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Stable Solutions of Elliptic Partial Differential Equations

By Louis Dupaigne

Series Editor: Haim Brezis

Published March 15th 2011 by Chapman and Hall/CRC – 335 pages

Series: Monographs and Surveys in Pure and Applied Mathematics

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Description

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.

Contents

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

Author Bio

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

Name: Stable Solutions of Elliptic Partial Differential Equations (Hardback) – Chapman and Hall/CRC 
Description: By Louis DupaigneSeries Editor: Haim Brezis. 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...
Categories: Differential Equations, Mathematical Physics