Stable Solutions of Elliptic Partial Differential Equations

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.

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

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 = 1Dimension N = 2 Dimension N ≥ 3 Summary
Extremal Solutions Weak solutions Stable weak solutionsThe 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 = 4Regularity 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 conjectureDimension 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 PrinciplesAppendix B: Regularity Theory for Elliptic OperatorsAppendix C: Geometric Tools

References
Index