multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

dgmres a gmres type algorithm for drazin inverse solution of singular non symmetric linear systems abstract in a recent work by the author linear algebra appl 298 1999 99 krylov subspace methods were derived for drazin inverse solution of consistent or inconsistent linear systems of the form ax b where a c n n is a singular and in general non hermitian matrix that has arbitrary index one of these methods modeled after the generalized conjugate residual method gcr and denoted dgcr is considered in the present work again it is shown that all of the approximations produced by dgcr exist and a gmres like algorithm denoted dgmres for its implementation is derived like gmres dgmres too is economical computationally and storagewise

a unified approach to krylov subspace methods for the drazin inverse solution of singular nonsymmetric linear systems abstract consider the linear system ax b where a c n n is a singular matrix in the present work we propose a general framework within which krylov subspace methods for drazin inverse solution of this system can be derived in a convenient way the krylov subspace methods known to us to date treat only the cases in which a is hermitian and its index ind a is unity necessarily in the present work a is not required to be hermitian it can have any type of spectrum and ind a is arbitrary we show that as is the case with nonsingular systems the krylov subspace methods developed here terminate in a finite number of steps that is at most n ind a for one of the methods derived here we also provide an analysis by which we are able to bound the errors the relevant bounds decreasing with increasing dimension of the krylov subspaces involved the results of this paper are applicable to consistent systems as well as to inconsistent ones an interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard krylov subspace approach to nonsingular systems indeed our approach here reduces to that relevant for nonsingular systems upon setting ind a 0 everywhere

singular sets for curvature equation of order k abstract we consider the removability of singular sets for the curvature equations of the form h k u which is determined by the kth elementary symmetric function in an n dimensional domain we prove that for 1 k n 1 and a compact set k whose n k dimensional hausdorff measure is zero any generalized solution to the curvature equation on k is always extendable to a generalized solution on the whole domain

conformal invariance of domino tiling let u be a multiply connected region in r 2 with smooth boundary let p be a polyomino in z 2 approximating u as 0 we show that for certain boundary conditions on p the height distribution on a random domino tiling dimer covering of p is conformally invariant in the limit as tends to 0 in the sense that the distribution of heights of boundary components or rather the difference of the heights from their mean values only depends on the conformal type of u the mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way the mean height and all the moments are explicitly evaluated

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

the semigroup property of value functions in lagrange problems the lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form the binary operation of inf composition applied to the value functions of a lagrange problem equates the principle of optimality with a semigroup property this paper finds the infinitesimal generator of the semigroup by differentiating at t 0 the type of limit is epigraphical convergence in a uniform sense moreover the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed

viability solutions to structured hamilton jacobi equations under constraints structured hamilton jacobi partial differential equations are hamilton jacobi equations where the time variable is replaced by a vector valued variable structuring the system it could be the time age pair hamilton jacobi mckendrick equations or candidates for initial or terminal conditions hamilton jacobi cournot equations among a manifold of examples here we define the concept of viability solution which always exists and can be computed by viability algorithms this is a constructive approach allowing us to derive from the tools of viability theory dealing with sets instead of functions some known as well as new properties of its solutions regarding functions as their epigraphs we bypass the regularity issues to arrive directly at the concept of barron jensen frankowska viscosity solutions above all we take into account viability constraints and extend classical boundary conditions to other internal conditions beyond that we use the fenchel legendre transform to associate a lagrangian with the hamiltonian and uncover an underlying variational problem it is proved that the viability solution to the structured hamilton jacobi equation is the valuation function of this variational principle the optimal evolutions of which are regulated by a regulation map which is constructed from the viability solution and can be computed by viability algorithms in other words the viability solution solves all at once a class of first order partial differential equations of intertemporal optimization problems and the regulation of their optimal evolutions

on the convergence rate of dgmres abstract the dgmres algorithm was designed to compute the drazin inverse solution of consistent or inconsistent linear systems a x b where a is a square singular matrix with arbitrary index in this paper we compare the convergence rate of dgmres with that of gmres applied to the nonsingular part of a

local limits of lozenge tilings are stable under bounded boundary height perturbations we show that bounded changes to the boundary of a lozenge tilings do not affect the local behaviour inside the domain as a consequence we prove the existence of a local limit in all domains with planar boundary the proof does not rely on any exact solvability of the model beyond its links with uniform spanning trees

continuity of the upper and lower value of slow growth differential games abstract some continuity properties are proved for a zero sum differential game with an unbounded control and no coercivity assumptions

