Elements of analysis for linear and nonlinear partial elliptic differential equations and systems

Abstract

Chapter 2 summarizes general linear, special semilinear, semilinear, quasilinear, and fully nonlinear elliptic differential equations and systems of order 2m, m ≥ 1, e.g. the above equations. Essential are existence, uniqueness, and regularity of their solutions and linearization. Many important arguments for linearization are discussed. It is assumed that the derivative of the nonlinear operator, evaluated in the exact (isolated) solution, is boundedly invertible, closely related to the numerically necessary condition of a (locally) well-conditioned problem. Bifurcation problems are delayed to the next book; ill-conditioned problems are not considered. Linearization is applicable to nearly all nonlinear elliptic problems. Its bounded invertibility yields the Fredholm alternative and the stability of space discretization methods. Some nonlinear, monotone problems exclude linearization.