convergent finite difference solvers for viscosity solutions of the elliptic monge ampere equation in dimensions two and higher the elliptic monge ampere equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail novel solution methods are required for stability and convergence to the weak viscosity solution in this article we build a wide stencil finite difference discretization for the monge ampere equation the scheme is monotone so the barles souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation solutions of the scheme are found using a damped newton s method we prove convergence of newton s method and provide a systematic method to determine a starting point for the newton iteration computational results are presented in two and three dimensions which demonstrates the speed and accuracy of the method on a number of exact solutions which range in regularity from smooth to nondifferentiable

to the theory of viscosity solutions for uniformly elliptic isaacs equations we show how a theorem about the solvability in c 1 1 c 1 1 of special isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate isaacs equations we apply it also to establish the c 1 c 1 regularity of viscosity solutions and show that finite difference approximations have an algebraic rate of convergence the main coefficients of the isaacs equations are supposed to be in c c with slightly less than 1 2

pointwise rates of convergence for the oliker prussner method for the monge ampere equation we study the oliker prussner method exploiting its geometric nature we derive discrete stability and continuous dependence estimates in the max norm by using a discrete alexandroff estimate and the brunn minkowski inequality we show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain nodes still conform to the domain boundary this gives a suitable notion of operator consistency which combined with stability leads to pointwise rates of convergence for classical and non classical solutions of the monge ampere equation

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

the fujita exponent for the cauchy problem in the hyperbolic space abstract it is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the euclidean space the main purpose of this paper is to study its effect on the positive solutions of cauchy problems with power nonlinearities existence and non existence results for local solutions are derived emphasis is put on their long time behavior and on fujita s phenomenon to have the same situation as for the cauchy problem in r n namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent we must introduce a weight depending exponentially on the time in this respect the situation is similar to problems in bounded domains with dirichlet boundary conditions important tools are estimates for the heat kernel in the hyperbolic space and comparison principles

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

mean value theorems for stochastic integrals the distributions of stochastic integrals are approximated by the distributions of stochastic integrals of piece wise constant processes the rate of approximation in some negative sobolev spaces is estimated generalizations are given for problems arising in control theory

handling convexity like constraints in variational problems we provide a general framework to construct finite dimensional approximations of the space of convex functions which also applies to the space of c convex functions and to the space of support functions of convex bodies we give precise estimates of the distance between the approximation space and the admissible set this framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite dimensional spaces such as tensor product splines we show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints our implementation relies on proximal algorithms and can be easily parallelized thus making it applicable to large scale problems in dimension two and three we illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation three dimensional denoising the principal agent problem and optimizat

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

morrey spaces and classification of global solutions for a supercritical semilinear heat equation in rn abstract we prove the boundedness of global classical solutions for the semilinear heat equation u t u u p 1 u in the whole space r n with n 3 and supercritical power p n 2 n 2 this is proved without any radial symmetry or sign assumptions unlike in all the previously known results for the cauchy problem and under spatial decay assumptions on the initial data that are essentially optimal in view of the known counter examples moreover we show that any global classical solution has to decay in time faster than t 1 p 1 which is also optimal and in contrast with the subcritical case the proof relies on nontrivial modifications of techniques developed by chou du and zheng 8 and by blatt and struwe 3 5 for the case of convex bounded domains they are based on weighted energy estimates of giga kohn type combined with an analysis of the equation in a suitable morrey space this method actually works for any convex bounded or unbounded smooth domain but at the same time captures some of the specific behaviors associated with the case of the whole space r n as a consequence we also prove that the set of initial data producing global solutions is open in suitable topologies and we show that the so called borderline global weak solutions blow up in finite time and then become classical again and decay as t all these results confirm the key role played by the morrey space m 2 4 p 1 in the understanding of the structure of the set of global solutions for p p s

on the numerical solution of the equation frac partial 2 z partial x 2 frac partial 2 z partial y 2 left frac partial 2 z partial x partial y right 2 f and its discretizations i the equation indicated in the title is the simplest representative of the class of nonlinear equations of monge ampere type equations with such nonlinearities arise in dynamic meteorology geometric optics elasticity and differential geometry in some special cases heuristic procedures for numerical solution are available but in order for them to be successful a good initial guess is required for a bounded convex domain nonnegativef and dirichlet data we consider a special discretization of the equation based on its geometric interpretation for the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence no special information about an initial guess is required and to initiate the iterates a routine step is made the method is self correcting and is structurally suitable for a parallel computer the computer program modules and several examples are presented in two appendices

