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.
Preface
Chapter 1 : Defining Stability
Chapter 2 : The Gelfand Problem
Chapter 3 : Extremal Solutions
Chapter 4 : Regularity Theory of Stable Solutions
Chapter 5 : Singular Stable Solutions
Chapter 6 : Liouville Theorems for Stable Solutions
Chapter 7 : A Conjecture of De Giorgi
Chapter 8 : Further Readings
Appendix A : Maximum Principles
Appendix B: Regularity Theory for Elliptic Operators
Appendix C : Geometric Tools
References
Index
