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Inégalités isopérimétriques et applications en physique. (French) Zbl 0537.35002

Travaux en Cours. Paris: Hermann. 182 p. FF 130.00 (1984).
This exposition deals mainly with a particular kind of rearrangement technique, and its applications to partial differential equations of elliptic type. This and more general rearrangements were introduced by Steiner in order to prove the geometrical isoperimetric inequality. They were then successfully applied by Pólya and Szegö to variational problems in mathematical physics. Because of their great flexibility they were afterwords used in various fields and described in several books. The author felt the need to adding a further treatise on this matter after having worked in plasma problems. The rearrangement seemed to be an appropriate technique to handle those problems but because of their lack of regularity the classical approach had to be modified. The book is organized as follows. At first the main theoretical tools are discussed. Emphasis is put on having them in the most general form. A selection of classical results is then presented and at the end more recent work on variational inequalities and on the plasma problems are included. The text has the character of a lecture series and contains stimulating material for specialists working in this area.
Reviewer: C.Bandle

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J25 Boundary value problems for second-order elliptic equations
49J20 Existence theories for optimal control problems involving partial differential equations
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)