Pucci P., Serrin J. The Maximum Principle

Basel, Boston, Berlin: Birkhäuser Verlag AG., 2007. — 240 p.Progress in Nonlinear Differential Equations and Their Applications. Volume 73ISBN: 978-3-7643-8144-8
e-ISBN: 978-3-7643-8145-5The maximum principles of Eberhard Hopf are classical and bedrock results of the theory of second order elliptic partial differential equations. They go back to the maximum principle for harmonic functions, already known to Gauss in 1839 on the basis of the mean value theorem. On the other hand, they carry forward to the maximum principles of Gilbarg, Trudinger and Serrin, and the maximum principles for singular quasilinear elliptic differential inequalities, a theory initiated particularly by Vazquez and Diaz in the 1980s, but with earlier intimations in the work of Benilan, Brezis and Crandall. The purpose of the present work is to provide a clear explanation of the various maximum principles available for second-order elliptic equations, from their beginnings in linear theory to recent work on nonlinear equations, operators and inequalities. While simple in essence, these results lend themselves to a quite remarkable number of subtle uses when combined appropriately with other notions.Introduction and PreliminariesNotation
Tangency and Comparison Theorems for Elliptic Inequalities
The contributions of Eberhard Hopf
Tangency and comparison principles for quasilinear inequalities
Maximum and sweeping principles for quasilinear inequalities
Comparison theorems for divergence structure inequalities
Tangency theorems via Harnack’s inequality
Uniqueness of the Dirichlet problem
The boundary point lemma
Appendix: Proof of Eberhard Hopf’s maximum principle
Notes
Problems
Maximum Principles for Divergence Structure Elliptic Differential Inequalities
Distribution solutions
Maximum principles for homogeneous inequalities
A maximum principle for thin sets
A comparison theorem in W1, p(Ω)
Comparison theorems for singular elliptic inequalities
Strongly degenerate operators
Maximum principles for non-homogeneous elliptic inequalities
Uniqueness of the singular Dirichlet problem
Appendix: Sobolev’s inequality
Notes
Problems
Boundary Value Problems for Nonlinear Ordinary Differential Equations
Preliminary lemmas
Existence theorems
Existence theorems on a half-line
The end point lemma
Appendix: Proof of Proposition 4.2.1
Problems
The Strong Maximum Principle and the Compact Support Principle
The strongmaximum principle
The compact support principle
A special case
Strongmaximum principle: Generalized version
A boundary point lemma
Compact support principle: Generalized version
Notes
Problems
Non-homogeneous Divergence Structure Inequalities
Maximum principles for structured inequalities
Proof of Theorems 6.1.1 and 6.1.2
Proof of Theorem 6.1.3 and the first part of Theorem 6.1.5
Proof of Theorem 6.1.4 and the second part of Theorem 6.1.5
The case p=1 and themean curvature equation
Notes
Problems
The Harnack Inequality
Local boundedness and the weak Harnack inequality
The Harnack inequality
Holder continuity
The case p > n
Appendix. The John–Nirenberg theorem
Notes
Problems
Applications
Cauchy–Liouville Theorems
Radial symmetry
Symmetry for overdetermined boundary value problems
The phenomenon of dead cores
The strong maximum principle for Riemannian manifolds
Problems
Bibliography
Subject Index
Author Index