on the rate of convergence of solutions in singular perturbation problems where the obstacle cp is a given smooth function satisfying that rp 0 on dq using a technique in the theory of variational inequalities j l lions 11 proved that 11 min o p i l cnj ce for some constant c we claim that we can never deduce the pointwise convergence of a from the above estimate on the other hand the notion of viscosity solotions was introduced by m g crandall and p l lions 2 for nonlinear first order pdes and then it was extended for nonlinear second order pdes by p l lions 12 141 recently it appeared that a technique in the theory of viscosity solutions could be applied to derive a pointwise convergence for some singular perturbation problems in fact by the same argument as in 7 we can show that u converges to min 0 cp uniformly on 0 243 0022 247x 91 3 00

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

wavefront propagation for reaction diffusion systems of pde m i freidlin has introduced probabilistic techniques to study propagation for systems of reaction diffusion pde the motivating idea is that should a reaction diffusion system possess only a single unstable and a single stable equilibrium then the solution u of the system will presumably tend to switch for large times from near the former to near the latter state this paper brings to bear purely pde techniques to this problem especially the theory of viscosity solutions on hamilton jacobi equations due to crandall lions

the hele shaw asymptotics for mechanical models of tumor growth models of tumor growth now commonly used present several levels of complexity both in terms of the biomedical ingredients and the mathematical description our first goal here is to formulate a free boundary model of hele shaw type a variant including growth terms starting from the description at the cell level and passing to the stiff limit in the pressure law of state in contrast with the classical hele shaw problem here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics a complete description requires the equation on the cell number density we then go on to consider a more complex model including the supply of nutrients through vasculature and we study the stiff limit for the involved coupled system

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

viscosity solutions for weakly coupled systems of first order partial differential equations abstract we consider viscosity solutions for weakly coupled systems of first order partial differential equations and apply these existence and uniqueness results to examples involving random evolutions

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

the behavior of the free boundary for reaction diffusion equations with convection in an exterior domain with neumann or dirichlet boundary condition let l a r d2dr2 b r ddr be a second order elliptic operator and consider the reaction diffusion equation with neumann boundary condition lu upforr r u r h u r h u 0is minimal where p 0 1 p 0 1 r 0r 0 h 0h 0 and r 0 r 0 this equation is the radially symmetric case of an equation of the form lu upinrd d u n hon d u 0is minimal where l i j 1dai j 2 xi xj i 1dbi xi is a second order elliptic operator and where d 2d 2 h 0h 0 is continuous d rdd rd is bounded and n is the unit inward normal to the domain rd d consider also the same equations with the neumann boundary condition replaced by the dirichlet boundary condition namely u r hu r h in the radial case and u hu h on d in the general case the solutions to the above equations may possess a free boundary in the radially symmetric case if r h inf r r u r 0 r u r 0 we call this the radius of the free boundary otherwise there is no free boundary we normalize the diffusion coefficient a to be on unit order consider the convection vector field b to be on order rmrm m rm r pointing either inward or outward and consider the reaction coefficient to be on order r jr j j rj r for both the neumann boundary case and the dirichlet boundary case we show for which choices of m and j a free boundary exists and when it exists we obtain its growth rate in h as a function of m and j these results are then used to study the free boundary in the non radially symmetric case

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

on hopf s formulas for solutions of hamilton jacobi equations soit h r n r on considere l equation de hamilton jacobi u t h du 0 dans r n 1 r n 0 on etudie les proprietes des formules de hopf pour les solutions

a viscosity solution approach to the asymptotic analysis of queueing systems we consider a system of several interconnected queues with a single class of customers and model the state xt of the system as a jump markov process the problem of interest is to estimate the large deviations behavior of the rescaled system xt ext s corresponding to large time and large excursions of the original unscaled system the techniques employed are those of the theory of viscosity solutions to hamilton jacobi equations from the point of view of large deviation theory the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty since an abrupt change in the jump measure occurs from the point of view of viscosity solutions the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations much of the paper is devoted to proving uniqueness of viscosity solutions for these equations and these sections are of independent interest while our use of test functions in proving the uniqueness is an adaptation of the usual technique the construction of the test functions themselves via the legendre transform is new we obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem which suggests the correct integrand in the large deviation rate functional since it is the treatment of the effects due to the boundaries that is novel we devote the majority of the paper to the detailed development of a simple two dimensional system that exhibits all the essential new features however the arguments may be applied to nueueing systems that are considerably more general and we attempt to indicate this generality as well

