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Numerical Methods for Nonlinear Elliptic Differential EquationsA Synopsis$
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Klaus Boehmer

Print publication date: 2010

Print ISBN-13: 9780199577040

Published to Oxford Scholarship Online: January 2011

DOI: 10.1093/acprof:oso/9780199577040.001.0001

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

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

Chapter:
(p.32) 2 Elements of analysis for linear and nonlinear partial elliptic differential equations and systems
Source:
Numerical Methods for Nonlinear Elliptic Differential Equations
Author(s):

Klaus Böhmer

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199577040.003.0002

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.

Keywords:   general linear problems, special semilinear problems, semilinear problems, quasi-linear problems, fully nonlinear elliptic problems, existence, uniqueness, regularity, linearization

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