J. Ferrera, J. Lopez-Gomez, F.R. Ruiz del Portal0444518614, 9780444518613, 9780080459196
This book covers a wide range of new mathematical results. Among them, the most advanced characterisations of very weak versions of the classical maximum principle, the very last results on global bifurcation theory, algebraic multiplicities, general dependencies of solutions of boundary value problems with respect to variations of the underlying domains, the deepest available results in rapid monotone schemes applied to the resolution of non-linear boundary value problems, the intra-history of the the genesis of the first general global continuation results in the context of periodic solutions of nonlinear periodic systems, as well as the genesis of the coincidence degree, some novel applications of the topological degree for ascertaining the stability of the periodic solutions of some classical families of periodic second order equations, the resolution of a number of conjectures related to some very celebrated approximation problems in topology and inverse problems, as well as a number of applications to engineering, an extremely sharp discussion of the problem of approximating topological spaces by polyhedra using various techniques based on inverse systems, as well as homotopy expansions, and the Bishop-Phelps theorem.
Key features:
– It contains a number of seminal contributions by some of the most world leading mathematicians of the second half of the 20th Century.
– The papers cover a complete range of topics, from the intra-history of the involved mathematics to the very last developments in Differential Equations, Inverse Problems, Analysis, Nonlinear Analysis and Topology.
– All contributed papers are self-contained works containing rather complete list of references on each of the subjects covered.
– The book contains some of the very last findings concerning the maximum principle, the theory of monotone schemes in nonlinear problems, the theory of algebraic multiplicities, global bifurcation theory, dynamics of periodic equations and systems, inverse problems and approximation in topology.
– The papers are extremely well written and directed to a wide audience, from beginners to experts. An excellent occasion to become engaged with some of the most fruitful mathematics developed during the last decades. · It contains a number of seminal contributions by some of the most world leading mathematicians of the second half of the 20th Century. · The papers cover a complete range of topics, from the intra-history of the involved mathematics to the very last developments in Differential Equations, Inverse Problems, Analysis, Nonlinear Analysis and Topology. · All contributed papers are self-contained works containing rather complete list of references on each of the subjects covered. · The book contains some of the very last findings concerning the maximum principle, the theory of monotone schemes in nonlinear problems, the theory of algebraic multiplicities, global bifurcation theory, dynamics of periodic equations and systems, inverse problems and approximation in topology. · The papers are extremely well written and directed to a wide audience, from beginners to experts. An excellent occasion to become engaged with some of the most fruitful mathematics developed during the last decades.
Table of contents :
Cover……Page 1
Title……Page 2
Title page……Page 3
Date-line……Page 4
Contents……Page 5
Preface……Page 9
1 Introduction……Page 13
2 Elliptic boundary value problems……Page 18
3 Notations and conventions……Page 20
4 Weak, maximum principles……Page 22
5 Nonhomogeneous problems……Page 24
6 The principal eigenvalue……Page 25
7 The strong maximum principle……Page 27
8 Mbnotonicily of the principal eigenvalue……Page 29
9 Continuity of the principal eigenvalue……Page 32
10 Minimax characterizations……Page 36
11 Concavity of the principal eigenvalue……Page 38
12 Preparatory considerations……Page 39
13 The strong maximum principle for the scalar case……Page 43
14 Strong and weak solutions……Page 44
15 Resolvent positivity……Page 47
16 Proofs of the weak maximum principles……Page 49
17 Bounded Domains……Page 50
18 Domain perturbations……Page 55
19 Elliptic comparison theorems……Page 64
References……Page 68
1 Anti-Cech Approximation……Page 73
1.2 Coarse category……Page 74
1.3 Cecil and anli-Cech approximations……Page 76
1.4 Geometry or nerves……Page 78
1.5 Property A……Page 81
1.6 Coarse approach to the Novikov conjecture……Page 86
1.7 Coarse embeddings……Page 87
1.8 Expanders……Page 89
1.9 Polynomial dimension growth……Page 91
1.10 Nonpositively curved manifolds……Page 93
2 Mapping Intersection Problem……Page 94
2.1 Cohomological dimension……Page 95
2.2 Extending maps to CW complexes……Page 97
2.3 Negligibility Criterion……Page 98
2.4 Reduction to other approximation problems……Page 100
2.5 The codimension three case……Page 103
References……Page 104
1 Introduction……Page 107
2 Regular domain perturbations……Page 108
2.1 Torsional rigidity……Page 110
2.2 Eigenvalues……Page 111
2.3 Bifurcation and generecity……Page 113
3.1 General elliptic operators……Page 115
3.2 Operators in divergence form……Page 117
4 Neumann conditions and irregular perturbations……Page 120
4.1 Perturbations near boundary points……Page 121
4.2 Dumbbell shaped domains……Page 124
4.3 Thin domains……Page 126
4.4 General variations……Page 130
References……Page 132
1 Introduction……Page 137
2 Successive approximations……Page 138
3 Personal circumstances……Page 141
4 Monotone approximations……Page 143
5 Rapid convergence……Page 152
References……Page 160
1 Introduction……Page 163
2 General assumptions and basic concepts……Page 165
3 A brief introduction to the topological degree……Page 167
4 Topological characterization of nonlinear eigenvalues……Page 171
5 Algebraic characterizations of nonlinear eigenvalues……Page 173
6 Global behaviour of compact components……Page 181
References……Page 185
1 Introduction……Page 189
2 Properties preserved under inverse limits……Page 193
3 Spaces as limits of polyhedral systems with additional properties……Page 196
4 Resolutions of spaces……Page 198
5 Approximate inverse systems……Page 200
6 Approximate resolutions of spaces……Page 202
7 Homotopy expansions of spaces……Page 203
References……Page 207
1 Introduction……Page 211
2 Weakly nonlinear systems……Page 212
3 Cesari’s method for strongly nonlinear systems……Page 214
4 topological degree and Cronin’s monograph……Page 216
5 Injecting Brouwer degree in Cesari’s method……Page 218
6 Applying Leray-Schauder’s degree……Page 219
7 Learning from history……Page 222
References……Page 224
1 Introduction……Page 227
2 Perpetual stability and discrete dynamical systems……Page 229
3 The linear equation and the symplectic group……Page 230
4 Degree theory and index of zeros……Page 234
5 The index of an equilibrium……Page 236
6 Stability and index……Page 240
7 The pendulum of variable length……Page 242
References……Page 245
1 Introduction……Page 247
3 Operators which attain their norm……Page 250
4 Topological and set-theoretic properties of support points……Page 251
5 Non-support points……Page 252
6 Generalizations of the Bishop-Phelps proof……Page 253
References……Page 254
1 Introduction……Page 257
1.1 Approximation of the derivative of noisy data……Page 258
1.2 Property C: completeness of the set of products of solutions to homogeneous PDE……Page 259
1.3 Approximation by entire functions of exponential type……Page 260
2 Stable approximation of the derivative from noisy data……Page 261
3.1 Genericity of Property C……Page 263
3.2 Approximating by scattering solutions……Page 266
4 Approximation by entire functions of exponential type……Page 269
References……Page 272
Index……Page 275
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