singular limit of the porous medium equation with a drift abstract we study the stiff pressure limit of a nonlinear drift diffusion equation where the density is constrained to stay below the maximal value one the challenge lies in the presence of a drift and the consequent lack of monotonicity in time in the limit a hele shaw type free boundary problem emerges which describes the evolution of the congested zone where density equals one we discuss pointwise convergence of the densities as well as the bv regularity of the limiting free boundary

a stochastic control approach to the pricing of options we construct a stochastic control model of a portfolio in which the investor may invest in stock call options on the stock or risk free bonds he may also borrow funds we assume that the investor is attempting to maximize expected utility of total wealth at expiration of the options where the utility function is given as wealth to the power 0 1 this includes both the risk averse and risk neutral cases we allow the borrowing interest rate r to be distinct from the lending interest rate r we make two fundamental assumptions i the investor cannot achieve infinite expected utility of wealth in finite time stable markets and ii the market will set the option price to minimize the investor s maximal expected utility of wealth i e options are priced using a minimax strategy the market is stable only when r r we prove in the case r r and 1 that the minimax european call option price ft s is given as the solution of the black scholes equation ft i i 2s2fss i s fs i f 0 where i r when r i 2 1 i r when i 2 1 r and i i 2 1 if r i 2 1 r the parameter is the stock return drift and i is the stock return diffusion if r r then the call option price must by stable markets always be the black scholes price with rate r in the risk neutral 1 case we must have r r for a stable market and the option price given by the black scholes equation but with i explicit solutions are presented and the optimal portfolios and returns are also derived comparable results are obtained for american call options which can be exercised at any time up to and including the expiration date in this case the option pricing function is shown to be the solution of an obstacle problem

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

homogenization of the hele shaw problem in periodic spatiotemporal media we consider the homogenization of the hele shaw problem in periodic media that are inhomogeneous both in space and time after extending the theory of viscosity solutions into this context we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous hele shaw type problem with a general possibly nonlinear dependence of the free boundary velocity on the gradient moreover the free boundaries converge locally uniformly in hausdorff distance

the dead core for reaction diffusion equations with convection and its connection with the first exit time of the related markov diffusion process on considere un probleme de reaction diffusion avec convection dans une region bornee d r n on demontre l existence d une solution

holder gradient estimates for fully nonlinear elliptic equations in this paper we prove interior and global holder estimates for lipschitz viscosity solutions of second order nonlinear uniformly elliptic equations the smoothness hypotheses on the operators are more general than previously considered for classical solutions so that our estimates are also new in this case and readily extend to embrace obstacle problems in particular isaac s equations of stochastic differential game theory constitute a special case of our results and moreover our techniques in combination with recent existence theorems of ishii lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic isaac s equation

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

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

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

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

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

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

a remark on a system of inequalities with bilateral obstacles up x k s up x i up l x k x e sz apup fp if up x k up x up x k x e q apup 5 fp if up x up l x k x e l2 1 1 apup l f if up x up x k x e q u x 0 onaq p 1 2 m where p ranges over 1 m and urn represents u is dealt with by the second author in 5 and the existence of a weak solution of 1 1 is established under the assumption ap s have sufficiently large zeroth order coefficients 1 2 the proof of the existence result in 5 is based on the iv estimate independent of e for the solution u 124 u of the following approximate system apu u p uf l k u k uf f in r 1 3 u 0 0naq p 1 2 m where p ranges over 1 ml p is the so called penalty function and uf represents u the gradient estimate for solutions u of 1 3 in 5 is obtained by using bernstein s method and assumption 1 2 is solely employed to make it possible in addition it should be remarked that standard variants of the bernstein trick like in 2 and 4 which are useful to obtain the iv estimate for solutions of the system of variational inequalities with unilateral constraints without 1 2 do not work in our case