radial fast diffusion on the hyperbolic space we consider positive radial solutions to the fast diffusion equation ut u on the hyperbolic space h for n 2 m ms 1 ms n 2 n 2 by radial we mean solutions depending only on the geodesic distance r from a given point o h we investigate their fine asymptotics near the extinction time t in terms of a separable solution of the form v r t 1 t t 1 1 m v r where v is the unique positive energy solution radial w r t o to v c v 1 m for a suitable c 0 a semilinear elliptic problem thoroughly studied in 29 7 we show that u converges to v in relative error in the sense that u t v t 1 0 as t t in particular the solution is bounded above and below near the extinction time t by multiples of 1 t t 1 1 m e n 1 r m solutions are smooth and bounds on derivatives are given as well in particular sharp convergence results as t t are shown for spatial derivatives again in the form of convergence in relative error

approximation of viscosity solutions of elliptic partial differential equations on minimal grids we obtain estimates on the minimal size of an equally spaced grid which can be used to construct monotone and consistent approximations to elliptic problems the estimates are given in terms of the ellipticity of the equation this problem is related to diophantine approximations see the appendix written by w m schmidt

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

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 note on semilinear heat equation in hyperbolic space abstract bandle et al 1 obtained a quite interesting result about a semilinear heat equation that the fujita exponent relative to the whole hyperbolic space is just the same as that relative to bounded domain in euclidean space and in addition the properties of solutions are different in the critical exponent case our purpose is to answer an open problem proposed by bandle et al for the critical exponent case and it together with the one obtained by them shows that the critical exponent case does belong to the non blow up case which is completely different from the case in euclidean space

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

blow up of solutions to semilinear parabolic equations on riemannian manifolds with negative sectional curvature abstract on riemannian manifolds with negative sectional curvature we study finite time blow up and global existence of solutions to semilinear parabolic equations where the power nonlinearity is multiplied by a time dependent positive function h t we show that depending on the behavior at infinity of h either every solution blows up in finite time or a global solution exists if the initial datum is small enough in particular if h 1 we have global existence for small initial data whereas for h t e t a fujita type phenomenon appears for certain values of 0 a key role will be played by the infimum of the l 2 spectrum of the operator on m

an intrinsic metric approach to uniqueness of the positive cauchy neumann problem for parabolic equations abstract we study uniqueness of nonnegative solutions of the cauchy neumann problem for parabolic equations in unbounded domains and give a sufficient condition for the uniqueness of nonnegative solutions to hold we also give a parabolic harnack inequality with neumann boundary conditions

harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations abstract we present a new harnack inequality for non negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations on rectangular lattices this estimate applies to all supersolutions and has the harnack constant depending on the graph distance on lattices for the proof we modify the proof of the weak harnack inequality applying the same idea to elliptic equations in a euclidean space we also derive a harnack type inequality for non negative viscosity supersolutions

robust control design for linear systems with uncertain parameters this paper presents an approach for the control of linear systems with parameter uncertainty the parameter uncertainty under consideration is assumed to be unknown but bounded and can exist in both state and input matrices using the linear quadratic optimal control formulation we consider a cost functional which is parameterized by uncertainties an upper bound on the cost which is incurred by state feedback control and system uncertainties is obtained and a linear state feedback law is found to minimize the upper bound furthermore under the conditions presented in this paper the steady state bound minimizing control gain is shown to exist and the stability of the resultant closed loop systems is guaranteed for all admissible uncertainties finally a gradient search algorithm is proposed to find the optimal values of the free parameters which affect the cost upper bound a numerical example is included to demonstrate the effectiveness of the proposed method

blow up and global existence for the porous medium equation with reaction on a class of cartan hadamard manifolds abstract we consider the porous medium equation with power type reaction terms u p on negatively curved riemannian manifolds and solutions corresponding to bounded nonnegative and compactly supported data if p m small data give rise to global in time solutions while solutions associated to large data blow up in finite time if p m large data blow up at worst in infinite time and under the stronger restriction p 1 1 m 2 all data give rise to solutions existing globally in time whereas solutions corresponding to large data blow up in infinite time the results are in several aspects significantly different from the euclidean ones as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation

