
Preface  Preview Material  Table of Contents  Supplementary Material 
Mathematical Surveys and Monographs 2013; 299 pp; hardcover Volume: 187 ISBN10: 0821891529 ISBN13: 9780821891520 List Price: US$98 Member Price: US$78.40 Order Code: SURV/187 See also: Topics in Optimal Transportation  Cedric Villani Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS  Pierpaolo Esposito, Nassif Ghoussoub and Yujin Guo  The book describes how functional inequalities are often manifestations of natural mathematical structures and physical phenomena, and how a few general principles validate large classes of analytic/geometric inequalities, old and new. This point of view leads to "systematic" approaches for proving the most basic inequalities, but also for improving them, and for devising new onessometimes at will and often on demand. These general principles also offer novel ways for estimating best constants and for deciding whether these are attained in appropriate function spaces. As such, improvements of Hardy and HardyRellich type inequalities involving radially symmetric weights are variational manifestations of Sturm's theory on the oscillatory behavior of certain ordinary differential equations. On the other hand, most geometric inequalities, including those of Sobolev and LogSobolev type, are simply expressions of the convexity of certain free energy functionals along the geodesics on the Wasserstein manifold of probability measures equipped with the optimal mass transport metric. CaffarelliKohnNirenberg and HardyRellichSobolev type inequalities are then obtained by interpolating the above two classes of inequalities via the classical ones of Hölder. The subtle MoserOnofriAubin inequalities on the twodimensional sphere are connected to Liouville type theorems for planar mean field equations. Readership Graduate students and research mathematicians interested in analysis, calculus of variations, and PDEs. 


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