the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

lasalle type theorems for stochastic differential delay equations the main aim of this paper is to establish the lasalle type asymptotic convergence theorems for the solutions of stochastic differential delay equations these stochastic versions are then applied to establish sufficient criteria for the stochastically asymptotic stability of the delay equations several examples are also given for illustration

asymptotic tracking of a reference state for systems with a feedforward structure we consider the problem of asymptotic tracking for a system which can be written in a feedforward form the data of a bounded reference state trajectory is assumed our solution relies on a lyapunov construction the time varying state feedbacks obtained are bounded and ensure the global uniform asymptotic stability when the signals are periodic and the local exponential stability of the reference state trajectory of the closed loop system

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

global stabilization of stochastic high order feedforward nonlinear systems with time varying delay in this paper we consider the problem of global stabilization for a class of stochastic high order feedforward nonlinear systems with time varying delay by introducing the homogeneous domination design method and constructing the appropriate lyapunov krasovskii functional a state feedback controller is constructed to drive the closed loop system to be globally asymptotically stable in probability

explicit gradient estimates for minimal lagrangian surfaces of dimension two we derive explicit uniform a priori interior hessian and gradient estimates for special lagrangian equations of all phases in dimension two

hessian and gradient estimates for three dimensional special lagrangian equations with large phase we obtain a priori interior hessian and gradient estimates for special lagrangian equations with phase larger than a critical value in dimension three gradient estimates are also derived for critical and super critical phases in general dimensions

fluctuating interfaces in microemulsion and sponge phases a simple ginzburg landau theory with a single scalar order parameter is used to study the microscopic structure of microemulsions and sponge phases the scattering intensity in both film and bulk contrast as well as averages of the internal area s the euler characteristic chi e and the mean curvature squared are calculated by monte carlo methods the results are compared with results obtained from a variational approach in combination with the theory of gaussian random fields and level surfaces the results for the location of the transition from the microemulsion to oil water coexistence for the scattering intensity in bulk contrast and for the dimensionless ratio chi e v 2 s 3 where v is the volume are found to be in good quantitative agreement however the variational approach fails to give a peak in the scattering intensity in film contrast at finite wavevector a peak which is observed both in the monte carlo simulations and in experiment also the variational approach fails to produce a transition from the microemulsion to the lamellar phase

calibrations associated to monge ampere equations we show the volume maximizing property of the special lagrangian submanifolds of a pseudo euclidean space these special lagrangian submanifolds arise locally as gradient graphs of solutions to monge ampere equations

stability of stochastic nonlinear systems with state dependent switching in this paper the problem of stability on stochastic systems with state dependent switching is investigated to analyze properties of the switched system by means of ito s formula and dynkin s formula it is critical to show switching instants being stopping times when the given active region set can be replaced by its interior the local solution of the switched system is constructed by defining a series of stopping times as switching instants and the criteria on global existence and stability of solution are presented by lyapunov approach for the case where the active region set can not be replaced by its interior the switched systems do not necessarily have solutions thereby quasi solution to the underlying problem is constructed and the boundedness criterion is proposed the significance of this paper is that all the results presented depend on some easily verified assumptions that are as elegant as those in the deterministic case and the proofs themselves provide design procedures for switching controls

angle theorems for the lagrangian mean curvature flow we prove that symplectic maps between riemann surfaces l m of constant nonpositive and equal curvature converge to minimal symplectic maps if the lagrangian angle alpha for the corresponding lagrangian submanifold in the cross product space l times m satisfies text osc alpha le pi if one considers a 4 dimensional kahler einstein manifold overline m of nonpositive scalar curvature that admits two complex structures j k which commute and assumes that l subset overline m is a compact oriented lagrangian submanifold w r t j such that the kahler form overline kappa w r t k restricted to l is positive and text osc alpha le pi then l converges under the mean curvature flow to a minimal lagrangian submanifold which is calibrated w r t overline kappa

an explicit formula of bounded feedback stabilizers for feedforward systems abstract this work constitutes continuation of a recent result of the same authors which provides an explicit formula for bounded feedback stabilizers for a wide class of triangular nonlinear systems in the present paper the previous result is extended for nonlinear systems having feedforward structure and whose linearization at the equilibrium is in general uncontrollable

the dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions let be a bounded c2 domain in n and m be a continuous map the dirichlet problem for the minimal surface system asks whether there exists a lipschitz map f m with f and with the graph of f a minimal submanifold in n m for m 1 the dirichlet problem was solved more than 30 years ago by jenkins and serrin 12 for any mean convex domains and the solutions are all smooth this paper considers the dirichlet problem for convex domains in arbitrary codimension m we prove that if m satisfies 8n sup d2 2 sup d 1 then the dirichlet problem for is solvable in smooth maps here is the diameter of such a condition is necessary in view of an example of lawson and osserman 13 in order to prove this result we study the associated parabolic system and solve the cauchy dirichlet problem with as initial data 2003 wiley periodicals inc

