function variational principles and coercivity abstract the function type extension of ekeland s variational principle j math anal appl 47 1974 324 353 due to zhong nonlinear anal 29 1997 1421 1431 is deductible in a simplified manner and in a larger functional context this is also true for his normed coercivity result based on palais smale techniques

on differential games for infinite dimensional systems with nonlinear unbounded operators abstract we consider a two player zero sum differential game governed by an abstract nonlinear differential equation of accretive type in an infinite dimensional space we prove that the value function of the game is the unique viscosity solution of the corresponding hamilton jacobi isaacs equation in the sense of 12 we also discuss some properties of this notion of solution

a comparison principle for hamilton jacobi equations related to controlled gradient flows in infinite dimensions we develop new comparison principles for viscosity solutions of hamilton jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures our examples are optimal control of ginzburg landau and fokker planck equations they arise in limit considerations of externally forced non equilibrium statistical mechanics models or through the large deviation principle for interacting particle systems our approach is based on two key ingredients an appropriate choice of geometric structure defining the gradient flow and a free energy inequality resulting from such gradient flow structure the approach allows us to handle hamiltonians with singular state dependency in the nonlinear term as well as hamiltonians with a state space which does not satisfy the radon nikodym property in the case where the state space is a hilbert space the method simplifies existing theories by avoiding the perturbed optimization principle

generalized caristi s fixed point theorems by bae and others abstract in this paper we extend the generalized caristi s fixed point theorems proved by bae j math anal appl 284 2003 690 697 and others

lyapunov functions for infinite dimensional systems abstract we study lyapunov functions for infinite dimensional dynamical systems governed by general maximal monotone operators we obtain a characterization of lyapunov pairs by means of the associated hamilton jacobi partial differential equations whose solutions are meant in the viscosity sense as evolved in works of tataru and crandall lions our approach also leads to a new sufficient condition for lyapunov pairs generalizing a classical result of pazy

large deviation for diffusions and hamilton jacobi equation in hilbert spaces large deviation for markov processes can be studied by hamilton jacobi equation techniques the method of proof involves three steps first we apply a nonlinear transform to generators of the markov processes and verify that limit of the transformed generators exists such limit induces a hamilton jacobi equation second we show that a strong form of uniqueness the comparison principle holds for the limit equation finally we verify an exponential compact containment estimate the large deviation principle then follows from the above three verifications this paper illustrates such a method applied to a class of hilbert space valued small diffusion processes the examples include stochastically perturbed allen cahn cahn hilliard pdes and a one dimensional quasilinear pde with a viscosity term we prove the comparison principle using a variant of the tataru method we also discuss different notions of viscosity solution in infinite dimensions in such context

ekeland s variational principle minimax theorems and existence of nonconvex equilibria in complete metric spaces abstract in this paper we introduce the concept of function which generalizes the concept of w distance studied in the literature we establish a generalized ekeland s variational principle in the setting of lower semicontinuous from above and functions as applications of our ekeland s variational principle we derive generalized caristi s common fixed point theorems a generalized takahashi s nonconvex minimization theorem a nonconvex minimax theorem a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces we also prove that these theorems also imply our ekeland s variational principle

mappings of finite distortion monotonicity and continuity we study mappings f f1 fn rn in the sobolev space w loc r n where is a connected open subset of rn with n 2 thus for almost every x we can speak of the linear transformation d f x rn rn called differential of f at x its norm is defined by d f x sup d f x h h sn 1 we shall often identify d f x with its matrix and denote by j x f det d f x the jacobian determinant thus using the language of differential forms we can write

second order hamilton jacobi bellman equations with an unbounded operator abstract this work is devoted to the study of a class of hamilton jacobi bellman equations associated to an optimal control problem where the state equation is a stochastic differential inclusion with a maximal monotone operator we show that the value function minimizing a bolza type cost functional is a viscosity solution of the hjb equation the proof is based on the perturbation of the initial problem by approximating the unbounded operator finally by providing a comparison principle we are able to show that the solution of the equation is unique

