Rearrangements and convexity of level sets in PDE. (English) Zbl 0593.35002

Kawohl, Bernhard

Lecture Notes in Mathematics. 1150. Berlin etc.: Springer-Verlag. V, 136 p. DM 21.50 (1985).

The first part of this work is a survey on rearrangement technique and its applications to partial differential equations and the calculus of variations. The author studies in detail the main types of rearrangements for functions of one and several variables and uses the general theory to prove convexity or symmetry properties of solutions of differential equations. The applications include the treatment of some free boundary problems such as the dam problem, the capacitary problem, the theory of jets and cavities.
The second part of the book concerns the applications of the maximum principle to obtain the same type of results. The work contains several original contributions of the author and solves many interesting problems and conjectures on the shape of free boundary of elliptic variational inequalities.

Reviewer: V.Barbu

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35R35	Free boundary problems for PDEs
35J25	Boundary value problems for second-order elliptic equations
26B25	Convexity of real functions of several variables, generalizations
26D10	Inequalities involving derivatives and differential and integral operators
35A15	Variational methods applied to PDEs
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50	Maximum principles in context of PDEs
35J20	Variational methods for second-order elliptic equations
35J65	Nonlinear boundary value problems for linear elliptic equations
49R50	Variational methods for eigenvalues of operators (MSC2000)

Keywords:

rearrangement technique; free boundary problems; dam problem; capacitary problem; theory of jets and cavities; maximum principle; elliptic variational inequalities