repeated games with lack of information on one side we introduce the dual differential game of a repeated game with lack of information on one side as the natural continuous time version of the dual game introduced by de meyer 1996 a traditional way to study the value of differential games is through discrete time approximations here we follow the opposite approach we identify the limit value of a repeated game in discrete time as the value of a differential game namely we use the recursive structure for the finitely repeated version of the dual game to construct a differential game for which the upper values of the uniform discretization satisfy precisely the same property the value of the dual differential game exists and is the unique viscosity solution of a first order derivative equation which appears implicitely in de meyer and rosenberg 1999 with a limit condition we identify the solution by translating viscosity properties in the primal

jensen s inequality for g convex function under g expectation a real valued function h defined on mathbb r is called g convex if it satisfies the generalized jensen s inequality for a given g expectation i e h mathbb e g x leq mathbb e g h x holds for all random variables x such that both sides of the inequality are meaningful in this paper we will give a necessary and sufficient condition for a c 2 function being g convex and study some more general situations we also study g concave and g affine functions and a relation between g convexity and backward stochastic viability property

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

solutions of vectorial hamilton jacobi equations and minimizers of nonquasiconvex functionals abstract we provide a unified approach to prove existence results for the dirichlet problem for hamilton jacobi equations and for the minimum problem for nonquasiconvex functionals of the calculus of variations with affine boundary data the idea relies on the definition of integro extremal solutions introduced in the study of nonconvex scalar variational problem

chaperoning state evolutions by variable durations mckendrick partial differential equations in demography provide the state of a population depending on chronological time t and age d and involve two time evolutions the chronological one t in 0 t mapsto t geq 0 and the calendar age t in t d t mapsto t t d geq 0 both with constant velocity equal to 1 the calendar age evolution chaperons so to speak the evolution t in t d t mapsto x t of the state of a system some physical biological and economic problems motivate the introduction of variable durations with variable velocities representing the fluidity of time offering mathematical metaphors of a subjective fleeting specious time passing more or less slowly variable durations are no longer prescribed but chosen among those available and regulated the joint evolution of the variable evolution and the state is assumed to be governed by a differential inclusion or control system and provides as a byproduct the unknown temporal windows on which they evolve toge

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

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

risk sensitive control with hara utility a control methodology based on the hyperbolic absolute risk averse hara utility function is presented as an alternative to the exponential of an integral approach to finding robust controllers the work is inspired by the intuition that hara controllers while being robust may give better performance than exponential controllers in normal situations the hara problem is shown to be equivalent to a certain differential game and the asymptotic properties of the hara problem and this differential game are studied as an example a linear quadratic hara problem is studied where the problem of finding a robust hara controller is proved to be equivalent to solving a standard linear quadratic problem for a system with a higher noise intensity this reveals an interesting relationship between robustness and uncertainty

anticipated backward doubly stochastic differential equations in this paper we deal with a new type of differential equations called anticipated backward doubly stochastic differential equations anticipated bdsdes the coefficients of these bdsdes depend on the future value of the solution y z we obtain the existence and uniqueness theorem and a comparison theorem for the solutions of these equations besides as an application we also establish a duality between the anticipated bdsdes and the delayed doubly stochastic differential equations delayed dsdes

remarks on hamilton jacobi equations with measurable time dependent hamiltonians on considere les equations de hamilton jacobi de la forme suivante u t h x t u u 0 dans r n 0 t ou h x t s p est continu en x s p et mesurable en t on considere une extension de la notion de solution viscosite au cas de h mesurable dependant du temps

bulk universality for random lozenge tilings near straight boundaries and for tensor products we prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains the result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in gelfand tsetlin bases of tensor products of representations of unitary groups in a weaker form our theorem also applies to random domino tilings

viability of moving sets for a nonlinear neumann problem abstract we prove that there exists a viscosity solution of a nonlinear neumann problem u i t x t a t u i t x f i t x u t x t x x u i t x 0 0 t t x d i 1 n u i t x n x g i t x u t x 0 t t x b d d i 1 n u t x h x x d which stays under some sufficient conditions in a prescribed set of time space dependent constraints k t x t x 0 t d the result is obtained by a stochastic approach we study the viability of the moving set k t x for a backward stochastic differential equation and we apply the feynman kac representation formula

generation and propagation of interfaces in reaction diffusion systems this paper is concerned with the asymptotic behavior as e0 of the solution u e e of the second initial boundary value problem of the reaction diffusion system u t e e u e 1 e f u e e 1 e u e 1 u e2 e t e e u e e where 0 is a constant when 2 3 9 2 3 9 f is bistable in the sense that the ordinary differential equation u t f u has two stable solutions u h and u h and one unstable solution u h 0 where h h 0 and h are three solutions of the algebraic equation f u 0

