Functional inequalities : new perspectives and new applications
著者
書誌事項
Functional inequalities : new perspectives and new applications
（Mathematical surveys and monographs, v. 187）
American Mathematical Society, c2013
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注記
Includes bibliographical references (p. 289-299)
内容説明・目次
内容説明
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 ones--sometimes 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 Hardy-Rellich 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 Log-Sobolev 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. Caffarelli-Kohn-Nirenberg and Hardy-Rellich-Sobolev type inequalities are then obtained by interpolating the above two classes of inequalities via the classical ones of Hölder. The subtle Moser-Onofri-Aubin inequalities on the two-dimensional sphere are connected to Liouville type theorems for planar mean field equations.
目次
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 Hardy-Rellich type inequalities: Improved Hardy-Rellich inequalities on $H^2_0(\Omega)$ Weighted Hardy-Rellich 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 non-self-adjoint problems Mass transport and optimal geometric inequalities: A general comparison principle for interacting gases Optimal Euclidean Sobolev inequalities Geometric inequalities Hardy-Rellich-Sobolev inequalities: The Hardy-Sobolev inequalities Domain curvature and best constants in the Hardy-Sobolev inequalities Aubin-Moser-Onofri inequalities: Log-Sobolev inequalities on the real line Trudinger-Moser-Onofri inequality on $\mathbb{S}^2$ Optimal Aubin-Moser-Onofri inequality on $\mathbb{S}^2$ Bibliography
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