boundedness of global solutions of a supercritical parabolic equation we study radially symmetric classical solutions of the dirichlet problem for a heat equation with a supercritical nonlinear source we give a sufficient condition under which blow up in infinite time cannot occur this condition involves only the growth rate of the source term at infinity we do not need the homogeneity property which played a key role in previous proofs of similar results we also establish the blow up rate for a class of solutions which blow up in finite time

estimation and control of discrete time stochastic systems having cone bounded non linearities this paper addresses the problems of suboptimal estimation and control of discrete time stochastic systems with cone bounded non linearities and state dependent noise upper bounds for the estimation error covariance as well as the quadratic control cost are derived the behaviour of these upper bounds over the infinite time interval is then considered and sufficient conditions for their stability are given finally sufficient conditions for the stability of the closed loop stochastic control system are obtained these results provide verifiable conditions for the performance evaluation of simple estimators and controllers for this class of cone bounded non linear stochastic systems

a nonconforming morley finite element method for the fully nonlinear monge ampere equation in this paper we study finite element approximations of the viscosity solution of the fully nonlinear monge ampere equation det d 2 u f 0 using the well known nonconforming morley element our approach is based on the vanishing moment method which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and feng j sci comput 38 1 74 98 2009 the vanishing moment method approximates the monge ampere equation by the fourth order quasilinear equation epsilon delta 2u epsilon rm det d 2u epsilon f with appropriate boundary conditions we develop a finite element scheme using the n dimensional morley element introduced in wang and xu numer math 103 155 169 2006 to approximate the regularized fourth order problem in two and three dimensions and then derive optimal order error estimates

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

robust stability and performance of systems with structured and bounded uncertainties an extension of the guaranteed cost control approach a method is presented for designing a full state feedback linear control law that will ensure the robust stability and performance of a given linear uncertain system the systems under consideration are described by state equations that depend on uncertain parameters these uncertain parameters may be time varying their values are constrained to lie within known compact bounding sets the method is based on the guaranteed cost control concept of chang and peng 1972 the controller gains result from the solution of a riccati equation in which the weighting matrices depend on the uncertainty bounds sufficient conditions for the existence of a solution arise from the standard lqg control theory

convergent finite difference methods for one dimensional fully nonlinear second order partial differential equations this paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations pdes in 1 d the goal of the paper is to extend the successful framework of monotone consistent and stable finite difference methods for first order fully nonlinear hamilton jacobi equations to second order fully nonlinear pdes such as monge ampere and bellman type equations new concepts of consistency generalized monotonicity and stability are introduced among them the generalized monotonicity and consistency which are easier to verify in practice are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear hamilton jacobi equations the main component of the proposed framework is the concept of a numerical operator and the main idea used to design consistent generalized monotone and stable finite difference methods is the concept of a numerical moment these two new concepts play the same roles the numerical hamiltonian and the numerical viscosity play in the finite difference framework for first order fully nonlinear hamilton jacobi equations in the paper two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear pdes the first class contains lax friedrichs like methods which also are proved to be stable and the second class contains godunov like methods numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper

on convex functions and the finite element method many problems of theoretical and practical interest involve finding a convex or concave function for instance optimization problems such as finding the projection on the convex functions in h k omega or some problems in economics in the continuous setting and assuming smoothness the convexity constraints may be given locally by asking the hessian matrix to be positive semidefinite but in making discrete approximations two difficulties arise the continuous solutions may be not smooth and an adequate discrete version of the hessian must be given in this paper we propose a finite element description of the hessian and prove convergence under very general conditions even when the continuous solution is not smooth working on any dimension and requiring a linear number of constraints in the number of nodes using semidefinite programming codes we show concrete examples of approximations to optimization problems

source type solutions of the porous media equation with absorption the fast diffusion case in this paper we consider the cauchy problem u t u m u p on s t r n 0 t u x 0 0 x r n 0 where n 1 p 1 and 0 m 1

