Existence of local minima for capillarity problems

In this paper we study problems in capillarity theory associated with energy functionals for which no lower bounds exist. We treat the cases of pendent and rotating liquid drops. In order to show the existence of local minima we prove suitablea priori estimates for their diameter. To obtain these results sufficient conditions on the included volume depending respectively on the centrifugal and gravitational potential must be satisfied.     

Existence of stationary solutions in kinetic models with Gaussian thermostats

The thermostatted kinetic framework has been recently proposed in [C. Bianca, Nonlinear Analysis: Real World Applications 13 (2012) 2593‐2608] for the modeling of complex systems in the applied sciences under the action of an external force field that moves out of equilibrium the system. The framework consists in an integro‐differential equation with quadratic nonlinearity coupled with the Gaussian isokinetic thermostat. This paper is concerned with the existence of stationary solutions proof. The main result is gained by fixed point and measure theory arguments. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of two local minima for functionals on reflexive Banach spaces

In this paper, we establish an existence theorem of two local minima for functionals on reflexive Banach space. As a consequence, we obtain an existence theorem of three critical points which improves the conclusion of a result in a recent paper of G.Bonanno concerning some remarks on a three-critical-points theorem established by B. Ricceri.     

Existence and Lipschitz regularity for minima

We prove the existence, uniqueness and Lipschitz regularity of the minima of the integral functional

on ( ) for a class of integrands that are convex in and for boundary data satisfying some barrier conditions. We do not impose regularity or growth assumptions on .

     

Existence and uniqueness of solutions of stationary transport equations

We consider the abstract kinetic equation dψ /dx = –JLψ, x ∈ [0, τ ], in a Hilbert space H. It is supposed that J = J * = J^–1, L = L * ≥ 0, ker L = 0. The following theorem is proved: if JL is similar to a self-adjoint operator, then an associated boundary problem has a unique solution. We apply this theorem to the stationary equation of Brownian motion (sgn μ)|μ |α (∂ψ /∂x) (x,μ) = (∂²ψ /∂μ²) (x,μ), 0 < x < τ, μ ∈ ℝ. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system

In this paper, we study the existence of stationary solutions to the Vlasov-Poisson-Boltzmann system when the background density function tends to a positive constant with a very mild decay rate as |x|→∞. In fact, the stationary Vlasov-Poisson-Boltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being (ln(e+|x|))^−α for some α>0, it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the Vlasov-Poisson-Fokker-Planck system, J. Math. Anal. Appl. 202 (1996) 1058-1075] where the decay rate is assumed.     

EXISTENCE OF NONNEGATIVE SOLUTIONS FOR DEGENERATE ECOLOGICAL MODELS

In this paper,nonnegative solutions for the degenerate elliptic systems are considered.First,nonnegative solutions for scalar equation with spatial discontinuities are studied. Then themethod developed for scalar equation is applied to study elliptic systems. At last,the existence criteria of nonnegative solutions of elliptic systems are given.     

On the stationary solutions of the 2D Doi-Onsager model

In this paper, we study the 2D Doi-Onsager model

(1)     

Existence and uniqueness of conservative solutions for nonlocal fragmentation models

The aim of the paper is to investigate the well‐posedness of an integrodifferential equation describing multiple fragmentation processes, where the fragmentation rate is size and position dependent and new particles are randomly distributed in the space according to some probability density. An old method of Reuter and Lederman based on some approximation techniques is modified and applied to analyse the dynamics of the problem. The results of earlier papers on local models are extended to nonlocal models and the conservativeness of the solutions is investigated. In particular we prove the uniqueness of conservative solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

Characterizations of strict local minima and necessary conditions for weak sharp minima

In nonlinear programming, sufficient conditions of orderm usually identify a special type of local minimizer, here termed a strict local minimizer of orderm. In this paper, it is demonstrated that, if a constraint qualification is satisfied, standard sufficient conditions often characterize this special sort of minimizer. The first- and second-order cases are treated in detail. Necessary conditions for weak sharp local minima of orderm, a larger class of local minima, are also presented.This paper was completed during a sabbatical leave at the University of Waterloo. The author is grateful for the help and support of the Department of Combinatorics and Optimization. The helpful comments of the referees are also appreciated.     

On the dynamics of fine structure

Summary We investigate models for the dynamical behavior of mechanical systems that dissipate energy as timet increases. We focus on models whose underlying potential energy functions do not attain a minimum, possessing minimizing sequences with finer and finer structure that converge weakly to nonminimizing states. In Model 1 the evolution is governed by a nonlinear partial differential equation closely related to that of one-dimensional viscoelasticity, the underlying static problem being of mixed type. In Model 2 the equation of motion is an integro—partial differential equation obtained from that in Model 1 by an averaging of the nonlinear term; the corresponding potential energy is nonlocal.After establishing global existence and uniqueness results, we consider the longtime behavior of the systems. We find that the two systems differ dramatically. In Model 1, for no solution does the energy tend to its global minimum ast . In Model 2, however, a large, dense set of solutions realize global minimizing sequences; in this case we are able to characterize, asymptotically, how energy escapes to infinity in wavenumber space in a manner that depends upon the smoothness of initial data. We also briefly discuss a third model that shares the stationary solutions of the second but is a gradient dynamical system.The models were designed to provide insight into the dynamical development of finer and finer microstructure that is observed in certain material phase transformations. They are also of interest as examples of strongly dissipative, infinite-dimensional dynamical systems with infinitely many unstable modes, the asymptotic fate of solutions exhibiting in the case of Model 2 an extreme sensitivity with respect to the initial data.     

Existence,uniqueness and positivity of solutions for BGK models for mixtures

We consider kinetic models for a multi component gas mixture without chemical reactions. In the literature, one can find two types of BGK models in order to describe gas mixtures. One type has a sum of BGK type interaction terms in the relaxation operator, for example the model described by Klingenberg, Pirner and Puppo [20] which contains well-known models of physicists and engineers for example Hamel [16] and Gross and Krook [15] as special cases. The other type contains only one collision term on the right-hand side, for example the well-known model of Andries, Aoki and Perthame [1]. For each of these two models [20] and [1], we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model [20] in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie [11].     

Existence and structure of solutions for a class of optimal control systems with discounting arising in economic dynamics

We study the existence and a turnpike property of solutions of a discrete-time control system with discounting and with a compact metric space of states. In our recent work for this discrete-time optimal control system without discounting we establish the turnpike property and show that it is stable under perturbations of an objective function. In the present paper we show that this turnpike property together with its stability also hold for the system with discounting.     

Existence and uniqueness of solutions for a diffusion model of host-parasite dynamics

Milner and Patton (J. Comput. Appl. Math., in press) introduced earlier a new approach to modeling host-parasite dynamics through a convection-diffusion partial differential equation, which uses the parasite density as a continuous structure variable. A motivation for the model was presented there, as well as results from numerical simulations and comparisons with those from other models. However, no proof of existence or uniqueness of solutions to the new model proposed was included there. In the present work the authors deal with the well posedness of that model and they prove existence and uniqueness of solutions, as well as establishing some asymptotic results.     

Existence of local solutions for the Boltzmann equation without angular cutoff

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff. We prove the existence and uniqueness of local classical solutions to the Cauchy problem, in the function space with Maxwellian type exponential decay with respect to the velocity variable. To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).