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This Brief is mainly devoted to two classical and related results: the existence of a right inverse of the divergence operator and the so-called Korn Inequalities. It is well known that both results are fundamental tools in the analysis of some classic differential equations, particularly in those arising in fluid dynamics and elasticity. Several connections between these two topics and improved Poincaré inequalities are extensively treated. From simple key ideas the book is growing smoothly in complexity. Beginning with the study of these problems on star-shaped domains the arguments are extended first to John domains and then to Hölder domains where the need of weighted spaces arises naturally. In this fashion, the authors succeed in presenting in an unified and concise way several classic and recent developments in the field. These features certainly makes this Brief useful for students, post-graduate students, and researchers as well.

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Introduction.- Divergence Operator.- Korn´s Inequalities.- Singular Domains.- Basic Equations of Continuum Mechanics.- Powers of the Distance to the Boundary as Ap Weights.- An Auxiliary Result on Holder a Domains.

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"This book is a very nice introduction to many central topics in PDEs regarding the existence of a right inverse of the divergence operator div p , which is strongly related to the Korn inequality, Stokes equation, etc. ... These features certainly make this Springer Briefs manuscript very useful for students, postgraduate students, and researchers as well. (Javier Soria, zbMATH 1394.35001, 2018)

"This book gives a detailed discussion of two basic inequalities that are of fundamental importance in a variety of fields and discusses how their validity connects to geometric properties of the underlying domain. ... The book is nicely written, hints at open problems and is supplemented with many informative figures." (Stefan Steinerberger, Mathematical Reviews, October, 2017)

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