Cabré X., Henrot A., Peralta-Salas D., Reichel W. Geometry of PDEs and Related Problems

Springer, 2018. — xi+198 p. — (Lecture Notes in Mathematics, 2220). — ISBN: 978-3-319-95186-7.The aim of this book is to present different aspects of the deep interplay between Partial Differential Equations and Geometry. It gives an overview of some of the themes of recent research in the field and their mutual links, describing the main underlying ideas, and providing up-to-date references.
Collecting together the lecture notes of the five mini-courses given at the CIME Summer School held in Cetraro (Cosenza, Italy) in the week of June 19-23, 2017, the volume presents a friendly introduction to a broad spectrum of up-to-date and hot topics in the study of PDEs, describing the state-of-the-art in the subject. It also gives further details on the main ideas of the proofs, their technical difficulties, and their possible extension to other contexts. Aiming to be a primary source for researchers in the field, the book will attract potential readers from several areas of mathematics.
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Цель этой книги - представить различные аспекты глубокого взаимодействия уравнений в частных производных и геометрии. В нем дается обзор некоторых тем последних исследований в этой области и их взаимных связей, описываются основные основополагающие идеи и приводятся современные справочные материалы.
Собрав воедино конспекты лекций пяти мини-курсов, проведенных в Летней школе CIME в Четраро (Козенца, Италия) в течение недели с 19 по 23 июня 2017 года, книга представляет собой дружественное введение в широкий спектр современных и горячих тем в изучении УрЧП, описывающих современное состояние в этой области. Он также дает более подробную информацию об основных идеях доказательств, их технических трудностях и их возможном распространении на другие контексты. В стремлении стать главным источником для исследователя в этой области, книга привлечет потенциальных читателей из нескольких областей математики.Stable Solutions to Some Elliptic Problems: Minimal Cones, the Allen-Cahn Equation, and Blow-Up Solutions Xavier Cabré and Giorgio Poggesi
Minimal Cones
The Simons Cone. Minimality
Hardy’s Inequality
Proof of the Simons Lemma
Comments On: Harmonic Maps, Free Boundary Problems, and Nonlocal Minimal Surfaces
The Allen-Cahn Equation
Minimality of Monotone Solutions with Limits±1
A Conjecture of De Giorgi
The Saddle-Shaped Solution Vanishing on the Simons Cone
Blow-Up Problems
Stable and Extremal Solutions: A Singular Stable Solution for n>10
Regularity of Stable Solutions. The Allard and Michael-Simon Sobolev InequalityIsoperimetric Inequalities for Eigenvalues of the Laplacian Antoine Henrot
Notation and Prerequisites
Notation and Sobolev Spaces
Eigenvalues and Eigenfunctions
Properties of Eigenvalues.
Some Examples
Min-Max Principles and Applications
Topological Derivative
The First Eigenvalue
The Faber-Krahn Inequality
A Quantitative Version of Faber-Krahn Inequality
The Case of Polygons
Domains in a Box
Multi-Connected Domains
The Second Eigenvalue
Minimizing λ₂
A Convexity Constrain
The Other Dirichlet Eigenvalues
Existence
Connectedness of Minimizers
Other Geometric ConstraintsTopological Aspects of Critical Points and Level Sets in Elliptic PDEs Alberto Enciso and Daniel Peralta-Salas
Introduction: Emergence of Topological Structures in Elliptic PDEs
Critical Points of Green’s Functions on Complete Manifolds
Li-Tam Green’s Functions
A Topological Upper Bound on Surfaces
Critical Points in Higher Dimensions
General Strategy and Two Technical Tools: Thom’s Isotopy Theorem and a Runge-Type Global Approximation Theorem
Thom’s Isotopy Theorem
A Runge-Type Global Approximation Theorem for the Helmholtz Equation with Optimal Decay at Infinity
Monochromatic Waves: Nodal Sets of Solutions to the Helmholtz Equation
Emergence of Knotted Structures in High-Energy Eigenfunctions: Berry’s Conjecture
The Linear Regime of Nonlinear Equations: Nodal Sets of the Allen-Cahn EquationSymmetry Properties for Solutions of Higher-Order Elliptic Boundary Value Problems Wolfgang Reichel
Linear Problems: Weak Solutions, Eigenvalues, Regularity, Green Functions
Symmetry, Simplicity and Positivity for First Eigenfunctions
Symmetry for Nonlinear Problems by Uniqueness and Non-resonance
Symmetry via Moving Plane Method: An Example from Potential Theory
Asymptotic Expansion and the Role of the Barycenter
The Moving Plane Method
An Example of Symmetry in an Overdetermined 4th Order ProblemRecent Trends in Free Boundary Regularity Henrik ShahgholianBackground
Overview of the Content
A Catalog of Semi-classical FBPs
A Melting Ice Block
Hele-Shaw Flow
An Optimal Stopping Problem
Modeling Financial Derivatives
Smash Sums and Internal DLA
Mathematical Theory of Obstacle Problem
Existence, and Uniqueness
Optimal Regularity and Non-degeneracy of Solutions
Regularity of FB: Local and Global Analysis
Other Types of FBPs
Bernoulli Type FB
Broken PDEs with FB
Non-variational Problems
Nonlocal Problems, Extensions and Thin Obstacles
System Case
Optimal Switching
Minimization Problems