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Methods in the Nonlinear Analysis for the Study of Boundary Value Problems

This Ph.D. thesis deals with the qualitative analysis of several nonlinear elliptic equations arising in various processes at the interplay between nonlinear functional analysis, variational calculus, and mathematical physics. This work is composed of five chapters. In the first three chapters we are concerned with the following single-valued semilinear problems: (i) classical solutions
on the whole space (entire solutions) for nonlinear eigenvalue problems or logistic type equations in anisotropic media; (ii) weak solutions for a sub-critical perturbation of a linear eigenvalue problem with sign-changing potential. The proofs combine the maximum principle, elliptic estimates, and variational methods.
Chapters 4 and 5 deal essentially with quasilinear partial differential equations. Chapter 4 is mainly devoted to the study of some classes of quasilinear
eigenvalue problems in Sobolev spaces with variable exponent. In Chapter 5
we establish several existence results for a multivalued Schrödinger equation
on the whole spaces and for a Schrödinger elliptic system with discontinuous
nonlinearity. Our results in the last chapter extend a theorem a Rabinowitz
for a single-valued Schrödinger equation on the whole space.

Mostra/Nascondi contenuto.
Introduction The strongest explosive is neither toluene nor the atomic bomb, but the human idea. Grigore Moisil (1906-1973) Partial differential equations are of crucial importance in the modeling and the description of natural phenomena. Many physical phenomena from fluid dynamics, continuum mechanics, aircraft simulation, computer graphics and weather prediction are modeled by various partial differential equations. The central equations of general relativity and quantum mechanics are also partial differential equations. The motion of planets, computers, electric light, the working of GPS (Global Positioning System) and the changing weather can all be described by differential equations. The goal of this work is to apply some basic methods of the nonlinear analysis in order to develop a qualitative study of some classes of stationary partial differential equations. Their nonlinearities are essential for a realistic description of several natural questions, such as existence and uniqueness of solutions, asymptotic behaviour, approximation and so on. However, the tools for solving the equations, in particular the numerical tools, are rather general in this work, but they may have future relevance for other applied problems. We discuss some classes of nonlinear elliptic equations from the perspective of three basic methods: the maximum principle, the calculus of variations, and nonlinear operator theory. Our starting point is related to the Laplace opera- tor, but we emphasize various generalizations of the linear Laplace equation, including linear perturbations of the Laplace operator or quasilinear problems involving variable exponents. That is why we are concerned with classical 3

International thesis/dissertation

Facoltà: Matematică

Autore: Teodora-Liliana Dinu Radulescu Contatta »

Composta da 127 pagine.

 

Questa tesi ha raggiunto 121 click dal 29/04/2008.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.