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.