Maximum principles for elliptic partial differential equations. (English) Zbl 1193.35024

Chipot, Michel (ed.), Handbook of differential equations: Stationary partial differential equations. Vol. IV. Amsterdam: Elsevier/North Holland (ISBN 978-0-444-53036-3/hbk). Handbook of Differential Equations, 355-483 (2007).
In this paper, the authors provide a clear explanation of the various maximum principles available for elliptic second order equations, from their beginnings in linear theory to recent work on nonlinear equations, operators and inequalities.
The first Chapter (6.2) concerns tangency and comparison theorems. Section 2.1 includes in particular a discussion of Hopf’s nonlinear contributions. The authors treat then the case of quasilinear equations and inequalities considering both non-singular and singular cases, that is, in the latter case, equations which lose ellipticity at special values of the gradient of solutions, particularly at critical point. The concern here with singular equations arises both from their importance in variational theory and applied mathematics, as well as their specific theoretical interest.
Section 2.4 and 2.5 are devoted specifically to \(C^1\) solutions of divergence structure inequalities, allowing both singular and non-singular operators.
The next Chapter (6.3) continues the study of divergence structure inequalities, but for more general operators for which the methods from (6.2) are inadequate. The principal results are:
the maximum principle for homogeneous inequalities (Section 3.2);
the “thin set” maximum principle (Section 3.3);
result for weakly singular inequalities (Section 3.5);
result for strong singular inequalities (Section 3.6)
The authors give also some maximum principles in Section 3.7 and a series of uniqueness results in Section 3.8, results that extend theorems of Gilbarg and Trudinger for the Dirichlet problem.
Chapter 6.4 is concerned with the strong maximum principle and the compact support principle for singular quasilinear differential inequalities
\[ \text{div}\left\{ A\left( \left| Du\right| \right) Du\right\} +B\left( x,u,Du\right) \leq 0 \]
in a domain (connected open set) \(\Omega \) in \(\mathbb{R}^n\), under mild conditions of ellipticity, which allow both singular and degenerate behavior of the function \(A\) at \(s=0\), that is at critical points of \(u\).
The final chapter includes recent applications of the maximum principle to Liouville theorems and dead core problems, and to differential inequalities on Riemannian manifolds.
For the entire collection see [Zbl 1179.35001].


35J15 Second-order elliptic equations
35B50 Maximum principles in context of PDEs
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B51 Comparison principles in context of PDEs
35J62 Quasilinear elliptic equations
35J75 Singular elliptic equations
35R45 Partial differential inequalities and systems of partial differential inequalities