Dupaigne Stable Solutions of Elliptic Partial Differential Equations

Erscheinungsjahr 2011
ISBN: 978-1-4200-6655-5
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 335 Seiten

Reihe: Monographs and Surveys in Pure and Applied Mathematics

Stable solutions are ubiquitous in differential equations. They represent meaningful solutions from a physical point of view and appear in many applications, including mathematical physics (combustion, phase transition theory) and geometry (minimal surfaces).

Stable Solutions of Elliptic Partial Differential Equations offers a self-contained presentation of the notion of stability in elliptic partial differential equations (PDEs). The central questions of regularity and classification of stable solutions are treated at length. Specialists will find a summary of the most recent developments of the theory, such as nonlocal and higher-order equations. For beginners, the book walks you through the fine versions of the maximum principle, the standard regularity theory for linear elliptic equations, and the fundamental functional inequalities commonly used in this field. The text also includes two additional topics: the inverse-square potential and some background material on submanifolds of Euclidean space.

Zielgruppe

Researchers and graduate students in partial differential equations, differential geometry, and mathematical physics.

Autoren/Hrsg.

Dupaigne, Louis

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Defining Stability

Stability and the variations of energy

Linearized stability

Elementary properties of stable solutions

Dynamical stability

Stability outside a compact set

Resolving an ambiguity

The Gelfand Problem

Motivation

Dimension N = 1
Dimension N = 2

Dimension N = 3

Summary

Extremal Solutions

Weak solutions

Stable weak solutions
The stable branch
Regularity Theory of Stable Solutions

The radial case

Back to the Gelfand problem

Dimensions N = 1, 2,3

A geometric Poincaré formula

Dimension N = 4
Regularity of solutions of bounded Morse index
Singular Stable Solutions

The Gelfand problem in the perturbed ball

Flat domains

Partial regularity of stable solutions in higher dimensions
Liouville Theorems for Stable Solutions

Classifying radial stable entire solutions

Classifying stable entire solutions

Classifying solutions that are stable outside a compact set
A Conjecture of E De Giorgi

Statement of the conjecture

Motivation for the conjecture
Dimension N = 2

Dimension N = 3
Further Readings

Stability versus geometry of the domain

Symmetry of stable solutions

Beyond the stable branch

The parabolic equation

Other energy functional
Appendix A: Maximum Principles
Appendix B: Regularity Theory for Elliptic Operators
Appendix C: Geometric Tools

References

Index

Über Autor(innen)

Louis Dupaigne is an assistant professor at Université Picardie Jules Verne in Amiens, France.