Recent Developments in Nonlinear Analysis

The following sections are included:

• PREFACE

• CONTENTS

Given a three dimensional closed contact manifold (M³, α) and a nowhere singular Morse vector-field υ in its kernel, we sketch the construction of the space M/υ discussed in [4]. We also introduce spaces of immersed curves in M/υ and an action functional on these spaces. This is the first step in the completion of a program aimed at computing the homology for contact forms defined in [2] and [7].

In this work, generalized Sobolev spaces are considered. Existence of solutions for strongly nonlinear equation of infinite order of the form Au + g(x, u) = f is established. Here A is an operator from a Sobolev space type to its dual and g(x, s) is a lower order term satisfying a sign condition on s. We consider the case where the data f belongs to L¹.

In this paper, we shall be concerned with the existence result of the quasilinear elliptic equations of the form,

, where A is a Leray-Lions operator from into its dual. On the nonlinear lower order term g(x, u, ∇u), we assume that it is a Carathéodory function having natural growth with respect to |∇u|, but without assuming the sign condition. The main novelty of our work is a new technique based on a Poincaré's inequality. The right hand side f belongs to W^-1,P′(Ω).

We prove that for any λ ∈ IR, there is a sequence of eigencurves (µ_k(λ))_k for the nonlinear coupled elliptic system

by using min-max methods. We prove also via an homogeneity type condition that the eigenvector corresponding to the principal µ₁(λ) is bounded, positive and smooth. We end this work by proving that µ₁(λ) is simple.

In this paper we consider the solvability of nonlinear parabolic differential equations with discontinuous nonlinearities, subjected to nonlocal conditions. We are concerned with the existence of solutions. Our technique is based on the Green's function for linear parabolic partial differential equations, the maximum principle, fixed point theorems for multivalued maps and the method of lower and upper solutions.

In this paper we are interested in the problem:

where f is a Carathéodory function with polynomial growth in |∇u|, λ is a real number, µ a bounded Radon measure and p ≥ 2.

We summarize recent lectures devoted to the study of the existence and L^∞-regularity results of some Dirichlet problems associated to equations having degenerated coercivity in the principal part.

For the obstacle problem:

for which the unknowns are u and the free boundary ∂[u > ψ]. It is known¹ that the coincidence set [u = ψ] is contained in the set [f - ∆ψ ≥ 0]. In this work we introduce µ = (f - ∆ψ)⁺_χ[u>ψ], that characterizes the domain of non contact,² and we analyze a mixed formulation of the obstacle problem where µ appears as a Lagrange multiplier. We show, in particular, that the solution µ_h of the approached (by linear finite element) mixed problem converges to µ with an order of convergence that is an O(h), which is optimal seen the equivalent result on the approximation of the free boundary³ and.⁴

In this paper, we shall concern with the existence result of unilateral parabolic degenerated problems associated to the equations of the form

where A is a classical Leray-Lions operator acting from the weighted Sobolev space into its dual L^p′ (0, T, W^-1,p′ (Ω, w*)), while the datum f is assumed in L¹(Q_T).

The proof is based on the penalty methods.

Most common methods used in engineering practice to assess the seismic stability of slope consists of a pseudo static approach or traditional approaches based on the limit equilibrium or on the hypothesis of failure calculation. Such methods are widely used because of their simplicity and there is no need of sophisticated software for their application. However, they neglect important elements such as the soil deformability, the dynamic amplification and non linear soil behavior. The present study concerns the analysis of the seismic slopes stability using global dynamic approach. It will mainly focus on dynamic amplification in the slope under real earthquake records with frequency content close to the natural frequency of the slope. Analysis is conducted by numerical modeling using Flac3D finite difference program. Results are presented in the case of linear visco-elastic then elastoplastic behaviour for earthfill materials in order to elucidate the influence of plasticity.

In this paper, we consider the initial-boundary value problem of a nonlinear parabolic equation with the data belong to L¹ and no growth assumption is made on the nonlinearities. We establish the existence and partial uniqueness theorems of renormalized solutions.

In this paper, we show, in the framework of Orlicz Sobolev spaces, the existence of renormalized and entropy solutions of some nonlinear equations and the existence and uniqueness of a unilateral problem. The existence of weak and renormalized solution of a nonlinear equation is also presented in the framework of weighted Sobolev spaces.

An existence result is proved for a variational degenerated unilateral problems associated to the following equations

where A is a Leray-Lions operator acting from the weighted Sobolev space into its dual W^-1,p′(Ω, w*), while g(x, s, ξ) is a nonlinear term which has a growth condition with respect to ξ and a sign condition on s, i.e. g(x, s, ξ).s ≥ 0 for every s ϵ IR and for every x and ξ in their respective domains. The source term f is supposed to belong to W^-1,p′(Ω, w*).

In this paper, using an equivalent variational approach to a recent Ricceri's three critical points theorem,¹³ we obtain the existence of at least three non-trivial solutions of a Neumann problem for elliptic equations with variable exponents.

In this paper, we study the existence for two positive solutions to a nonhomogeneous elliptic equation of fourth order with a parameter λ such that . The first solution has a negative energy while the energy of the second one is positive for 0 < λ < λ₀ and negative for . The values λ₀ and are given under variational form and we show that every corresponding critical point is solution of the nonlinear elliptic problem (with a suitable multiplicative term).

In this survey we expose the fundamentals of the strongly nonlinear potential theory and relate some of its components. As applications, we establish a relation between this theory and Partial Differential Equations, and show whether the equation Δ_Au + h = 0 possesses a solution or not, for a fixed function h. Here Δ_A is the A-Laplacian which is the p-Laplacian Δ_p, when the Orlicz space L_A reduces to the Lebesgue space L^p.

Our aim in this note is to give a simplified proof of the convergence of Evans-Rezakhanlou Stochastic Model to the evolution surfaces model of sandpile.

This work deals with the existence of radial solutions for a nonlinear eigenvalue problem involving the p-biharmonic operator.

Floquet theory together with the resonating averages method (RAM) were used to solve the Schrödinger equation of periodically-time-dependent harmonic oscillators. The approach gives another way of solving this equation, and lead to identical analytical solutions to those published in the literature based on different other methods.

In this paper, we study the existence and uniqueness of renormalized solutions for the nonlinear degenerated parabolic problem

where a : Ω×[0, T]×ℝ^N → ℝ^N is a Carathéodory function satisfying the coercivity condition, the general growth condition and only the large monotonicity. The second term f belongs to L¹(Q) and u₀ ϵ L¹(Ω).

In this paper we prove the existence of solutions for quasilinear degenerated elliptic operators A(u) + g(x, u, ∇u) = f, where A is a Leray-Lions operator from into its dual, while g(x, s, ξ) is a nonlinear term which has a growth condition with respect to ξ and no growth with respect to s, but it satisfies a sign condition on s. The right hand side f is assumed to belong to L¹(Ω).

In this study, we present an adaptive combined fuzzy sliding mode controller for a class of nonlinear systems with unknown nonlinear dynamics. The proposed control strategy is based on the fuzzy logic and sliding mode control technique (SMC). The adaptive fuzzy logic system type Takagi-Sugeno (T-S) is used to approximate the unknown dynamics of system. In order to assure the robustness on stability and tracking performance, the sliding mode control term is added in the control law. The latter consists to suppress the influence of external disturbances and attenuate fuzzy approximation error. Simulation results illustrate the good performances of the system when the proposed control technique is applied.

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