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E-grāmata: Nonlinear Analysis and Semilinear Elliptic Problems

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Nonlinear Analysis and Semilinear Elliptic ProblemsPalielināt
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Graduate text explaining how methods of nonlinear analysis can be used to tackle nonlinear differential equations.

Many problems in science and engineering are described by nonlinear differential equations, which can be notoriously difficult to solve. Through the interplay of topological and variational ideas, methods of nonlinear analysis are able to tackle such fundamental problems. This graduate text explains some of the key techniques in a way that will be appreciated by mathematicians, physicists and engineers. Starting from elementary tools of bifurcation theory and analysis, the authors cover a number of more modern topics from critical point theory to elliptic partial differential equations. A series of Appendices give convenient accounts of a variety of advanced topics that will introduce the reader to areas of current research. The book is amply illustrated and many chapters are rounded off with a set of exercises.

Recenzijas

'In the reviewer's opinion, this book can serve very well as a textbook in topological and variational methods in nonlinear analysis. Even researchers working in this field might find some interesting material (at least the reviewer did).' Zentralblatt MATH

Papildus informācija

Graduate text explaining how methods of nonlinear analysis can be used to tackle nonlinear differential equations.
Preface ix
Preliminaries
1(12)
Differential calculus
1(3)
Function spaces
4(1)
Nemitski operators
5(2)
Elliptic equations
7(6)
Part I Topological methods
13(62)
A primer on bifurcation theory
15(11)
Bifurcation: definition and necessary conditions
15(3)
The Lyapunov--Schmidt reduction
18(1)
Bifurcation from the simple eigenvalue
19(7)
Topological degree, I
26(29)
Brouwer degree and its properties
26(4)
Application: the Brouwer fixed point theorem
30(1)
An analytic definition of the degree
31(7)
The Leray--Schauder degree
38(5)
The Schauder fixed point theorem
43(1)
Some applications of the Leray--Schauder degree to elliptic equations
44(8)
The Krasnoselski bifurcation theorem
52(2)
Exercises
54(1)
Topological degree, II: global properties
55(20)
Improving the homotopy invariance
55(2)
An application to a boundary value problem with sub- and super-solutions
57(3)
The Rabinowitz global bifurcation theorem
60(5)
Bifurcation from infinity and positive solutions of asymptotically linear elliptic problems
65(8)
Exercises
73(2)
Part II Variational methods, I
75(66)
Critical points: extrema
77(12)
Functionals and critical points
77(1)
Gradients
78(2)
Existence of extrema
80(2)
Some applications
82(4)
Linear eigenvalues
86(2)
Exercises
88(1)
Constrained critical points
89(11)
Differentiable manifolds, an outline
89(4)
Constrained critical points
93(2)
Manifolds of codimension one
95(2)
Natural constraints
97(3)
Deformations and the Palais--Smale condition
100(16)
Deformations of sublevels
100(1)
The steepest descent flow
101(4)
Deformations and compactness
105(2)
The Palais--Smale condition
107(2)
Existence of constrained minima
109(1)
An application to a superlinear Dirichlet problem
109(5)
Exercises
114(2)
Saddle points and min-max methods
116(25)
The mountain pass theorem
117(6)
Applications
123(6)
Linking theorems
129(6)
The Pohozaev identity
135(3)
Exercises
138(3)
Part III Variational methods, II
141(92)
Lusternik--Schnirelman theory
143(14)
The Lusternik--Schnirelman category
143(4)
Lusternik--Schnirelman theorems
147(8)
Exercises
155(2)
Critical points of even functionals on symmetric manifolds
157(20)
The Krasnoselski genus
157(3)
Existence of critical points
160(4)
Multiple critical points of even unbounded functionals
164(6)
Applications to Dirichlet boundary value problems
170(6)
Exercises
176(1)
Further results on elliptic Dirichlet problems
177(27)
Radial solutions of semilinear elliptic equation on Rn
177(3)
Boundary value problems with critical exponent
180(8)
Discontinuous nonlinearities
188(10)
Problems with concave-convex nonlinearities
198(5)
Exercises
203(1)
Morse theory
204(29)
A short review of basic facts in algebraic topology
204(8)
The Morse inequalities
212(12)
An application: bifurcation for variational operators
224(5)
Morse index of mountain pass critical points
229(6)
Exercises
235
Part IV Appendices
233(76)
Appendix 1 Qualitative results
241(11)
Appendix 2 The concentration compactness principle
252(10)
Appendix 3 Bifurcation for problems on Rn
262(12)
Appendix 4 Vortex rings in an ideal fluid
274(12)
Appendix 5 Perturbation methods
286(16)
Appendix 6 Some problems arising in differential geometry
302(7)
References 309(6)
Index 315


Antonio Ambrosetti is a Professor at SISSA, Trieste. Andrea Malchiodi is an Associate Professor at SISSA, Trieste.
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