Functional inequalities : new perspectives and new applications
 Author/Creator
 Ghoussoub, N. (Nassif), 1953
 Language
 English.
 Publication
 Providence, Rhode Island : American Mathematical Society, [2013]
 Physical description
 xxiv, 299 pages ; 26 cm.
 Series
 Mathematical surveys and monographs ; no. 187.
Access
Available online

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QA3 .A4 V.187

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QA3 .A4 V.187
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Contributors
 Contributor
 Moradifam, Amir, 1980
Contents/Summary
 Bibliography
 Includes bibliographical references and index.
 Contents

 Hardy type inequalities: Bessel pairs and Sturm's oscillation theory The classical Hardy inequality and its improvements Improved Hardy inequality with boundary singularity Weighted Hardy inequalities The Hardy inequality and second order nonlinear eigenvalue problems HardyRellich type inequalities: Improved HardyRellich inequalities on $H^2_0(\Omega)$ Weighted HardyRellich inequalities on $H^2(\Omega)\cap H^1_0(\Omega)$ Critical dimensions for $4^{\textrm{th}}$ order nonlinear eigenvalue problems Hardy inequalities for general elliptic operators: General Hardy inequalities Improved Hardy inequalities for general elliptic operators Regularity and stability of solutions in nonselfadjoint problems Mass transport and optimal geometric inequalities: A general comparison principle for interacting gases Optimal Euclidean Sobolev inequalities Geometric inequalities HardyRellichSobolev inequalities: The HardySobolev inequalities Domain curvature and best constants in the HardySobolev inequalities AubinMoserOnofri inequalities: LogSobolev inequalities on the real line TrudingerMoserOnofri inequality on $\mathbb{S}^2$ Optimal AubinMoserOnofri inequality on $\mathbb{S}^2$ Bibliography.
 (source: Nielsen Book Data)
 Publisher's Summary
 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 willand 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 Holder. The subtle MoserOnofriAubin inequalities on the twodimensional sphere are connected to Liouville type theorems for planar mean field equations.
(source: Nielsen Book Data)
Subjects
 Subject
 Inequalities (Mathematics)
 Harmonic analysis.
 Harmonic analysis on Euclidean spaces  Harmonic analysis in several variables  Maximal functions, LittlewoodPaley theory.
 Partial differential equations  General topics  Inequalities involving derivatives and differential and integral operators, inequalities for integrals.
 Real functions  Inequalities  Inequalities involving derivatives and differential and integral operators.
 Partial differential equations  General topics  Variational methods.
 Functional analysis  Linear function spaces and their duals  Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems.
Bibliographic information
 Publication date
 2013
 Responsibility
 Nassif Ghoussoub, Amir Moradifam.
 Series
 Mathematical surveys and monographs ; v. 187
 ISBN
 9780821891520 (alk. paper)
 0821891529 (alk. paper)