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

the porous medium equation on riemannian manifolds with negative curvature the large time behaviour abstract we consider nonnegative solutions of the porous medium equation pme on cartan hadamard manifolds whose negative curvature can be unbounded we take compactly supported initial data because we are also interested in free boundaries we classify the geometrical cases we study into quasi hyperbolic quasi euclidean and critical cases depending on the growth rate of the curvature at infinity we prove sharp upper and lower bounds on the long time behaviour of the solutions in terms of corresponding bounds on the curvature in particular we estimate the location of the free boundary a global harnack principle follows we also present a change of variables that allows to transform radially symmetric solutions of the pme on model manifolds into radially symmetric solutions of a corresponding weighted pme on euclidean space and back this equivalence turns out to be an important tool of the theory

numerical methods for fully nonlinear elliptic equations of the monge ampere type abstract in this article we discuss the numerical solution of the dirichlet problem for the monge ampere equation in two dimensions the solution of closely related problems is also discussed these include a family of pucci s equations the equation prescribing the harmonic mean of the eigenvalues of the hessian of a smooth function of two variables and a minimization problem from nonlinear elasticity where the cost functional involves the determinant of the gradient of vector valued functions to solve the monge ampere equation we consider two methods the first one reduces the monge ampere equation to a saddle point problem for a well chosen augmented lagrangian to solve this saddle point problem we advocate an uzawa douglas rachford algorithm the second method combines nonlinear least squares and operator splitting this second method being simpler to implements we apply variants of it to the solution of the other problems for the space discretization we use mixed finite element approximations closely related to methods already used for the solution of linear and nonlinear bi harmonic problems through these approximations the solution of the above problems is essentially reduced to the solution of discrete poisson problems the methods discussed in this article are validated by the results of numerical experiments

nonexistence of solutions to parabolic differential inequalities with a potential on riemannian manifolds we are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact riemannian manifolds in particular we highlight the interplay between the geometry of the underlying manifold the power nonlinearity and the behavior of the potential at infinity

linear elliptic difference inequalities with random coefficients we prove various pointwise estimates for solutions of linear ellip tic difference inequalities with random coefficients these estimates include discrete versions of the maximum principle of aleksandrov and harnack in equalities and holder estimates of krylov and safonov for elliptic differential operators with bounded coefficients

the design of servomechanism controllers for uncertain plants abstract the design of multivariable servomechanism controllers is considered here for plants whose parameters are uncertain or perturbed within a cone bounding criterion the solution comes from a controller design problem as an lq optimal control problem in which the gain matrices result in a quadratic performance upper bound similar to that described by gilman and rhodes 1976 which depends on the uncertainty under conditions on the nominal plant and on the uncertainty a resulting linear time invariant controller is guaranteed to achieve the objectives of robust tracking and regulation with closed loop stability

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

on finite element methods for fully nonlinear elliptic equations of second order for the first time we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard c 1 finite element method fem we consider throughout the paper two cases in parallel for convex bounded polyhedral domains in bf r n or for c 2 bounded domains in bf r 2 we prove stability and convergence for the corresponding conforming or nonconforming c 1 fem respectively the results for equations and systems of orders 2 and 2m and quadrature approximations appear elsewhere the classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator the consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated the stability has to be proved in an unusual way this is the hard core of the paper essential tools are linearization a compactness argument the interplay between the weak and strong form of the linearized operator and a new regularity result for solutions of finite element equations an essential basis for our proofs are davydov s results for c 1 fes on polyhedral domains in bf r n or of local degree 5 for c 2 domains in bf r 2 better convergence and extensions to bf r n for c 2 domains are to be expected from his forthcoming results on curved domains our proof for the second case in bf r n includes the first essentially as a special case the method applies to quasi linear elliptic problems not in divergence form as well a discrete newton method is shown to converge locally quadratically essentially independently of the actual grid size by the mesh independence principle