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

remarks regarding the gap between continuous lipschitz and differentiable storage functions for dissipation inequalities appearing in h control abstract this paper deals with the regularity of solutions of the hamilton jacobi inequality which arises in h control it shows by explicit counterexamples that there are gaps between existence of continuous and locally lipschitz positive definite and proper solutions and between lipschitz and continuously differentiable ones on the other hand it is shown that it is always possible to smooth out solutions provided that an infinitesimal increase in gain is allowed

asymptotics of random lozenge tilings via gelfand tsetlin schemes a gelfand tsetlin scheme of depth n is a triangular array with m integers at level m m 1 ldots n subject to certain interlacing constraints we study the ensemble of uniformly random gelfand tsetlin schemes with arbitrary fixed n th row we obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble and also of its q deformation this provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice or equivalently of random stepped surfaces we work with a class of polygons which allows arbitrarily large number of sides we show that the local limit behavior of random tilings as all dimensions of the polygon grow is directed by ergodic translation invariant gibbs measures the slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by kenyon and okounkov acta math 199 2 263 302 2007 we also prove that at the edge of the limit shape the asymptotic behavior of random tilings is given by the airy process in particular our results cover the most investigated case of random boxed plane partitions when the polygon is a hexagon

an asymptotic analysis of hierarchical control of manufacturing systems under uncertainty this paper presents an asymptotic analysis of a hierarchical manufacturing system with machines subject to breakdown and repair the rate of change in machine states is much larger than the rate of fluctuation in demand and the rate of discounting of costs and this gives rise to a limiting problem in which the stochastic machine availability is replaced by the equilibrium mean availability the value function for the original problem converges to the value function of the limiting problem moreover the control for the original problem can be constructed from the optimal controls of the limiting problem in a way which guarantees asymptotic optimality of the value function the limiting problem is computationally more tractable and sometimes has a closed form solution

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

limit shapes and local statistics for the stochastic six vertex model in this paper we consider the stochastic six vertex model on a cylinder with arbitrary initial data first we show that it exhibits a limit shape in the thermodynamic limit whose density profile is given by the entropy solution to an explicit non linear conservation law that was predicted by gwa spohn phys rev lett 68 725 728 1992 and by reshetikhin sridhar commun math phys 363 741 765 2018 then we show that the local statistics of this model around any continuity point of its limit shape are given by an infinite volume translation invariant gibbs measure of the appropriate slope

existence of solutions to the hamilton jacobi bellman equation under quadratic growth conditions abstract in this paper we show the existence of solutions with quadratic growth to hamilton jacobi bellman equations we assume that the hamiltonian has the quadratic growth both in x and p we obtain the explicit lipschitz bound of solutions in the weighted sup norm also we discuss the vanishing viscosity limit of solutions and show the existence of viscosity solutions to the resulting hamilton jacobi equation

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

layered viscosity solutions of nonautonomous hamilton jacobi equations semiconvexity and relations to characteristics abstract we construct an explicit representation of viscosity solutions of the cauchy problem for the hamilton jacobi equation h on a given domain 0 t r n it is known that if the hamiltonian h h t p is not a convex or concave function in p or h p may change its sign on 0 t then the hopf type formula does not define a viscosity solution on under some assumptions for h t p on the subdomains t i t i 1 r n we are able to arrange partial solutions given by the hopf type formula to get a viscosity solution on then we study the semiconvexity of the solution as well as its relations to characteristics

