Singularly Perturbed Nonlinear Elliptic Problems Of Second Order

A systematic introduction to the singularly perturbed methods in the study of concentration solutions for nonlinear elliptic problems.

Singularly perturbed second order elliptic partial differential equations with Neumann boundary conditions arise in many areas of application. These problems rarely have smooth limit solutions. In this paper, the author characterize the limit solution for a wide class of such problems. They also give an abstract rate of convergence theorem and apply the abstract theorem to certain finite difference approximations. Keywords: Viscosity solution; and Viscosity inequalities.

Numerical Solution of Nonlinear Elliptic Problems Via Preconditioning Operators - Theory & Applications

No detailed description available for "Singularly Perturbed Differential Equations".

The subject of this textbook is the mathematical theory of singular perturbations, which despite its respectable history is still in a state of vigorous development. Singular perturbations of cumulative and of boundary layer type are presented. Attention has been given to composite expansions of solutions of initial and boundary value problems for ordinary and partial differential equations, linear as well as quasilinear; also turning points are discussed. The main emphasis lies on several methods of approximation for solutions of singularly perturbed differential equations and on the mathematical justification of these methods. The latter implies a priori estimates of solutions of differential equations; this involves the application of Gronwall's lemma, maximum principles, energy integrals, fixed point theorems and Gåding's theorem for general elliptic equations. These features make the book of value to mathematicians and researchers in the engineering sciences, interested in the mathematical justification of formal approximations of solutions of practical perturbation problems. The text is selfcontained and each chapter is concluded with some exercises.

This accessible monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. It provides rapid access to recent results and references.

14 lectures by the invited speakers and 14 shorter contributions from the other speakers (pref.)