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

representation of solutions of hamilton jacobi equations on etudie le probleme de cauchy pour l equation de hamilton jacobi u t h t x du 0 dans o t r n avec la condition initiale u o x u o x pour x r n on a t 0 hi o t r n r n r et u o r n r sont donnes on represente la solution viscosite comme la fonction valeur d un jeu differentiel approprie

a regularity result for viscosity solutions of hamilton jacobi equations in one space dimension on demontre que dans le cas particulier d un probleme stationnaire a une dimension sous des hypotheses tres generales les solutions viscosite continues de lipschitz ont des derivees a droite et a gauche en chaque point

incompressible limit of the navier stokes model with a growth term starting from isentropic compressible navier stokes equations with growth term in the continuity equation we rigorously justify that performing an incompressible limit one arrives to the two phase free boundary fluid system

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

on the complexity of real root isolation using continued fractions we present algorithmic complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers one motivation is to explain the method s good performance in practice we derive an expected complexity bound of email protected b d 6 d email protected 2 where d is the polynomial degree and t bounds the coefficient bit size using a standard bound on the expected bit size of the integers in the continued fraction expansion thus matching the current worst case complexity bound for real root isolation by exact methods sturm descartes and bernstein subdivision moreover using a homothetic transformation we improve the expected complexity bound to email protected b d email protected we compute the multiplicities within the same complexity and extend the algorithm to non square free polynomials finally we present an open source c implementation in the algebraic library synaps and illustrate its completeness and efficiency as compared to some other available software for this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000

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

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

existence of viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games recently m g crandall and p l lions trans amer math soc 277 1983 1 42 introduced the class of viscosity solutions of these equations and proved uniqueness within this class this paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness

hele shaw limit for a system of two reaction cross diffusion equations for living tissues multiphase mechanical models are now commonly used to describe living tissues including tumour growth the specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation including cross reaction terms we study the incompressible limit when the pressure becomes stiff which generates a free boundary problem we establish the complementarity relation and also a segregation result several major mathematical difficulties arise in the two species case firstly the system structure makes comparison principles fail secondly segregation and internal layers limit the regularity available on some quantities to bv thirdly the aronson benilan estimates cannot be established in our context we are lead as it is classical to add correction terms this procedure requires technical manipulations based on bv estimates only valid in one space dimension another novelty is to establish an l1 version in place of the standard upper bound

the dynamic programming equation for the time optimal control problem in infinite dimensions the existence and uniqueness of a viscosity solution for the bellman equation associated with the time optimal control problem for a semilinear evolution equation in hilbert space is provided applications to time optimal control problems governed by parabolic equations are given

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

uniqueness and existence results on the hele shaw and the stefan problems in this paper we introduce a notion of viscosity solutions for the one phase hele shaw and stefan problems when there is no surface tension we prove the uniqueness and existence of the solutions for both problems and the uniform convergence of solutions of porous medium equation to those of the hele shaw problem

complexity of real root isolation using continued fractions in this paper we provide polynomial bounds on the worst case bit complexity of two formulations of the continued fraction algorithm in particular for a square free integer polynomial of degree n with coefficients of bit length l we show that the bit complexity of akritas formulation is o n 8l 3 and the bit complexity of a formulation by akritas and strzebonski is o n 7l 2 here o indicates that we are omitting logarithmic factors the analyses use a bound by hong to compute the floor of the smallest positive root of a polynomial which is a crucial step in the continued fraction algorithm we also propose a modification of the latter formulation that achieves a bit complexity of o n 5l 2

an asymptotic formula for solutions of hamilton jacobi bellman equations on demontre que la solution de l equation de hamilton jacobi bellman converge exponentiellement vers 0 quand e 0 avec un taux de decroissance i x 0 1 e et on donne une formule de representation pour i x

improved bounds for the cf algorithm we consider the problem of isolating the real roots of a square free polynomial with integer coefficients using the classic variant of the continued fraction algorithm cf introduced by akritas we compute a lower bound on the positive real roots of univariate polynomials using an exponential search this allows us to derive a worst case bound of email protected b d email protected 2 for isolating the real roots of a polynomial with integer coefficients using the classic variant of cf where d is the degree of the polynomial and t the maximum bitsize of its coefficients this improves the previous bound of sharma by a factor of d 3 and matches the bound derived by mehlhorn and ray for another variant of cf which is combined with subdivision it also matches the worst case bound of the classical subdivision based solvers sturm descartes and bernstein

