }$ norm of small solution of the Cauchy problem for the generalized Boussinesq equation decays to zero as t$t$ tends to the infinity.     

A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton–Jacobi equations

We investigate the large-time behavior of the value functions of the optimal control problems on the n-dimensional torus which appear in the dynamic programming for the system whose states are governed by random changes. From the point of view of the study on partial differential equations, it is equivalent to consider viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations. The large-time behavior of viscosity solutions of this problem has been recently studied by the authors and Camilli, Ley, Loreti, and Nguyen for some special cases, independently, but the general cases remain widely open. We establish a convergence result to asymptotic solutions as time goes to infinity under rather general assumptions by using dynamical properties of value functions.     

Asymptotically self-similar solutions of the damped wave equation

We consider the damped hyperbolic equation

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