**Mathematical Surveys and Monographs**

Volume: 187;
2013;
299 pp;
Hardcover

MSC: Primary 42; 35; 26; 46;

**Print ISBN: 978-0-8218-9152-0
Product Code: SURV/187**

List Price: $104.00

AMS Member Price: $83.20

MAA Member Price: $93.60

**Electronic ISBN: 978-1-4704-0947-0
Product Code: SURV/187.E**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

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#### Supplemental Materials

# Functional Inequalities: New Perspectives and New Applications

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*Nassif Ghoussoub; Amir Moradifam*

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.

#### Readership

Graduate students and research mathematicians interested in analysis, calculus of variations, and PDEs.

#### Table of Contents

# Table of Contents

## Functional Inequalities: New Perspectives and New Applications

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Introduction xiii14 free
- Part 1. Hardy type inequalities 126 free
- Bessel pairs and Sturm’s oscillation theory 328
- The classical Hardy inequality and its improvements 1944
- Improved Hardy inequality with boundary singularity 3156
- Weighted Hardy inequalities 4570
- The Hardy inequality and second order nonlinear eigenvalue problems 5984
- Part 2. Hardy-Rellich type inequalities 6994
- Improved Hardy-Rellich inequalities on 𝐻²₀(Ω) 7196
- Weighted Hardy-Rellich inequalities on 𝐻²(Ω)∩𝐻¹₀(Ω) 93118
- Critical dimensions for 4^{𝑡ℎ} order nonlinear eigenvalue problems 109134
- Part 3. Hardy inequalities for general elliptic operators 123148
- General Hardy inequalities 125150
- Improved Hardy inequalities for general elliptic operators 143168
- Regularity and stability of solutions in non-self-adjoint problems 157182
- Part 4. Mass transport and optimal geometric inequalities 169194
- A general comparison principle for interacting gases 171196
- Optimal Euclidean Sobolev inequalities 181206
- Geometric inequalities 191216
- Part 5. Hardy-Rellich-Sobolev inequalities 199224
- The Hardy-Sobolev inequalities 201226
- Domain curvature and best constants in the Hardy-Sobolev inequalities 213238
- Part 6. Aubin-Moser-Onofri inequalities 245270
- Log-Sobolev inequalities on the real line 247272
- Trudinger-Moser-Onofri inequality on 𝕊² 263288
- Optimal Aubin-Moser-Onofri inequality on 𝕊² 275300
- Bibliography 289314
- Back Cover Back Cover1330