Inequalities for differential forms. (English) Zbl 1184.53001

Agarwal, Ravi P.; Ding, Shusen; Nolder, Craig

Berlin: Springer (ISBN 978-0-387-36034-8/hbk; 978-0-387-68417-8/ebook). xvi, 387 p. (2009).

This is a book to present a series of estimates and inequalities for differential forms, in particular the forms satisfying the homogeneous A-harmonic equations, the non-homogeneous A-harmonic equations, and the conjugate A-harmonic equations. The Hardy-Littlewood inequality, the Poincaré inequality, the Caccioppoli inequality, the Sobolev imbedding inequalities, the reverse Hölder inequalities etc. are extended. Particular integral estimates are devoted to the homotopy operator, Laplace-Beltrami operator, the gradient operator, Jacobian of a quasiconformal mappings. All the proofs are carefully written in details.
This material contains the results obtained by these authors in the last few years. As in the title of the book, only inequalities for differential forms are studied, nothing is concerned with the topology of the underlying manifolds.

Reviewer: Chang Kungching (Beijing)

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
44-02	Research exposition (monographs, survey articles) pertaining to integral transforms
49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
58-02	Research exposition (monographs, survey articles) pertaining to global analysis

Keywords:

differential forms; Laplace-Beltrami operator; Jacobian; Hardy-Littlewood inequality