Umfang:
Online-Ressource (XX, 476 p. 18 illus., 1 illus. in color, digital)
Ausgabe:
2nd ed. 2013
ISBN:
9783034805131
Serie:
International Series of Numerical Mathematics 153
Inhalt:
Preface -- Preface to the 2nd edition -- Notational conventions -- 1 Preliminary general material -- I Steady-state problems -- 2 Pseudomonotone or weakly continuous mappings -- 3 Accretive mappings -- 4 Potential problems: smooth case -- 5 Nonsmooth problems; variational inequalities -- 6. Systems of equations: particular examples -- II Evolution problems -- 7 Special auxiliary tools -- 8 Evolution by pseudomonotone or weakly continuous mappings -- 9 Evolution governed by accretive mappings -- 10 Evolution governed by certain set-valued mappings -- 11 Doubly-nonlinear problems -- 12 Systems of equations: particular examples -- References -- Index.
Inhalt:
This book primarily concerns quasilinear and semilinear elliptic and parabolic partial differential equations, inequalities, and systems. The exposition leads the reader through the general theory based on abstract (pseudo-) monotone or accretive operators as fast as possible towards the analysis of concrete differential equations, which have specific applications in continuum (thermo-) mechanics of solids and fluids, electrically (semi-) conductive media, modelling of biological systems, or in mechanical engineering. Selected parts are mainly an introduction into the subject while some others form an advanced textbook. The second edition simplifies and extends the exposition at particular spots and augments the applications especially towards thermally coupled systems, magnetism, and more. The intended audience is graduate and PhD students as well as researchers in the theory of partial differential equations or in mathematical modelling of distributed parameter systems. ------ The monograph contains a wealth of material in both the abstract theory of steady-state or evolution equations of monotone and accretive type and concrete applications to nonlinear partial differential equations from mathematical modeling. The organization of the material is well done, and the presentation, although concise, is clear, elegant and rigorous. (…) this book is a notable addition to the existing literature. Also, it certainly will prove useful to engineers, physicists, biologists and other scientists interested in the analysis of (...) nonlinear differential models of the real world. (Mathematical Reviews).
Anmerkung:
Description based upon print version of record
,
Nonlinear Partial Differential Equations with Applications; Contents; Preface; Preface to the 2nd edition; Notational conventions; Chapter 1 Preliminary general material; 1.1 Functional analysis; 1.1.1 Normed spaces, Banach spaces, locally convex spaces; 1.1.2 Functions and mappings on Banach spaces, dual spaces; 1.1.3 Convex sets; 1.1.4 Compactness; 1.1.5 Fixed-point theorems; 1.2 Function spaces; 1.2.1 Continuous and smooth functions; 1.2.2 Lebesgue integrable functions; 1.2.3 Sobolev spaces; 1.3 Nemytski.i mappings; 1.4 Green formula and some inequalities; 1.5 Bochner spaces
,
1.6 Some ordinary differential equationsPart I STEADY-STATE PROBLEMS; Chapter 2 Pseudomonotone or weakly continuous mappings; 2.1 Abstract theory, basic definitions, Galerkin method; 2.2 Some facts about pseudomonotone mappings; 2.3 Equations with monotone mappings; 2.4 Quasilinear elliptic equations; 2.4.1 Boundary-value problems for 2nd-order equations; 2.4.2 Weak formulation; 2.4.3 Pseudomonotonicity, coercivity, existence of solutions; 2.4.4 Higher-order equations; 2.5 Weakly continuous mappings, semilinear equations; 2.6 Examples and exercises; 2.6.1 General tools
,
2.6.2 Semilinear heat equation of type -div(A(x, u)∇u) = g2.6.3 Quasilinear equations of type -div |∇u|p-2∇u +c(u,∇u)=g; 2.7 Excursion to regularity for semilinear equations; 2.8 Bibliographical remarks; Chapter 3 Accretive mappings; 3.1 Abstract theory; 3.2 Applications to boundary-value problems; 3.2.1 Duality mappings in Lebesgue and Sobolev spaces; 3.2.2 Accretivity of monotone quasilinear mappings; 3.2.3 Accretivity of heat equation; 3.2.4 Accretivity of some other boundary-value problems; 3.2.5 Excursion to equations with measures in right-hand sides; 3.3 Exercises
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3.4 Bibliographical remarksChapter 4 Potential problems: smooth case; 4.1 Abstract theory; 4.2 Application to boundary-value problems; 4.3 Examples and exercises; 4.4 Bibliographical remarks; Chapter 5 Nonsmooth problems; variational inequalities; 5.1 Abstract inclusions with a potential; 5.2 Application to elliptic variational inequalities; 5.3 Some abstract non-potential inclusions; 5.4 Excursion to quasivariational inequalities; 5.5 Exercises; 5.6 Some applications to free-boundary problems; 5.6.1 Porous media flow: a potential variational inequality
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5.6.2 Continuous casting: a non-potential variational inequality5.7 Bibliographical remarks; Chapter 6 Systems of equations: particular examples; 6.1 Minimization-type variational method: polyconvex functionals; 6.2 Buoyancy-driven viscous flow; 6.3 Reaction-diffusion system; 6.4 Thermistor; 6.5 Semiconductors; Part II EVOLUTION PROBLEMS; Chapter 7 Special auxiliary tools; 7.1 Sobolev-Bochner space W1,p,q(I; V1, V2); 7.2 Gelfand triple, embedding W1,p,p (I; V ,V *)⊂C(I; H); 7.3 Aubin-Lions lemma; Chapter 8 Evolution by pseudomonotone or weakly continuous mappings
,
8.1 Abstract initial-value problems
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Preface -- Preface to the 2nd edition -- Notational conventions -- 1 Preliminary general material -- I Steady-state problems -- 2 Pseudomonotone or weakly continuous mappings -- 3 Accretive mappings -- 4 Potential problems: smooth case -- 5 Nonsmooth problems; variational inequalities -- 6. Systems of equations: particular examples -- II Evolution problems -- 7 Special auxiliary tools -- 8 Evolution by pseudomonotone or weakly continuous mappings -- 9 Evolution governed by accretive mappings -- 10 Evolution governed by certain set-valued mappings -- 11 Doubly-nonlinear problems -- 12 Systems of equations: particular examples -- References -- Index.
Weitere Ausg.:
ISBN 9783034805124
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe Roubíček, Tomáš, 1956 - Nonlinear partial differential equations with applications Basel : Birkhäuser Verl., 2013 ISBN 3034805128
Weitere Ausg.:
ISBN 9783034805124
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Nichtlineare partielle Differentialgleichung
;
Nichtlineare partielle Differentialgleichung
;
Lehrbuch
;
Bibliografie
DOI:
10.1007/978-3-0348-0513-1
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Mehr zum Autor:
Roubíček, Tomáš 1956-
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