Chapter

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

Klaus Böhmer

in Numerical Methods for Nonlinear Elliptic Differential Equations

Published in print October 2010 | ISBN: 9780199577040
Published online January 2011 | e-ISBN: 9780191595172 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199577040.003.0002

Series: Numerical Mathematics and Scientific Computation

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

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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

Chapter.  39239 words. 

Subjects: Mathematical and Statistical Physics

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