Regularity theory for a class of quasilinear equations

This paper addresses the analysis of the weak solution of in a bounded domain Ω subject to the boundary condition on , when the data f belongs to and . We prove existence and uniqueness of solution for this problem in the Nikolskii space . Moreover, we obtain energy estimates regarding the Nikolskii norm of ω in terms of the norm of f.     

Comparison principle for equations of the Hamilton-Jacobi type in control theory

This paper deals with the comparison principle for the first-order ODEs of the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Bellman-Isaacs type which describe solutions to the problems of reachability and control synthesis under complete as well as under limited information on the system disturbances. Since the exact solutions require fairly complicated calculation, this paper presents the upper and lower bounds to these solutions, which in some cases may suffice for solving such problems as the investigation of safety zones in motion planning, verification of control strategies or of conditions for the nonintersection of reachability tubes, etc. For systems with original linear structure it is indicated that present among the suggested estimates are those of ellipsoidal type, which ensure tight approximations of the convex reachability sets as well as of the solvability sets for the problem of control synthesis.     

Entropy penalization methods for Hamilton-Jacobi equations

In this paper we present a new entropy penalization problem and we discuss its relations with approximate solutions of Hamilton-Jacobi equations, the convergence of associated discrete schemes, as well as several applications, such as: a generalization of the Hopf-Cole transformation which converts non-linear Hamilton-Jacobi equations into linear evolution equations, the study of fixed point problems, approximation of certain linear evolution equations, and the construction of entropy penalized Mather measures.     

Externality and Hamilton-Jacobi equations

The relationship between optimal control problems and Hamilton-Jacobi-Bellman equations is well known [9]. In fact the value function, defined as the infimum of the cost functional, satisfies in the viscosity sense an appropriate Hamilton-Jacobi-Bellman equation. In this paper we consider several control problems such that the cost functional associated to each problem depends explicitly on the value functions of the other problems. This leads to a system of Hamilton-Jacobi-Bellman equations. This is known, in economic context [14] cap XI, as an externality problem. In these problems may occur a lack of uniqueness of the value functions. We give conditions to ensure existence, uniqueness of the value functions and an implicit integral representation formula. Moreover, under uniqueness assumption, we prove that the variational solutions of the associated Hamilton-Jacobi system converge asymptotically to the value functions. We prove also an uniqueness theorem in the case of viscosity solutions of Hamilton-Jacobi-Bellman system.     

Uniqueness results for nonlocal Hamilton-Jacobi equations

We are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal, non-monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.     

On shock generation for Hamilton-Jacobi equations

The subject of this paper is the generation of singularities of solutions of Hamilton-Jacobi equations set in (0, ∞) × ? fordataofclass C∞. Shockwaves originate from conjugate points. To show sharpness of a known Hausdorff estimate, an example is given in which the set of conjugate, regular points includes uncountably many affine subspaces of dimension n − 1.     

Regularity theory in Orlicz spaces for the parabolic polyharmonic equations

In this paper we generalize classical L ^p estimates to Orlicz spaces for the parabolic polyharmonic equations. Our argument is based on the iteration-covering procedure. Received: 10 September 2007     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

The solution of evolutionary games using the theory of Hamilton-Jacobi equations

A dynamical model of a non-antagonistic evolutionary game for two coalitions is considered. The model features an infinite time span and discounted payoff functionals. A solution is presented using differential game theory. The solution is based on the construction of a value function for auxiliary antagonistic differential games and uses an approximate grid scheme from the theory of generalized solutions of the Hamilton-Jacobi equations. Together with the value functions the optimal guaranteeing procedures for control on the grid are computed and the Nash dynamic equilibrium is constructed. The behaviour of trajectories generated by the guaranteeing controls is investigated. Examples are given.     

Hamilton-Jacobi theory for periodic control problems

The general problem of periodic optimization is considered in this paper. The contribution consists in stating sufficient conditions for the optimality of a control satisfying the periodicity constraint. The result has been achieved by means of the classical Hamilton-Jacobi equation, suitably modified in order to consider the peculiar constraint of the problem. Finally, an interesting application of the theory to the case of linear, time-invariant systems is given.This work was supported by CNR (Consiglio Nazionale delle Ricerche), Rome, Italy.     

Regularity theory in Orlicz spaces for elliptic equations in Reifenberg domains

In this paper, we establish the global estimates for divergence form elliptic equations in the Orlicz space L?(Ω), where Ω is a bounded Reifenberg domain in Rn. Our result generalizes the W1,p estimates.     

Metric character of Hamilton-Jacobi equations

We deal with the metrics related to Hamilton-Jacobi equations of eikonal type. If no convexity conditions are assumed on the Hamiltonian, these metrics are expressed by an - formula involving certain level sets of the Hamiltonian. In the case where these level sets are star-shaped with respect to 0, we study the induced length metric and show that it coincides with the Finsler metric related to a suitable convexification of the equation.

     

Numerical schemes for Hamilton-Jacobi equations on unstructured meshes

Summary.  In this paper, a numerical scheme is presented by applying the finite element method to the viscosity equations of the Hamilton-Jacobi equations on unstructured meshes. By improving the finite element scheme, another numerical scheme is constructed. Under certain limitations, the numerical solutions of the two schemes converge to the viscosity solutions of the Hamilton-Jacobi equations. The latter numerical scheme has weaker restrictions than the former scheme for convergence. Numerical examples are provided to test the stability, convergence and sensitivity to different meshes. Received November 5, 2001 / Revised version received March 5, 2002 / Published online October 29, 2002 RID="*" ID="*" Current address: Department of Applied Mathematics, University of Petroleum, Dongying 257062, Shandong, P.R.China; e-mail: xianggui_li@sina.com Mathematics Subject Classification (1991): 65M60     

Boundary-value problems for stationary Hamilton-Jacobi and Bellman equations

We introduce solutions of boundary-value problems for the stationary Hamilton-Jacobi and Bellman equations in functional spaces (semimodules) with a special algebraic structure adapted to these problems. In these spaces, we obtain representations of solutions in terms of “basic” ones and prove a theorem on approximation of these solutions in the case where nonsmooth Hamiltonians are approximated by smooth Hamiltonians. This approach is an alternative to the maximum principle.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

Implicit difference methods for Hamilton-Jacobi functional differential equations

Classical solutions of initial boundary value problems are approximated by solutions of associated implicit difference functional equations. A stability result is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used to numerically solve nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent whereas the corresponding explicit difference methods are not. The results obtained can be applied to differential integral problems and differential equations with deviated variables.     

Uniqueness for first-order Hamilton-Jacobi equations and Hopf formula

We study uniqueness properties for a certain class of Cauchy problems for first-order Hamilton-Jacobi equations for which a solution is given by the Hopf formula. We prove various comparison and characterisation results concerning both convex generalized solutions and viscosity solutions. In particular, we show that the Hopf solution is the maximum convex generalized subsolution and the unique convex viscosity solution of the Cauchy problem.     

Relaxation Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations

This paper presents a relaxation Lax-Friedrichs sweeping scheme to approximate viscosity solutions of static Hamilton Jacobi equations in any number of spatial dimensions. It is a generalization of the scheme proposed in Kao et al. (J Comput Phys 196:367–391, 2004). Numerical examples suggest that the relaxation Lax-Friedrichs sweeping scheme has smaller number of iterations than the original Lax-Friedrichs sweeping scheme when the relaxation factor ω is slightly larger than one. And first order convergence is also demonstrated by numerical results. A theoretical analysis for our scheme in a special case is given.