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

the necessary conditions for optimal control in hilbert spaces obtention du principe maximum de pontryagin pour une classe generale de problemes de commande de type lagragien afin qu associe avec un ensemble de conditions necessaires une commande soit satisfaite le principe est demontre en utilisant la programmation dynamique et l equation resultante d hamilton jacobi bellman de dimension finie

the pontryagin maximum principle from dynamic programming and viscosity solutions to first order partial differential equations on demontre le principe de maximum de pontryagin pour le probleme de lagrange de commande optimale en utilisant le fait que la fonction valeur du probleme est la solution viscosite de l equation de hamilton jacobi bellman associee

perturbed dynamical systems with an attracting singularity and weak viscosity limits in hamilton jacobi equations nous donnons une nouvelle preuve par des methodes edp du theoreme de wentzell freidlin concernant les petites perturbations d un systeme dynamique l e u e e 2 u e b u e 0 dans u e sur nous prouvons que si b a un seul point singulier attractif alors u e converge vers une constante uniformement sur tout compact et avec une vitesse exponentielle nous determinons nous traitons aussi le cas de conditions aux limites de neumann pour cela nous faisons l analyse asymptotique d une mesure ergodique intervenant naturellement dans le probleme ce qui revient a etudier la limite par viscosite evanescente dans une equation de hamilton jacobi

viscosity solutions for weakly coupled systems of hamilton jacobi equations we obtain comparison results for viscosity solutions of weakly coupled systems of hamilton jacobi equations perron s method is extended to give existence proofs for systems a stronger definition of sub and supersolutions is introduced leading to existence results for continuous solutions the needed sub and supersolutions are constructed for continuous boundary data crandall s and lions notion of viscosity solutions for first order hamilton jacobi equations can be extended in a componentwise way to systems a key tool in obtaining uniqueness comparison and existence results is the maximum principle we obtain such results for weakly coupled systems of hamilton jacobi equations in which the coupling has such a structure to allow the use of maximum principle techniques to motivate the coupling assumptions consider this linear system n

nonlinear elliptic parabolic problems we introduce a notion of viscosity solutions for a general class of elliptic parabolic phase transition problems these include the richards equation which is a classical model in filtration theory existence and uniqueness results are proved via the comparison principle in particular we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data these results are new even in the linear case where we also show that viscosity solutions coincide with the regular weak solutions introduced in alt and luckhaus math z 183 311 341 1983

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

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

efficient isolation of polynomial s real roots this paper revisits an algorithm isolating the real roots of a univariate polynomial using descartes rule of signs it follows work of vincent uspensky collins and akritas johnson krandick our first contribution is a generic algorithm which enables one to describe all the known algorithms based on descartes rule of sign and the bisection strategy in a unified framework using that framework a new algorithm is presented which is optimal in terms of memory usage and as fast as both collins and akritas algorithm and krandick s variant independently of the input polynomial from this new algorithm we derive an adaptive semi numerical version using multi precision interval arithmetic we finally show that these critical optimizations have important consequences since our new algorithm still works with huge polynomials including orthogonal polynomials of degree 1000 and more which were out of reach of previous methods

a hele shaw problem for tumor growth abstract we consider weak solutions to a problem modeling tumor growth under certain conditions on the initial data solutions can be obtained by passing to the stiff incompressible limit in a porous medium type problem with a lotka volterra source term describing the evolution of the number density of cancerous cells we prove that such limit solutions solve a free boundary problem of hele shaw type we also obtain regularity properties both for the solution and for its free boundary the main new difficulty arises from the competition between the growth due to the source which keeps the initial singularities and the free boundary which invades the domain with a regularizing effect new islands can be generated at singular times