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a dynamic programming approach to nonlinear boundary control problems of parabolic type abstract in this paper we study a hamilton jacobi equation related to the boundary control of a parabolic equation with neumann boundary conditions the state space of this problem is a hilbert space and the equation is defined classically only on a dense subset of the state space moreover the hamiltonian appearing in the equation contains fractional powers of an unbounded operator these facts render the problem difficult in this paper we give a revised definition of a viscosity solution to accommodate the unboundness of the hamiltonian we then obtain existence and uniqueness results for viscosity solutions in particular we show that under suitable assumptions the value function of the boundary control problem is the unique viscosity solution of the related hamilton jacobi equation

some estimates in gγ p m w spaces abstract we consider a class of rearrangement invariant banach function spaces recently appeared in a paper by the same authors containing at the same time some lorentz spaces classical lebesgue spaces and small lebesgue spaces we discuss the main properties coming directly from the norm and for certain values of the involved parameters we prove some estimates of the norm of the associate space

contractions of banach tarafdar meir keeler ciric jachymski matkowski and suzuki types and fixed points in uniform spaces with generalized pseudodistances in the uniform spaces sequentially complete and not sequentially complete we introduce the concept of the j families of generalized pseudodistances we apply it to construct j generalized contractions and we provide the conditions guaranteeing the existence and uniqueness of fixed points of these contractions and the convergence to these fixed points of each iterative sequence of these contractions our j generalized contractions essentially extend banach tarafdar meir keeler ciric jachymski matkowski and suzuki contractions to uniform spaces with j families of generalized pseudodistances examples showing a difference between our results and the well known ones are given the definitions the results and the method to investigate uniqueness and iterative approximation of fixed points of the maps presented here are new for maps in uniform and locally convex spaces and even in metric spaces

contractivity of leader type and fixed points in uniform spaces with generalized pseudodistances abstract recently jachymski and jo wik proved that among various classes of contractions which are introduced and studied in the metric fixed point theory the leader contractions are greatest general contractions in this article we want to show how generalized pseudodistances in uniform spaces can be used to obtain new and general results of leader type without complete graph assumptions about maps and without sequentially complete assumptions about spaces which was not done in the previous publications on this subject the definitions results and methods presented here are new for maps in uniform and locally convex spaces and even in metric spaces examples showing a difference between our results and the well known ones are given

viscosity solutions of hamilton jacobi equations in infinite dimensions iv hamiltonians with unbounded linear terms in this work we show how to interpret s and e in the viscosity sense when the solutions may be expected to be weakly sequentially continuous in particular we introduce notions which allow us to handle the terms and for which we can obtain existence and uniqueness results in a variety of circumstances

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

on the relation between the weak palais smale condition and coercivity given by zhong abstract in this paper we discuss zhong s result that the weak palais smale condition implies coercivity under some assumption given in c k zhong a generalization of ekeland s variational principle and application to the study of the relation between the weak p s condition and coercivity nonlinear anal 29 1997 1421 1431 we also give a simple proof of zhong s result further we generalize the result of caklovic li and willem l caklovic s j li m willem a note on palais smale condition and coercivity differential integral equations 3 1990 799 800

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

a viscosity solution approach to the infinite dimensional hjb equation related to a boundary control problem in a transport equation the paper concerns the infinite dimensional hamilton jacobi bellman equation related to an optimal control problem regulated by a linear transport equation with boundary control a suitable viscosity solution approach is needed in view of the presence of the unbounded control related term in the state equation in the hilbert setting an existence and uniqueness result is obtained

a viscosity approach to infinite dimensional hamilton jacobi equations arising in optimal control with state constraints we consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous we prove that the value function is a viscosity solution of the bellman equation and is in fact the minimal nonnegative supersolution

hamilton jacobi equations in infinite dimensions ii existence of viscosity solutions on demontre des resultats d existence generaux pour 2 problemes 1 le probleme de cauchy u t h x t u du 0 dans v 0 t u x 0 x dans v et 2 le probleme stationnaire u h x u du 0 dans v ou h et sont donnees

