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

strong uniqueness for sdes in hilbert spaces with nonregular drift we prove pathwise uniqueness for a class of stochastic dierential equations sde on a hilbert space with cylindrical wiener noise whose non linear drift parts are sums of the subdierential of a convex function and a bounded part this generalizes a classical result by one of the authors to innite dimensions our results also generalize

large amplitude time periodic solutions of a semilinear wave equation abstract this paper studies the existence of periodic solutions for a family of semilinear wave equations where the restoring force is independent of time monotone and grows at a more rapid rate than linear near infinity with appropriate technical assumptions it is shown that there is an unbounded sequence of such free vibrations i e there are solutions of arbitrarily large amplitude if the restoring force is independent of x the monotonicity assumption can be omitted

total variation flow and sign fast diffusion in one dimension abstract we consider the dynamics of the total variation flow tvf u t div d u d u and of the sign fast diffusion equation sfde u t sign u in one spatial dimension we find the explicit dynamic and sharp asymptotic behaviour for the tvf and we deduce the one for the sfde by an explicit correspondence between the two equations

the first initial boundary value problem for fully nonlinear parabolic equations generated by functions of the eigenvalues of the hessian abstract for a class of elliptic hessian operators raised by caffarelli nirenberg spruck the corresponding parabolic monge ampere equation was studied the existence and uniqueness of the admissible solution to the first initial boundary value problem for the equation were established which extended a result of ivochkina ladyzhenskaya

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

interior gradient estimates for solutions of certain fully nonlinear elliptic equations korevaar has given a new proof for interior gradients estimates for solutions of prescribed mean curvature equations in n j korevaar proc sympos pure math 45 part 2 1986 interior gradient estimates for elliptic weingarten equations are also obtained in n j korevaar ann inst h poincare 43 1987 1 17 in this paper we apply the normal perturbation method introduced in n j korevaar proc sympos pure math 45 part 2 1986 n j korevaar ann inst h poincare 43 1987 1 17 to generalize the result in n j korevaar ann inst h poincare 43 1987 1 17 to more general fully nonlinear elliptic equations

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

a remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system abstract in this work motivated by t f wu the nehari manifold for a semilinear elliptic system involving sign changing weight function nonlinear anal 68 2008 1733 1745 and using recent ideas from brown and wu k j brown t f wu a semilinear elliptic system involving nonlinear boundary condition and sign changing weight function j math anal appl 337 2008 1326 1336 we prove the existence of nontrivial nonnegative solutions to the nonlinear elliptic system p u m x u p 2 u u 2 u c x u 2 u v x p v m x v p 2 v v 2 v c x u v 2 v x u 0 v x here p denotes the p laplacian operator defined by p z div z p 2 z p 2 r n is a bounded domain with smooth boundary 1 1 2 p p p p n n p if n p p if n p n is the outer normal derivative r 2 0 0 the weight m x is a bounded function with m 0 and c x is a continuous function which changes sign in

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

some existence results for fully nonlinear elliptic equations of monge ampere type we study the existence and uniqueness of solutions of monge ampere type equations this type of equations has been studied extensively by caffarelli nirenberg spruck and many others see 5 through 8 and the references therein we present some existence and uniqueness results for this type of equations on compact riemannian manifolds with non negative sectional curvature we have also generalized some results in 7

existence of coupled quasi solutions of systems of nonlinear reaction diffusion equations systems of nonlinear parabolic initial boundary value problems arise in many applications such as epidemies ecology biochemistry biology chemical and nuclear engineering constructive methods of proving existence results for such problems which can also provide numerical procedures for the computation of solutions are of greater value than theoretical existence results the method of upper and lower solutions coupled with monotone iterative technique has been employed successfully to prove existence of multiple solutions of nonlinear reaction diffusion equations in special cases by various authors 335 10 11 15 181 recently in 6 171 weakly coupled systems of reaction diffusion equations when the nonlinear terms are independent of gradient terms are discussed and some special type of results are obtained we in this paper investigate general systems of nonlinear reaction diffusion problems when the nonlinear terms possess a mixed quasi monotone property we discuss a very general situation and obtain coupled extremal quasi solutions which in special cases reduce to minimal and maximal solutions we shall also indicate how one step cyclic monotone iterative schemes can be generated which yield accelerated rate of convergence of iterates this work is in the spirit of our recent paper 12 for elliptic systems

