Stable Solutions of Elliptic Partial Differential Equations

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