Singularities in Nonlinear Evolution Phenomena and Applications

, by Novaga, Matteo; Orlandi, Giandomenico

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ISBN: 9788876423437 | 8876423435
Cover: Paperback
Copyright: 8/3/2009

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This volume collects some contributed papers by the participants to the workshop "Singularities in nonlinear evolution phenomena and applications", which was held at the Centro di Ricerca Matematica E. De Giorgi, from May 26th to May 30th, 2008.The topic of the workshop was the formation and the evolution of singular structures, like systems of points, curves and surfaces.Such questions naturally arise in many models from physics, biology, image processing and applied mathematics in general, and have attracted a lot of attention in recent years. Their analysis requires sophisticated tools and an interdisciplinary approach, and poses new challenging mathematical problems.

Preface	p. ix
Authors' addresses	p. xi
Entire solutions to nonconvex variational elliptic systems in the presence of a finite symmetry group	p. 1
References	p. 24
Tissue repair modeling	p. 27
Introduction	p. 27
Wound healing	p. 30
Simple mathematical model	p. 30
Numerical simulation	p. 34
Dorsal closure	p. 39
Appendix: material and methods	p. 44
Movie captions	p. 44
References	p. 45
Asymptotic behavior of energy-bounded local minimizers of the Ginzburg-Landau functionals	p. 47
Introduction	p. 47
Improved compactness and ¿-convergence	p. 51
References	p. 56
Some applications of the least squares method to differential equations and related problems	p. 59
Introduction	p. 59
Least squares method for ODEs	p. 65
Least squares method and homogenization of transport equations	p. 69
The case without space derivatives	p. 72
The case with space derivatives	p. 75
Least squares method and scalar conservation laws	p. 76
The quasi-potential problem for Burgers' equation	p. 80
Least squares method and singular perturbations to mean curvature flow	p. 84
References	p. 85
Plane-like minimizers in periodic media: the cell problem	p. 89
Introduction	p. 89
A new construction for the plane-like minimizers	p. 94
Elimination of the external field and weaker coercivity	p. 99
The coercive case in equivalent to the case g = 0	p. 100
Weaker coercivity	p. 102
References	p. 104
Three long wave asymptotic regimes for the Nonlinear-Schröet;dinger Equation	p. 107
Introduction	p. 107
The Euler regime for (NLS)	p. 113
The approach of E. Grenier	p. 113
The modulated energy functional	p. 115
Linearizing around an approximate solution	p. 118
Beyond the breaking time	p. 123
The linear wave regine for (NLS)	p. 125
The KdV/KP-I regime for (NLS)	p. 128
The KdV regime for (NLS)	p. 128
The KdV/KP-I regime for (NLS) in smooth norms	p. 131
References	p. 133
A transport problem with queue penalization effect	p. 139
Introduction	p. 139
Optimal mass transportation: a very quick review	p. 141
Existence, uniqueness and characterization of the optimum	p. 142
Existence, uniqueness and characterization of the equilibrium	p. 144
The concept of equilibrium	p. 145
A direct proof in the case k = 2	p. 146
A comparison with Pareto optimum	p. 147
Dynamical evolution and convergence to the equilibrium	p. 147
Standard evolution	p. 148
Evolution with prudence and evolution with memory	p. 150
A proof in a simplified setting	p. 152
References	p. 155
Some remarks on Huisken's monotonicity formula for mean curvature flow	p. 157
Maximizing Huisken's monotonicity formula	p. 157
Applications	p. 164
A no-breathers result	p. 164
Singularities	p. 166
References	p. 168
Geometric strong segregation theory for compositionally asymmetric diblock copolymer melts	p. 171
References	p. 179
Symmetry of solutions of the Ginzburg-Landau equations: some new results and open problems	p. 183
Introduction and main results	p. 183
Sketch of the proof and further results related to Theorem 1.3	p. 186
Applications and open problems	p. 192
References	p. 196
Total variation minimization with an H-1 constraint	p. 201
Introduction and motivations	p. 201
Total variation minimization in image processing	p. 202
TV-H-1 minimization for image denoising and image decomposition	p. 205
TV-H-1 inpainting	p. 206
Numerical solution for TV-H-1 minimization	p. 209
An algorithm for TV-H-1 minimization	p. 212
Preliminaries	p. 212
Chambolle's algorithm for total variation minimization	p. 214
A generalization of Chambolle's algorithm for TV-H-1 minimization	p. 214
Applications	p. 222
Image denoising and decomposition	p. 222
Image inpainting	p. 223
Domain decomposition for TV-H-1 minimization	p. 227
References	p. 232
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