noise prevents singularities in linear transport equations a stochastic linear transport equation with multiplicative noise is considered and the question of no blow up is investigated the drift is assumed only integrable to a certain power opposite to the deterministic case where smooth initial conditions may develop discontinuities we prove that a certain sobolev degree of regularity is maintained which implies holder continuity of solutions the proof is based on a careful analysis of the associated stochastic flow of characteristics

speed of propagation for hamilton jacobi equations with multiplicative rough time dependence and convex hamiltonians we show that the initial value problem for hamilton jacobi equations with multiplicative rough time dependence typically stochastic and convex hamiltonians satisfies finite speed of propagation we prove that in general the range of dependence is bounded by a multiple of the length of the skeleton of the path that is a piecewise linear path obtained by connecting the successive extrema of the original one when the driving path is a brownian motion we prove that its skeleton has almost surely finite length we also discuss the optimality of the estimate

monotone enclosure for elliptic and parabolic sytems with nonmonotone nonlinearities abstract monotone iteration techniques for weakly coupled systems having nonmonotone nonlinearities have been developed recently by m khavanin and v lakshmikantham and by s carl in this paper an existence and enclosing theorem is given in terms of upper and lower solutions and a new monotone iteration technique which in a natural way extends the monotone iteration technique used in the monotone case to the nonmonotone case considered here is proposed the maximum and minimum functions that occur in the iteration scheme allow one to work under low regularity conditions particularly no local lipschitz conditions on the nonlinearities are required for the iteration process a crucial role in the analysis is played by a measurable selection theorem which is usually known from optimization theory furthermore conditions which ensure that the iteration process converges to the uniquely defined solution are given

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

regularization by noise for stochastic hamilton jacobi equations we study regularizing effects of nonlinear stochastic perturbations for fully nonlinear pde more precisely path by path l infty bounds for the second derivative of solutions to such pde are shown these bounds are expressed as solutions to reflected sde and are shown to be optimal

on the regularization effect of space time white noise on quasi linear parabolic partial differential equations we show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi linear parabolic equation in one space dimension even if the nonlinearity is only measurable and not even locally bounded

the existence of positive solution of elliptic system by a linking theorem on product space abstract this paper is devoted to study the existence of nontrivial positive solution of a class of elliptic system with dirichlet data by using the abstract linking theorems on product space we established in zhao et al nonlinear anal 49 2002 431 we obtain the existence of three nonnegative solutions for a class of elliptic systems and the existence of a nontrivial positive solution for the problem related to the model of competing species systems involving critical sobolev exponents

on strongly indefinite functionals with applications recently in their remarkable paper critical point theory for indefinite functionals v benci and p rabinowitz gave a direct approach avoiding finitedimensional approximationsto the existence theory for critical points of indefinite functionals in this paper we develop under weaker hypotheses a simpler but more general theory for such problems in the second part of the paper the abstract results are applied to a class of resonance problems of the landesman and lazer type and moreover they are illustrated by an application to a wave equation problem 0 introduction this paper is devoted to the critical point theory of strongly indefinite functionals in a real hilbert space h here a strongly indefinite functional is a cl map h r which is neither bounded from above or below not even modulo a finite dimensional subspace such functionals arise for example in the study of periodic solutions of the one dimensional wave equation in the study of periodic solutions of hamiltonian systems of ordinary differential equations in the existence theory for systems of elliptic equations etc to find the critical points of strongly indefinite functionals different methods were used for example rabinowitz studied the restrictions of the functional to finite dimensional subspaces 1 2 and then he tried to find the critical points of the original problem by passing to the limit for a suitable subsequence of the critical points of the restricted problems amann 3 reduced the indefinite functionals by means of a saddlepoint reduction to a finite dimensional problem of course this is only possible under some strong hypotheses under certain assumptions the functional can be replaced by a dual variational problem having some better properties see for example 4 or 5 the first direct approach was given by benci and rabinowitz 6 the essential in their paper is the observation that more detailed information about the special form of the deformations a substitute for the gradient flow associated to the functional is needed they constructed deformations having special representations by solving appropriate differential equations approximately by time discretisation this method has some disadvantages because one needs strong hypotheses in order to get the approximations uniformly on bounded sets a main part of our paper will be the construction of flows and deformations having special representations by solving appropriate differential equations exactly received by the editors march 30 1981 and in revised form november 30 1981 1980 mathematics subject classification primary 58e05 secondary 47h25 35l05

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness