zbMATH — the first resource for mathematics

Monotone flows and rapid convergence for nonlinear partial differential equations. (English) Zbl 1017.35001
Series in Mathematical Analysis and Applications. 7. London: Taylor and Francis. x, 318 p. (2003).
This book is a treatise on methods for proving the existence of solutions to nonlinear partial differential equations. Under some monotonicity assumptions on $$F$$ $$(Lu=F(x,u))$$, one obtains the existence of maximal and minimal solutions. This is combined with an iterative technique $$(Lu_{n+1}= F(x,u_n))$$. Using quasilinearization one gets the rapid convergence of monotone sequences. Here, an improvement over previous results is the relaxation of a convexity assumption. The book is divided in two parts where the first one deals with classical cases and the second with a variational approach and generalized solutions. Each part deals in a unified fashion with elliptic, parabolic (including impulsive parabolic in the first part) and hyperbolic equations. The appendices remind basics on Sobolev spaces and PDEs. One finds at the end a bibliography (85 entries) and a small index.

MSC:
 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35A35 Theoretical approximation in context of PDEs 35A15 Variational methods applied to PDEs 35D05 Existence of generalized solutions of PDE (MSC2000) 35R12 Impulsive partial differential equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)