on the second boundary value problem for lagrangian mean curvature flow abstract we consider a fully nonlinear parabolic equation with nonlinear neumann type boundary condition and show the long time existence and convergence of the flow finally we apply this study to the boundary value problem for minimal lagrangian graphs

the neumann problem of special lagrangian equations with supercritical phase abstract in this paper we establish the global c 2 estimates of the neumann problem of special lagrangian equations with supercritical phase and the existence theorem by the method of continuity

mean curvature flows and isotopy of maps between spheres let f be a smooth map between unit spheres of possibly different dimensions we prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions a corollary is that any area decreasing map between unit spheres of possibly different dimensions is isotopic to a constant map c 2004 wiley periodicals inc

a bernstein problem for special lagrangian equations where is are the eigenvalues of the hessian d2u namely any global convex solution to 1 1 in rn must be a quadratic polynomial recall the classical result any global convex solution in rn to the laplace equation u 1 n c or the monge ampere equation log det d2u log 1 log n c must be quadratic equation 1 1 originates from special lagrangian geometry hl the lagrangian graph x u x rn rn is called special when the argument of the complex number 1 1 1 1 1 n is constant c or u satisfies 1 1 and it is special if and only if x u x is a minimal surface in rn rn hl theorem 2 3 proposition 2 17 in terms of minimal surface our result is the following

interior gradient estimates for mean curvature equations the interior gradient estimate for the prescribed mean curvature equation has been extensively studied see 9 and the references therein for high order mean curvature equations it has also been obtained in 11 18 in most articles such estimates were obtained by carrying out analysis on the graphs of solutions and so the arguments depend on the invariance of the equations under rigid motion from the view point of partial differential equations such estimate should hold for equations with similar structural conditions in 13 6 the gradient estimate was obtained for certain fully nonlinear elliptic equations under various conditions different proofs for the gradient estimate for mean curvature equations have been given in 1 3 in this note we show that for curvature and hessian equations the interior gradient estimate can be obtained very easily our proof which is based on suitable choice of auxiliary functions is elementary and avoids geometric computations on the graph of solutions the technique in this note has actually been widely used in literature see e g 9

output feedback regulation of stochastic nonlinear systems with stochastic iiss inverse dynamics this paper develops a unifying framework for output feedback regulation of stochastic nonlinear systems with more general stochastic inverse dynamics the contributions of this work are characterized by the following novel features 1 motivated by the concept of integral input to state stability iiss in deterministic systems and stochastic input to state stability siss using lyapunov function in stochastic systems a concept of stochastic integral input to state stability siiss using lyapunov function is first introduced two important properties of siiss are obtained i siiss is strictly weaker than siss using lyapunov function ii siiss is stronger than the minimum phase property however only under the minimum phase assumption there is no dynamic output feedback control law for global stabilization in probability 2 almost sure boundedness a reasonable and stronger concept than boundedness in probability is introduced the purpose of introducing the concept is to prove the boundedness and convergence of some signals in the closed loop control system 3 some important mathematical tools which play an essential role in the boundedness and convergence analysis of the closed loop system are established 4 a unifying framework is proposed to design a dynamic output feedback control law which drives the states to the origin almost surely while maintaining all the closed loop signals bounded almost surely

non parametric mean curvature evolution with boundary conditions soit un domaine borne de r n n 2 de frontiere lisse et soit a l operateur de surface minimale donne par a d i a i p a i p i 1 p 2 1 2 on considere des surfaces dont la vitesse en direction de leur normale unite est egale a la courbure moyenne on etudie l equation d evolution qui en resulte u 1 du 2 1 2 au 0 dans 0 t

time varying state feedback stabilisation of stochastic feedforward nonlinear systems with unknown growth rate abstractwe consider the time varying state feedback stabilisation problem for a class of stochastic feedforward nonlinear systems with unknown growth rate in this paper a new lasalle type theorem for stochastic time varying systems is firstly established by using the generalized weakly positive definite function as an application to deal with serious uncertainties in the unknown growth rate a time varying approach rather than an adaptive one is adopted to design the scheme of a state feedback controller for stochastic feedforward systems based on the established lasalle type theorem it is shown that all signals of the resulting closed loop system converge to zero almost surely illustrative examples are given to verify the theoretical findings

brief paper feedback stabilization for high order feedforward nonlinear time delay systems this paper investigates the problem of global strong stabilization by state feedback for a family of high order feedforward nonlinear time delay systems the uncertain nonlinearities are assumed to satisfy a polynomial growth assumption with an input or delayed input dependent rate with the help of the appropriate lyapunov krasovskii functionals and a rescaling transformation with a gain to be tuned online by a dynamic equation we propose a dynamic low gain state feedback control scheme a simulation example is given to demonstrate the effectiveness of the proposed design procedure