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

blow up time involved with perturbed hamilton jacobi equations in this paper we consider the evolution of positive bounded uniformly continuous data u 0 by perturbed equations like under general assumptions on u 0 existence uniqueness and regularity of the evolution u in the set are studied where the blow up function is given by the exact blow up rate of u is obtained uniqueness regularity decay at infinity of the function as well a s a representation formula for the case m 1 are also proved

generalized viscosity solutions for hamilton jacobi equations with time measurable hamiltonians the objective of this paper is to extend the idea of viscosity solution for nonlinear first order hamilton jacobi equations u h t x d u 0 with time continuous h to time measurable hamiltonians we are primarily motivated by the fact that control problems for which the value function is the viscosity solution of a hamilton jacobi equation should not be restric ted to continuity in time in barron and jensen 1 we were confronted with a linear hamilton jacobi equation arising in the proof of the pon tryagin which had time measurable coeflicients the existing theory of viscosity solutions did not apply to even this case moreover important models of controlled first order hamiltonjacobi equations require time measurable controls for example models of controlled traffic flow or queueing processes might be of this type we are about then to extend the definition of viscosity solution to generalized viscosity solution applying to time measurable hamiltonians in this paper we give the definition and derive the corresponding uni queness result with an implicit domain of dependence consequent the fun damental technique used in the proof is the so called blow up method which has also been used elsewhere s although not in the generality discussed here this method effectively freezes the point to x0 under consideration and essentially removes it from the problem of course t is

dissipative control system for the stochastic nonlinear h problems abstract we characterize functions satisfying a dissipative inequality associated with a stochastic control problem such a characterization is provided in terms of an upper generalized gaussian solution or a viscosity supersolution to a partial differential equation called hamilton jacobi equation h j links between upper generalized gaussian solutions and viscosity supersolutions to hamilton jacobi equation are studied finally it shows that generalized gaussian solutions is identical to viscosity solutions to hamilton jacobi equation

scaling limits of random skew plane partitions with arbitrarily sloped back walls the paper studies scaling limits of random skew plane partitions confined to a box when the inner shapes converge uniformly to a piecewise linear function v of arbitrary slopes in 1 1 it is shown that the correlation kernels in the bulk are given by the incomplete beta kernel as expected as a consequence it is established that the local correlation functions in the scaling limit do not depend on the particular sequence of discrete inner shapes that converge to v a detailed analysis of the correlation kernels at the top of the limit shape and of the frozen boundary is given it is shown that depending on the slope of the linear section of the back wall the system exhibits behavior observed in either okounkov and reshetikhin commun math phys 269 3 571 609 2007 or boutillier et al http arxiv org abs 0912 3968v2 math ph 2009

isolated singularities for some types of curvature equations abstract we consider the removability of isolated singularities for the curvature equations of the form h k u 0 which is determined by the k th elementary symmetric function in an n dimensional domain we prove that for 1 k n 1 isolated singularities of any viscosity solutions to the curvature equations are always removable provided the solution can be extended continuously at the singularities we also consider the class of generalized solutions and prove the removability of isolated singularities

l p solutions of multidimensional bsdes with weak monotonicity and general growth generators abstract in this paper we first establish the existence and uniqueness of l p p 1 solutions for multidimensional backward stochastic differential equations bsdes under a weak monotonicity condition together with a general growth condition in y for the generator g then we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way finally we put forward and prove a stability theorem and a comparison theorem of l p p 1 solutions for this kind of bsdes

global structure and regularity of solutions to the eikonal equation the evolutionary eikonal equation is a hamilton jacobi equation with hamiltonian h p p which is not strictly convex nor smooth the regularizing effect of hamiltonian for the eikonal equation is much weaker than that of strictly convex hamiltonians therefore leading to new phenomena in this paper we study the set of singularity points of solutions in the upper half space for c1 or c2 initial data with emphasis on the countability of connected components of the set the regularity of solutions in the complement of the set of singularity points is also obtained

quasi variational inequalities and first order hamilton jacobi equations on considere des inegalites quasi variationnelles de hamilton jacobi dans r n on introduit pour ce probleme la notion de solution viscosite

new methods for computing the drazin inverse solution of singular linear systems the dgmres method is an iterative method for computing the drazin inverse solution of consistent or inconsistent linear systems of the form a x b where a c n n is a singular and in general non hermitian matrix that has an arbitrary index this method is generally used with restarting but the restarting often slows down the convergence and dgmres often stagnates based on the lgmres and gmres e methods we present two new techniques for accelerating the convergence of restarted dgmres by adding some approximate error vectors or approximate eigenvectors corresponding to a few of the smallest eigenvalues to the krylov subspace we derive the implementation of these methods and present some numerical examples to show the advantages of these methods

persistence and stability of solutions of hamilton jacobi equations abstract given a convergent sequence of hamiltonians h n and a convergent sequence of initial data g n for the first order evolutionary hamilton jacobi equation we look for conditions ensuring that the sequences u n and v n of lax solutions and hopf solutions respectively converge the convergences we deal with are variational convergences we take advantage of several recent results giving criteria for the continuity of usual operations

