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El. knyga: Geometric Partial Differential Equations

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This book is the outcome of a conference held at the Centro De Giorgi of the Scuola Normale of Pisa in September 2012. The aim of the conference was to discuss recent results on nonlinear partial differential equations, and more specifically geometric evolutions and reaction-diffusion equations. Particular attention was paid to self-similar solutions, such as solitons and travelling waves, asymptotic behaviour, formation of singularities and qualitative properties of solutions. These problems arise in many models from Physics, Biology, Image Processing and Applied Mathematics in general, and have attracted a lot of attention in recent years.
Preface ix
Authors' affiliations xi
On the structure of phase transition maps for three or more coexisting phases
1(32)
Nicholas D. Alikakos
1 Introduction
1(1)
2 The basics for general potentials
2(2)
3 Symmetric phase transition potentials Existence of equivariant connection maps
4(3)
4 A related Bernstein-type theorem -- Background
7(9)
4.1 Minimizing partitions
9(1)
4.2 Flat chains with coefficients in a group
10(1)
4.3 Flat chains of top dimension
11(1)
4.4 The group of surface tension coefficients
12(1)
4.5 Basics on minimizing chains
13(3)
5 A related Bernstein-type theorem -- Statements and proofs
16(8)
6 The hierarchical structure of equivariant connection maps
24(9)
References
28(5)
The nonlinear multidomain model: a new formal asymptotic analysis
33(42)
Stefano Amato
Giovanni Bellettini
Maurizio Paolini
1 Introduction
33(4)
2 Star-shaped combination of star bodies and of anisotropies
37(5)
2.1 On the hessian of the combined anisotropy
41(1)
3 The bidomain model
42(4)
4 The nonlinear multidomain model
46(2)
5 Formal asymptotics of the multidomain model
48(27)
5.1 Outer expansion
49(2)
5.2 Inner expansion
51(21)
References
72(3)
Existence and qualitative properties of isoperimetric sets in periodic media
75(18)
Antonin Chambolle
Michael Goldman
Matteo Novaga
1 Introduction
75(2)
2 Plane-like minimizers
77(1)
3 Strict convexity and differentiability properties of the stable norm
78(7)
4 Existence and asymptotic behavior of isoperimetric sets
85(8)
References
90(3)
Minimizing movements and level set approaches to nonlocal variational geometric flows
93(12)
Antonin Chambolle
Massimiliano Morini
Marcello Ponsiglione
1 Introduction
93(1)
2 Generalized perimeters and curvatures
94(3)
2.1 Notion of generalized perimeter
94(1)
2.2 Definition of the curvature
95(1)
2.3 Assumptions on the curvature κ
96(1)
2.4 Semi-continuous extensions
96(1)
3 Examples of generalized curvatures
97(3)
3.1 The Euclidean perimeter
97(1)
3.2 The fractional perimeter
98(1)
3.3 The pre-Minkowski content
98(2)
4 The curvature flow: notion of viscosity solutions
100(2)
4.1 The curvature flow
100(1)
4.2 Definition of the viscosity solutions
100(1)
4.3 Existence of viscosity solutions
101(1)
5 Variational curvature flows
102(1)
5.1 The time-discrete scheme
102(1)
5.2 Level by level minimizing movements
102(1)
6 Shrinking zebras
103(2)
References
103(2)
Homogenization with oscillatory Neumann boundary data in general domain
105(14)
Sunhi Choi
Inwon C. Kim
1 Introduction
105(4)
1.1 Main results and the discussion of main ideas
106(3)
2 The problem in the strip domain
109(2)
2.1 Preliminaries
109(1)
2.2 The problem in the strip domain
110(1)
3 The continuity of the homogenized slope
111(4)
3.1 Description of the perturbation of boundary data and a sketch of the proof
113(2)
4 Perturbed test function method
115(4)
References
117(2)
The analysis of shock formation in 3-dimensional fluids
119(20)
Demetrios Christodoulou
References
138(1)
Regularity of the extremal solutions for the Liouville system
139(6)
Louis Dupaigne
Alberto Farina
Boyan Sirakov
References
144(1)
On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force
145(26)
Mi-Ho Giga
Yoshikazu Giga
Atsushi Nakayasu
1 Introduction
145(3)
2 Definitions of generalized solutions
148(6)
2.1 Faceted functions
149(1)
2.2 Nonlocal curvature with a nonuniform driving force
150(2)
2.3 Admissible functions and definition of a generalized solution
152(2)
3 Effective region and canonical modification
154(4)
4 Perron type existence theorem
158(7)
5 Existence theorem for periodic initial data
165(6)
References
168(3)
On representation of boundary integrals involving the mean curvature for mean-convex domains
171(18)
Yoshikazu Giga
Giovanni Pisante
1 Introduction
171(2)
2 Preliminaries
173(2)
3 The measure -- div(ƒ(k)Δd)
175(3)
4 Main result
178(7)
5 The anisotropic case
185(4)
References
186(3)
Boundary regularity for the Poisson equation in reifenberg-flat domains
189(22)
Antoine Lemenant
Yannick Sire
1 The monotonicity Lemma
194(7)
2 An elementary computation
201(1)
3 Interior estimate
202(2)
4 Boundary estimate
204(2)
5 Global decay result
206(1)
6 Conclusion and main result
207(4)
References
208(3)
Limiting models in condensed matter Physics and gradient flows of 1-homogeneous functionals
211(16)
Matteo Novaga
Giandomenico Orlandi
1 Introduction
211(3)
1.1 Some examples
212(1)
1.2 Gradient flows
212(1)
1.3 Formulation for differential forms
213(1)
2 Gradient flow of Jk
214(4)
2.1 Dual formulation
215(1)
2.2 Non local obstacle-type problems
216(1)
2.3 Some properties of the gradient flow of Jn-1
217(1)
3 The functional I1
218(9)
3.1 Asymptotics for the Gross-Pitaevskii model
218(3)
3.2 Rotational symmetry and weighted TV minimization
221(1)
3.3 Contact Curves and vortex curves
222(3)
References
225(2)
Maximally localized Wannier functions: existence and exponential localization
227(24)
Adriano Pisante
1 Introduction
227(2)
2 Wannier functions and Bloch bundles
229(6)
3 The Marzari-Vanderbilt localization functional
235(4)
4 Regularity of minimizers and exponential localization
239(5)
5 Harmonic maps into U(m) and Liouville theorem
244(7)
References
247(4)
Flows by powers of centro-affine curvature
251
Alina Stancu
1 Introduction
251(3)
2 Gauss curvature bounds
254(5)
3 Analysis of the singularity
259(2)
4 Asymptotic behavior in dimension three
261
References
264
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