Showing papers by "Paris Dauphine University published in 1988"

Morse index of some min‐max critical points. I. Application to multiplicity results

Abstract: On donne un resultat general sur des bornes inferieures pour des indices de Morse de points critiques obtenus par des principes de min-max. En combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques, a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d'equations elliptiques semi-lineaires d'ordre 2

On a non linear partial differential equation having natural growth terms and unbounded solution

Abstract: We prove the existence of a solution of the nonlinear elliptic equation: A(u) + g(x, u, Du) = h(x), where A is a Leray-Lions operator from W 0 1 , p ( Ω ) into W−1, p′(Ω) and g is a nonlinear term with “natural” growth with respect to Du [i.e. such that |g(x, u, ξ)| ≦ b(|u|) (|ξ|p + c(x))], satisfying the sign condition g(x, u, ξ)u ≧ 0 but no growth condition with respect to u. Here h belongs to W−1, p′(Ω), thus the solution u of the problem does not in general be more smooth than W 0 1 , p ( Ω ) . The existence of a solution is also proved for the corresponding obstacle problem.

On the Fokker-Planck-Boltzmann equation

Abstract: We consider the Boltzmann equation perturbed by Fokker-Planck type operator. To overcome the lack of strong a priori estimates and to define a meaningful collision operator, we introduce a notion of renormalized solution which enables us to establish stability results for sequences of solutions and global existence for the Cauchy problem with large data. The proof of stability and existence combines renormalization with an analysis of a defect measure.

Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: the case of bounded stochastic evolutions

Abstract: 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

On Positive Solutions of Semilinear Elliptic Equations in Unbounded Domains

Abstract: We give a short survey of some existence results concerning positive solutions of semilinear elliptic equations in unbounded domains. We consider only solutions which vanish at infinity. We also investigate the existence of “minimum-type” solutions as well as solutions given by “higher” critical points.

Viscosity Solutions of Second-Order Equations, Stochastic Control and Stochastic Differential Games

Abstract: In this note we review, explain and detail some recent results concerning the possible relations between various value functions of general optimal stochastic control and stochastic differential games and the viscosity solutions of the associated Hamilton-Jacobi-Bellman HJB and Bellman-Isaacs BI equations It is well-known that the derivation of these equations is heuristic and it is justified only when the value functions are smooth enough (WH Fleming and R Richel [15]) On the other hand, the equations are fully nonlinear, second-order, elliptic but possibly degenerate Smooth solutions do not exist in general and nonsmooth solutions (like Lipschitz continuous solutions in the deterministic case) are highly nonunique (For some simple examples we refer to P-L Lions [24]) As far as the first-order Hamilton-Jacobi equations are concerned, to overcome these typical difficulties and related ones like numerical approximations, asymptotic problems etc MG Crandall and P-L Lions [8] introduced the notion of viscosity solutions and proved general uniqueness results A systematic exploration of several equivalent formulations of this notion and an easy and readable account of the typical uniqueness results may be found in MG Crandall, LC Evans and P-L Lions [6] It was also observed in P-L Lions [24] that the classical derivation of the Bellman equation for deterministic control problems can be easily adapted to yield the following general fact: Value functions of deterministic control problems are always viscosity solutions of the associated Hamilton-Jacobi-Bellman equations The uniqueness of viscosity solutions and the above fact imply then a complete characterization of the value functions This observation was then extended to differential games by EN Barron, LC Evans and R Jensen [3], PE Souganidis [36] and LC Evans and PE Souganidis [14]

On Some Approximation Techniques in Non Linear Filtering

Abstract: The following problem has been considered by J. Picard [3]. Let a signal be governed by the equation $$ ^{\begin{array}{*{20}{c}} {dx = b\left( x \right)dt + {\varepsilon ^r}\sigma \left( x \right)d{w^1}} \\ {x\left( 0 \right) = \xi } \end{array}} $$ (1.1) This signal is “observed” by the process $$ \begin{array}{*{20}{c}} {dy = h\left( x \right)dt + {\varepsilon ^{1 - }}{}^rd{w^2}} \\ {y\left( 0 \right) = 0} \end{array} $$ (1.2) where wl, w2 are two independent Wiener processes, and ξ is a random variable independent of wl, w2. The parameter e is small and 0 < γ < ½. This assumption means that there is a high signal to noise ratio. Let $$ \phi \left( t \right) = E\left[ {\phi \left( {{x_t}} \right)\left| { y\left( s \right), 0 \leqslant s \leqslant t} \right.} \right] $$ (1.3)

Skyrmions and symmetry

Abstract: It was already known that when the topological charge k is larger than 1, the ground state energy for Skyrme's problem can never be achieved by a hedge-hog type function. In this paper we prove that in fact the ratio of the ground state energy to the lower bound of the energy for the hedge-hog functions converges to 0 as k- 1•