asymptotic behavior of the first order obstacle problem on considere l equation aux derivees partielles d ordre 1 non lineaire max u x x u x g x du x f x 0 x ou est un sous ensemble ouvert de r d est un parametre positif g g 1 g d est un champ vectoriel sur r d du est le gradient de la fonction inconnue u et f et sont des fonctions a valeurs reelles donnees on etudie le comportement de cette equation quand 0

value functions for differential games and control systems with discontinuous terminal cost this paper deals with mayer s problem for control systems and differential games with discontinuous terminal cost there are two main results in the paper the first one says that the value function for control systems can be characterized as the unique solution in suitable sense to the hamilton jacobi bellman equation without any regularity assumptions on the terminal cost for differential games satisfying isaacs s minmax condition the second main result says that the value function is the unique solution to the hamilton jacobi isaacs equation when the terminal cost is semicontinuous this allows to prove the existence of the value under isaacs s condition this paper extends some results already well known in the continuous case

on existence and uniqueness of solutions of hamilton jacobi equations abstract the theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years this development occurred because the authors established uniqueness criteria for generalized solutions called viscosity solutions which correctly identify the solutions sought in areas of application including control theory differential games and the calculus of variations the concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept a feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results the current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs it is also felt that the presentation of the comparison results which imply uniqueness continuous dependence and are used to estimate moduli of continuity has something to offer over earlier presentations in special cases author

a semi iterative method for real spectrum singular linear systems with an arbitrary index in this paper we develop a semi iterative method for computing the drazin inverse solution of a singular linear system ax b where the spectrum of a is real but its index i e the size of its largest jordan block corresponding to the eigenvalue zero is arbitrary the method employs a set of polynomials that satisfy certain normalization conditions and minimize some well defined least squares norm we develop an efficient recursive algorithm for implementing this method that has a fixed length independent of the index of a following that we give a complete theory of convergence in which we provide rates of convergence as well we conclude with a numerical application to determine eigenprojections onto generalized eigenspaces our treatment extends the work of hanke and hochbruck 1993 that considers the case in which the index of a is 1 keyword singular systems iterative methods polynomial acceleration

cauchy dirichlet problem for first order nonlinear systems abstract let r nbe open u r mand thus the gradient matrixdu r m n we lete r m nbe compact and denote byrcoeandpcoethe rank one convex and polyconvex hull ofe respectively we show that ifrcoe pcoe and two other hypotheses named the segment property and the extreme points property and if c1 r m is such that d x e int pcoe x then there exists a dense set of u w1 r m such that formula we apply this existence theorem to some relevant examples studied in the literature as well as to problems with x u dependence

generalized one sided estimates for solutions of hamilton jacobi equations and applications on etudie les proprietes locales des solutions de l equation de hamilton jacobi u t h t x u 0 et plus generalement de l equation u t h t x u u 0 ou t o t et x r n

weak kam from a pde point of view viscosity solutions of the hamilton jacobi equation and aubry set we introduce the notion of a viscosity solution for the first order hamilton jacobi equation in the more general setting of manifolds to obtain a weak kam theory using only tools from partial differential equations this work should be accessible to people with no prior knowledge of the subject

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

an adaptive method with rigorous error control for the hamilton jacobi equations part ii the two dimensional steady state case in this paper we devise and study an adaptive method for finding approximations to the viscosity solution of hamilton jacobi equations the method which is an extension to two space dimensions of a similar method previously proposed for one space dimension is studied in the framework of steady state hamilton jacobi equations with periodic boundary conditions it seeks numerical approximations whose l distance to the viscosity solution is no bigger than a prescribed tolerance a thorough numerical study is carried out which shows that a strict error control is achieved and that the method exhibits an optimal computational complexity which does not depend on the value of the tolerance or on the type of hamiltonian

viability property for a backward stochastic differential equation and applications to partial differential equations in the present paper we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints this property is the so called viability property in a separate section this condition is translated to a class of partial differential equations

adjoint and compensated compactness methods for hamilton jacobi pde we investigate the vanishing viscosity limit for hamilton jacobi pde with nonconvex hamiltonians and present a new method to augment the standard viscosity solution approach the main idea is to introduce a solution e of the adjoint of the formal linearization and then to integrate by parts with respect to the density e this procedure leads to a natural phase space kinetic formulation and also to a new compensated compactness technique