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Elliptic Systems of Phase Transition Type
This book focuses on the vector Allen-Cahn equation, which models coexistence of three or more phases and is related to Plateau complexes – non-orientable objects with a stratified structure. The minimal solutions of the vector equation exhibit an analogous structure not present in the scalar Allen-Cahn equation, which models coexistence of two phases and is related to minimal surfaces. The 1978 De Giorgi conjecture for the scalar problem was settled in a series of papers: Ghoussoub and Gui (2d), Ambrosio and Cabré (3d), Savin (up to 8d), and del Pino, Kowalczyk and Wei (counterexample for 9d and above). This book extends, in various ways, the Caffarelli-Córdoba density estimates that pl...
Modern Problems in PDEs and Applications
The principal aim of the volume is gathering all the contributions given by the speakers (mini courses) and some of the participants (short talks) of the summer school "Modern Problems in PDEs and Applications" held at the Ghent Analysis and PDE Center from 23 August to 2 September 2023. The school was devoted to the study of new techniques and approaches for solving partial differential equations, which can either be considered or arise from the physical point of view or the mathematical perspective. Both sides are extremely important since theories and methods can be developed independently, aiming to gather each other in a common objective. The aim of the summer school was to progress and advance in the problems considered. Note that real-world problems and their applications are classical study trends in physical or mathematical modelling. The summer school was organised in a friendly atmosphere and synergy, and it was an excellent opportunity to promote and encourage the development of the subject in the community.
Une société hors de soi
L'Empire ottoman rassemblait des populations tres diverses par la confession, par l'ethnie, par la nationalite. Comment les habitants, par-dela ces appartenances distinctes, vivaient-ils et percevaient-ils la coexistence dans leur vie concrete? Que retiraient-ils de cette experience quotidienne de la difference? Quels moyens utilisaient-ils pour definir leur identite? Ce sont les habitants de la Smyrne des XVIIIe et XIXe siecles - principal port de la Mediterranee orientale et deuxieme ville ottomane en termes de population dans cette periode -, qui, a travers leurs histoires de vie, les liens qu'ils nouent, leurs manieres d'utiliser l'espace, leurs rattachements institutionnels, vont guider...
Instabilities and Nonequilibrium Structures
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perbaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van GuIik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate ar...
A Course in Minimal Surfaces
Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science. The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle.
The Greek Revolution
Winner of the 2022 London Hellenic Prize On the bicentennial of the Greek Revolution, an essential guide to the momentous war for independence of the Greeks from the Ottoman Empire. The Greek war for independence (1821–1830) often goes missing from discussion of the Age of Revolutions. Yet the rebellion against Ottoman rule was enormously influential in its time, and its resonances are felt across modern history. The Greeks inspired others to throw off the oppression that developed in the backlash to the French Revolution. And Europeans in general were hardly blind to the sight of Christian subjects toppling Muslim rulers. In this collection of essays, Paschalis Kitromilides and Constantin...
The Maximum Principle
Maximum principles are bedrock results in the theory of second order elliptic equations. This principle, simple enough in essence, lends itself to a quite remarkable number of subtle uses when combined appropriately with other notions. Intended for a wide audience, the book provides a clear and comprehensive explanation of the various maximum principles available in elliptic theory, from their beginning for linear equations to recent work on nonlinear and singular equations.
Blow-Up in Quasilinear Parabolic Equations
The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk ...
A Geometric Approach to Free Boundary Problems
We hope that the tools and ideas presented here will serve as a basis for the study of more complex phenomena and problems."--Jacket.
