**CRM Proceedings & Lecture Notes**

Volume: 44;
2008;
267 pp;
Softcover

MSC: Primary 35; 47;

**Print ISBN: 978-0-8218-4350-5
Product Code: CRMP/44**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-3958-3
Product Code: CRMP/44.E**

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

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# Singularities in PDE and the Calculus of Variations

Share this page *Edited by *
*Stanley Alama; Lia Bronsard; Peter J. Sternberg*

A co-publication of the AMS and Centre de Recherches Mathématiques

This book contains papers presented at the "Workshop on Singularities in PDE and the Calculus of Variations" at the CRM in July 2006. The main theme of the meeting was the formation of geometrical singularities in PDE problems with a variational formulation. These equations typically arise in some applications (to physics, engineering, or biology, for example) and their resolution often requires a combination of methods coming from areas such as functional and harmonic analysis, differential geometry and geometric measure theory. Among the PDE problems discussed were: the Cahn-Hilliard model of phase transitions and domain walls; vortices in Ginzburg-Landau type models for superconductivity and superfluidity; the Ohna-Kawasaki model for di-block copolymers; models of image enhancement; and Monge-Ampère functions. The articles give a sampling of problems and methods in this diverse area of mathematics, which touches a large part of modern mathematics and its applications.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students and research mathematicians interested in PDEs and calculus of variation.

# Table of Contents

## Singularities in PDE and the Calculus of Variations

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Variational problems arising in biology 110
- On the Cauchy problem for phase and vortices in the parabolic Ginzburg-Landau equation 1120
- Nonlocal Cahn-Hilliard and isoperimetric problems: Periodic phase separation induced by competing long- and short-range interactions 3342
- On a generalized Ginzburg-Landau energy for superconducting/normal composite materials 4756
- Global questions for map evolution equations 6170
- Pohožaev-type identities for an elliptic equation 7584
- Some remarks on Monge-Ampère functions 8998
- Variational versus pde-based approaches in mathematical image processing 113122
- On the energy of a Chern-Simons-Higgs vortex lattice 127136
- Some recent results about a class of singularly perturbed elliptic equations 153162
- The dipole problem for 𝐻^{1/2}(𝕊²;𝕊¹)-maps and application 165174
- Hodge decompositions, Γ-convergence and the Gross-Pitaevskii energy 179188
- Bifurcation of vortex solutions to a Ginzburg-Landau equation in an annulus 187196
- An Allen-Cahn type problem with curvature modification 201210
- Rare events, action minimization, and sharp interface limits 217226
- The Gauss-Green theorem for weakly differentiable vector fields 233242
- Back Cover Back Cover1280