mean curvature flow of surfaces in einstein four manifolds let be a compact oriented surface immersed in a four dimensional kahler einstein manifold m w we consider the evolution of in the direction of its mean curvature vector it is proved that being symplectic is preserved along the flow and the flow does not develop type i singularity when m has two parallel kahler forms w and w that determine different orientations and is symplectic with respect to both w and w we prove the mean curvature flow of exists smoothly for all time in the positive curvature case the flow indeed converges at infinity

absolute stabilization and minimax optimal control of uncertain systems with stochastic uncertainty this paper is concerned with existence and optimality properties of so called guaranteed cost controllers for an uncertain system subject to structured uncertainty the uncertainty in the system is assumed to have a stochastic character and to satisfy certain stochastic integral constraints it is shown that a minimax optimal guaranteed cost state feedback controller for a stochastic system can be synthesized as a state feedback controller absolutely stabilizing this system for each initial state of the system this controller can be found by parametric optimization of solutions of a parameter dependent generalized matrix riccati equation arising in stochastic h infty theory

dissipativity based sliding mode control of switched stochastic systems this technical brief is concerned with dissipativity analysis and dissipativity based sliding mode control smc of continuous time switched stochastic systems firstly a sufficient condition is proposed to guarantee the mean square exponential stability and strict dissipativity for the switched stochastic system then an integral type sliding surface function is designed for establishing a sliding mode dynamics which can be formulated by a switched stochastic system with an external disturbance uncertainty dissipativity analysis and synthesis are both investigated for the sliding mode dynamics and consequently sufficient conditions are derived which pave the way for solving the dissipativity analysis and control problems moreover a smc law is synthesized to drive the system trajectories onto the predefined sliding surface in a finite time finally the efficiency of the theoretical findings is demonstrated by an illustrative example

a new type of stability theorem for stochastic systems with application to stochastic stabilization in this note a new type of stability theorem for stochastic systems is established firstly based on this stability theorem and its corollaries stochastic stabilization and destabilization by noise are further investigated in the note the local lipschitz condition is weakened to the generalized local lipschitz condition the commonly used linear growth condition or one side linear growth condition is weakened to the generalized one side linear growth condition which is local variable and nonlinear admits nearly arbitrary variability in the time and real nonlinearity in the state as an application a simple and direct design method is proposed for finding a noise strength hat mathbb g t x so that the added noise hat mathbb g t x d hat b t stabilizes an unstable stochastic system or destabilizes a stable one a numerical example is presented at the end of the note to illustrate the usage and efficiency of the proposed design method of the note

long time existence and convergence of graphic mean curvature flow in arbitrary codimension let f 1 2 be a map between compact riemannian manifolds of constant curvature this article considers the evolution of the graph of f in 1 2 by the mean curvature flow under suitable conditions on the curvature of 1 and 2 and the differential of the initial map we show that the flow exists smoothly for all time at each instant t the flow remains the graph of a map ft and ft converges to a constant map as t approaches infinity this also provides a regularity estimate for lipschitz initial data

detectability observability and lyapunov type theorems of linear discrete time varying stochastic systems with multiplicative noise abstractthe objective of this paper is to study detectability observability and related lyapunov type theorems of linear discrete time time varying stochastic systems with multiplicative noise some new concepts such as uniform detectability k exact detectability resp kwft exact detectability kft exact detectability kn exact detectability and k exact observability resp kwft exact detectability kft exact detectability kn exact detectability are introduced respectively and nice properties associated with uniform detectability exact detectability and exact observability are also obtained moreover some lyapunov type theorems associated with generalised lyapunov equations and exponential stability in mean square sense are presented under uniform detectability kn exact observability and kn exact detectability respectively

singular solutions to special lagrangian equations with subcritical phases and minimal surface systems we construct singular solutions to special lagrangian equations with subcritical phases and minimal surface systems a priori estimate breaking families of smooth solutions are also produced correspondingly a priori estimates for special lagrangian equations with certain convexity are largely known by now

stability analysis of a class of hybrid stochastic retarded systems under asynchronous switching the stability of a class of hybrid stochastic retarded systems hsrss with an asynchronous switching controller is investigated in this model the controller design relies on the observed jumping parameters which are however delayed and thus can not be measured in real time precisely this delayed time interval referred to as the asynchronous switching interval is markovian and dependent on the actual switching signal the sufficient conditions under which the system is either stochastically asymptotic stable or input to state stable are obtained by applying the extended razumikhin type theorem to the asynchronous switching interval these results are less conservative as it is only required that the designed lyapunov function is non decreasing it is shown that the stability of the considered system can be guaranteed by a sufficiently small mode transition rate of the underlying markov process which is a conclusion similar to that in asynchronous deterministic switched systems the effectiveness and correctness of the obtained results are finally verified by a numerical example