new properties of small lebesgue spaces and their applications we prove some new properties of the small lebesgue spaces introduced by fiorenza in f combining these properties with the poincare sobolev inequalities for the relative rearrangement see r1 we derive some new and precises estimates either for small lebesgue sobolev spaces or for quasilinear equations with data in the small lebesgue spaces

the strong ekeland variational principle abstract in this paper we consider the strong ekeland variational principle due to georgiev p g georgiev the strong ekeland variational principle the strong drop theorem and applications j math anal appl 131 1988 1 21 we discuss it for functions defined on banach spaces and on compact metric spaces we also prove the distance version of it

generalized distance and existence theorems in complete metric spaces abstract in this paper we first introduce the concept of distance on a metric space which is a generalized concept of both w distance and tataru s distance we also improve the generalizations of the banach contraction principle caristi s fixed point theorem ekeland s variational principle and the nonconvex minimization theorem according to takahashi further we discuss the relation between w distance and tataru s distance

limit theorems and the support of sdes with oblique reflections on nonsmooth domains abstract in this paper we consider the wong zakai type approximation and the support problem of stochastic differential equations with oblique reflections in nonsmooth domains satisfying conditions introduced by dupuis and ishii 4 we first establish the limit theorem for reflected diffusions by proving that the sequence of solutions of approximating equations converge uniformly in l p to the unique solution of the reflected stochastic differential equation we also prove the approximate continuity and thus characterize the support of diffusions described by reflected sdes

a generalized distance and enhanced ekeland s variational principle for vector functions we propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation this result is shown to contain probably a large part of existing versions of ekeland s variational principle evp we introduce the notion of a weak function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced evp for various settings under relaxed lower semicontinuity assumptions these principles conclude the existence not only of p strict minimizers of p perturbations of the considered vector function but also p sharp and p strong minimizers our results are proved to be stronger than the classical evp and many generalizations in the literature even in the usual one dimensional case by numerous corollaries and examples we include equivalent formulations of our enhanced evp as well

viscosity solutions of dynamic programming equations for the optimal control of the two dimensional navier stokes equations in this paper we study the infinite dimensional hamilton jacobi equation associated with the optimal feedback control of viscous hydrodynamics we resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution

a direct approach to the duality of grand and small lebesgue spaces abstract in this paper we show by elementary methods that the quasinorms of the grand and small lebesgue spaces f l p sup 0 t 1 1 log t 1 p t 1 f s p d s 1 p 1 p f l q 0 1 1 log t 1 q 0 t f s q d s 1 q d t t q p p 1 found by fiorenza and karadzhov in a fiorenza g e karadzhov grand and small lebesgue spaces and their analogs z anal anwendungen 23 4 2004 657 681 by using deeply extrapolation interpolation techniques are associate each other in other terms the sharp holder s type inequality 0 1 f g d x c p f l q g l p is proven where the sharpness means that l q is the smallest quasinorm up to equivalences such that the inequality holds the method is based entirely on integral estimates makes use of asymptotic properties of euler s gamma function and gives an explicit estimate of the constant c p all the results are expressed in terms of the more general spaces l p and l q 0

support d un processus de reflexion let d be a regular open set in n we describe in this paper the support of d valued diffusion processes with reflecting boundary condition generalizing the results of stroock and varadhan obtained when d o

viscosity solutions for a class of hamilton jacobi equations in hilbert spaces we study hamilton jacobi equations with an unbounded term in hilbert spaces we introduce a new variant of the notion of viscosity solution for a class of hamilton jacobi equations and obtain comparison and existence results for viscosity solutions we also examine the usefulness of the notion of viscosity solution in optimal control

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

noncoalescence for the skorohod equation in a convex domain of ℝ2 given a convex domain of 2 we show that a s the paths of two solutions of the skorohod equations driven by the same brownian motion but starting at different points do not meet at the same time

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

subrahmanyam s fixed point theorem abstract we give a sufficient and necessary condition for the convergence of the sequence of successive approximations to a fixed point which is the conclusion of subrahmanyam s fixed point theorem