Homogenization with jumping nonlinearities

Summary Let a bounded regular open set of R ^N (N1),and {A ,0} be a sequence of second order, uniformly elliptic operators, which G-converges to A ₀.Let gC(R, R)be a nonlinear function with «jumping nonlinearities» (that is ).For h L ² () given, we obtain some results of convergence of the (eventual) solutions of the equation A u =g(u) +h.For instance, we study the case so-called «Ambrosetti-Prodi equation», that is when –< ^– < ₁ < ⁺ < ₂ where ₁ and ₂ are the firts and the second) eigenvalues of A ₀.

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Résumé Soient un ouvert borné régulier de R ^N (N1), et {A ,0} une suite d'opérateurs sur , du 2éme ordre, uniformément elliptiques et qui G-converge vers A ₀.Soit gC (R, R)une fonction demi-linéaire à l'infini (c'est à dire telle que ).Pour hL ² () donné, on obtient des résultats de convergence pour les solutions (éventuelles) de l'équation A u =g(u) +h.Par exemple, on étudie le cas de «l'équation d' Ambrosetti-Prodi», c'est à dire le cas – _– < ₁) ⁺ < ₂,où ₁ et ₂ sont les lère et 2éme valeurs propres de A ₀.

     

Blow-up with logarithmic nonlinearities

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition,

     

Bell-shaped nonstationary refinable ripplets

We study the approximation properties of the class of nonstationary refinable ripplets introduced in Gori and Pitolli (2008). These functions are the solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions, convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the C ² nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented.     

Vector-valued nonstationary Gabor frames

As an extension of Gabor frames, nonstationary Gabor (NSG) frames were recently introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper we generalize the notion of NSG frames from $L^2(\mathbb{R})$ to the vector-valued Hilbert space $L^2(\mathbb{R},{\mathbb{C}}^L)$, and investigate the resulting vector-valued NSG frames. We derive a Walnut's representation of the mixed frame operator, and provide some necessary/sufficient conditions for a vector-valued NSG system to be a frame for $L^2(\mathbb{R},{\mathbb{C}}^L)$. Furthermore, we show the existence of painless vector-valued NSG frames, and of vector-valued NSG frames with fast decaying window functions.     

Queues with nonstationary inputs

In this paper we study single server queues with independent and identically distributed service times and a general nonstationary input stream. We discuss several notions of being in equilibrium. For queues with a doubly stochastic Poisson input we survey continuity and bounds of moments of some performance characteristics. We also discuss conjectures posed by Ross [34] to the effect that for a more stationary input we have a better performance characteristics. Some results are reviewed to typify a problem and then it is followed by a discussion, questions and related bibliography.     

Ten Issues About Hysteresis

In this note some basic models of hysteresis are reviewed: the linear and generalized plays, the relay and the Preisach model. Some related questions are also illustrated.     

Polynomial nonlinearities in differential equations

Adomian and his collaborators have found solutions for nonlinear stochastic (or deterministic in the limiting case) differential equations involving polynomial non-linearities. Using the first author's A_n polynomials and recently developed methods of calculating these polynomials, it becomes very easy to write solutions for non-linear terms such as y^m for positive integers m. The cases of negative integer m or even decimal powers will appear elsewhere.     

Multiplicity and eigenvalue intersecting nonlinearities

Summary Several results on the existence of multiple solutions are proved for nonlinear Sturm-Liouville equations with nonlinearities which cross asymptotically some eigenvalues of the linear operator (so-called jumping nonlinearities).By a change of variable the problem is transformed into a bifurcation equation. The bifurcation branches of this equation are seen to start in linear eigenvalues and to end asymptotically in nonlinear eigenvalues (split eigenvalues). This allows to deduce in a unified way several multiplicity results.     

Variational inequalities with strong nonlinearities

We study a connection between critical values and topological characteristics of non-smooth functionals. We establish analogs of theorems about regular interval and “nek”. We also find lower estimates of solutions to variational inequalities with odd potential operators.