brief paper small gain control method for stochastic nonlinear systems with stochastic iiss inverse dynamics this paper investigates the small gain type conditions on stochastic iiss siiss systems and makes full use of these conditions in the design and analysis of the controller the contributions are as follows 1 a new proof of the stochastic lasalle theorem is provided 2 the small gain type conditions on siiss are developed and their relationship is discussed 3 based on the stochastic lasalle theorem and siiss small gain type conditions the adaptive controllers are designed to guarantee that all of the closed loop signals are bounded almost surely and the stochastic closed loop systems are globally asymptotically stable in probability

triply periodic surfaces and multiply continuous structures from the landau model of microemulsions we present a method for the generation of periodic embedded surfaces of nonpositive gaussian curvature and multiply continuous phases the structures are related to the local minima of the scalar order parameter landau ginzburg hamiltonian for microemulsions in the bicontinuous structure the single surface separates the volume into two disjoint subvolumes in some of our phases multiply continuous there is more than one periodic surface that disconnects the volume into three or more disjoint subvolumes we show that some of these surfaces are triply periodic minimal surfaces we have generated known minimal surfaces e g schwarz primitive p diamond d and schoen luzatti gyroid g and many surfaces of high genus we speculate that the structure of microemulsion can be related to the high genus gyroid structures since the high genus surfaces were most easily generated in the phase diagram close to the microemulsion stability region we study in detail the geometrical characteristics of these phases such as genus per unit cell surface area per unit volume and volume fraction occupied by oil or water in such a structure our discovery calls for new experimental techniques which could be used to discern between bicontinuous and multiply continuous structures we observe that multiply continuous structures are most easily generated close to the water oil coexistence region s1063 651x 96 01311 6

mean curvature flows of lagrangian submanifolds with convex potentials this article studies the mean curvature flow of lagrangian submanifolds in particular we prove the following global existence and convergence theorem if the potential function of a lagrangian graph in t 2n is convex then the flow exists for all time and converges smoothly to a flat lagrangian submanifold we also discuss various conditions on the potential function that guarantee global existence and convergence

state feedback stabilization for stochastic feedforward nonlinear systems with time varying delay this paper investigates a class of stochastic feedforward nonlinear systems with time varying delay by introducing the homogeneous domination approach to stochastic systems a state feedback controller is constructed to render the closed loop system globally asymptotically stable in probability

interior gradient bounds for solutions to the minimal surface system in this article we generalize the classical gradient estimate for the minimal surface equation to higher codimension we consider a vector valued function u cl c rn rm that satisfies the minimal surface system see equation 1 1 in 1 the graph of u is then a minimal submanifold of rn m we prove an a priori gradient bound under the assumption that the jacobian of du rn rm on any two dimensional subspace of rn is less than or equal to one this assumption is automatically satisfied when du is of rank one and thus the estimate covers the case when m 1 i e the original minimal surface equation this is applied to bernstein type theorems for minimal submanifolds of higher codimension 1 introduction the interior gradient bound for solutions to the minimal surface equation was discovered in the case of two variables by finn 3 and in the general case by bombieri di giorgi and miranda 1 the a priori bound is a key step in the existence and regularity of minimal surface theory the estimate has been generalized to other curvature equations and proved by different methods we refer to the note at the end of chapter 16 of 5 for literature in these directions recall a function m q c r e is a solution to the minimal surface equation if

global weak solutions to a higher order nonlinear degenerate parabolic equation abstract our aim in this paper is to investigate the existence for a sixth order parabolic equation which arises in oil water surfactant mixtures more precisely under suitable assumptions first we study the existence and regularity of weak solutions to an initial boundary problem for a non degenerate sixth order parabolic equation second considering the limits of the solutions obtained in the first step we prove the existence of weak solutions

stabilization of stochastic nonlinear systems driven by noise of unknown covariance this paper poses and solves a new problem of stochastic nonlinear disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise this is a natural stochastic counterpart of the problem of input to state stabilization in the sense of sontag 1989 our development starts with a set of new global stochastic lyapunov theorems for an exemplary class of stochastic strict feedback systems with vanishing nonlinearities where the equilibrium is preserved in the presence of noise we develop an adaptive stabilization scheme based on tuning functions that requires no a priori knowledge of a bound on the covariance next we introduce a control lyapunov function formula for stochastic disturbance attenuation finally we address optimality and solve a differential game problem with the control and the noise covariance as opposing players for strict feedback systems the resulting isaacs equation has a closed form solution

on graphic bernstein type results in higher codimension let e be a minimal submanifold of r n m that can be represented as the graph of a smooth map f r n r m we apply a formula that we derived in the study of mean curvature flow to obtain conditions under which must be an affine subspace our result covers all known ones in the general case the conditions are stated in terms of the singular values of df