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

a numerical approach to variational problems subject to convexity constraint we describe an algorithm to approximate the minimizer of an elliptic functional in the form int omega j x u nabla u on the set cal c of convex functions u in an appropriate functional space x such problems arise for instance in mathematical economics 4 a special case gives the convex envelope u 0 of a given function u 0 let t n be any quasiuniform sequence of meshes whose diameter goes to zero and i n the corresponding affine interpolation operators we prove that the minimizer over cal c is the limit of the sequence u n where u n minimizes the functional over i n cal c we give an implementable characterization of i n cal c then the finite dimensional problem turns out to be a minimization problem with linear constraints

c 0 penalty methods for the fully nonlinear monge ampere equation in this paper we develop and analyze c 0 penalty methods for the fully nonlinear monge ampere equation det d 2 u f in two dimensions the key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric stable and consistent with the continuous linearization we are then able to show the well posedness of the penalty method as well as quasi optimal error estimates using the banach fixed point theorem as our main tool numerical experiments are presented which support the theoretical results

discrete methods for fully nonlinear elliptic equations this paper exhibits and proves the stability of discrete approximations for the solution of the dirichlet problem for fully nonlinear uniformly elliptic partial differential equations in situations where classical solutions are not known to exist the resulting solutions are characterized in the viscosity sense of crandall and lions traps amer math soc 177 1983 pp 1 42 and our stability analysis depends on estimates for nonlinear difference equations of positive type which are consequences of earlier work on linear difference equations with random coefficients

fast finite difference solvers for singular solutions of the elliptic monge ampere equation the elliptic monge ampere equation is a fully nonlinear partial differential equation which originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail in this article we build a finite difference solver for the monge ampere equation which converges even for singular solutions regularity results are used to select a priori between a stable provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain this allows singular solutions to be computed using a stable method and regular solutions to be computed more accurately the resulting nonlinear equations are then solved by newton s method computational results in two and three dimensions validate the claims of accuracy and solution speed a computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities

smoothing effects for the porous medium equation on cartan hadamard manifolds abstract we prove three sharp bounds for solutions to the porous medium equation posed on riemannian manifolds or for weighted versions of such equation firstly we prove a smoothing effect for solutions which is valid on any cartan hadamard manifold whose sectional curvatures are bounded above by a strictly negative constant this bound includes as a special case the sharp smoothing effect recently proved by vazquez on the hyperbolic space in vazquez 2015 which is similar to the absolute bound valid in the case of bounded euclidean domains but has a logarithmic correction secondly we prove a bound which interpolates between such smoothing effect and the euclidean one supposing a suitable quantitative sobolev inequality holds showing that it is sharp by means of explicit examples finally assuming a stronger functional inequality of sub poincare type we prove that the above mentioned sharp absolute bound holds and provide examples of weighted porous media equations on manifolds of infinite volume in which it holds in contrast with the non weighted euclidean situation it is also shown that sub poincare inequalities cannot hold on cartan hadamard manifolds

blow up and global existence for solutions to the porous medium equation with reaction and slowly decaying density abstract we study existence of global solutions and finite time blow up of solutions to the cauchy problem for the porous medium equation with a variable density x and a power like reaction term x u p with p 1 this is a mathematical model of a thermal evolution of a heated plasma see 29 the density decays slowly at infinity in the sense that x x q as x with q 0 2 we show that for large enough initial data solutions blow up in finite time for any p 1 on the other hand if the initial datum is small enough and p p for a suitable p depending on m n then global solutions exist in addition if p p for a suitable p p depending on m n then the solution blows up in finite time for any nontrivial initial datum we need the extra hypothesis that q 0 for 0 small enough when m p p observe that p p if x is a multiple of x q for x large enough such results are in agreement with those established in 48 where x 1 and are related to some results in 32 33 the case of fast decaying density at infinity i e q 2 is examined in 36

a numerical method for variational problems with convexity constraints we consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex this problem is at the interface between convex analysis convex optimization variational problems and partial differential equation techniques the approach is to approximate the nonpolyhedral cone of convex functions by a polyhedral cone which can be represented by linear inequalities this approach leads to an optimization problem with linear constraints which can be computed efficiently hundreds of times faster than existing methods

mean field games numerical methods mean field type models describing the limiting behavior as the number of players tends to infty of stochastic differential game problems have been recently introduced by j m lasry and p l lions c r math acad sci paris 343 2006 pp 619 625 c r math acad sci paris 343 2006 pp 679 684 jpn j math 2 2007 pp 229 260 numerical methods for the approximation of the stationary and evolutive versions of such models are proposed here in particular existence and uniqueness properties as well as bounds for the solutions of the discrete schemes are investigated numerical experiments are carried out

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft