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Hardcover ISBN:  9780821891520 
Product Code:  SURV/187 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470409470 
Product Code:  SURV/187.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821891520 
eBook ISBN:  9781470409470 
Product Code:  SURV/187.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 187; 2013; 299 ppMSC: Primary 42; 35; 26; 46;
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 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.
ReadershipGraduate students and research mathematicians interested in analysis, calculus of variations, and PDEs.

Table of Contents

Hardy type inequalities

1. Bessel pairs and Sturm’s oscillation theory

2. The classical Hardy inequality and its improvements

3. Improved Hardy inequality with boundary singularity

4. Weighted Hardy inequalities

5. The Hardy inequality and second order nonlinear eigenvalue problems

HardyRellich type inequalities

6. Improved HardyRellich inequalities on $H^2_0(\Omega )$

7. Weighted HardyRellich inequalities on $H^2(\Omega )\cap H^1_0(\Omega )$

8. Critical dimensions for $4^{\textrm {th}}$ order nonlinear eigenvalue problems

Hardy inequalities for general elliptic operators

9. General Hardy inequalities

10. Improved Hardy inequalities for general elliptic operators

11. Regularity and stability of solutions in nonselfadjoint problems

Mass transport and optimal geometric inequalities

12. A general comparison principle for interacting gases

13. Optimal Euclidean Sobolev inequalities

14. Geometric inequalities

HardyRellichSobolev inequalities

15. The HardySobolev inequalities

16. Domain curvature and best constants in the HardySobolev inequalities

AubinMoserOnofri inequalities

17. LogSobolev inequalities on the real line

18. TrudingerMoserOnofri inequality on $\mathbb {S}^2$

19. Optimal AubinMoserOnofri inequality on $\mathbb {S}^2$


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

Hardy type inequalities

1. Bessel pairs and Sturm’s oscillation theory

2. The classical Hardy inequality and its improvements

3. Improved Hardy inequality with boundary singularity

4. Weighted Hardy inequalities

5. The Hardy inequality and second order nonlinear eigenvalue problems

HardyRellich type inequalities

6. Improved HardyRellich inequalities on $H^2_0(\Omega )$

7. Weighted HardyRellich inequalities on $H^2(\Omega )\cap H^1_0(\Omega )$

8. Critical dimensions for $4^{\textrm {th}}$ order nonlinear eigenvalue problems

Hardy inequalities for general elliptic operators

9. General Hardy inequalities

10. Improved Hardy inequalities for general elliptic operators

11. Regularity and stability of solutions in nonselfadjoint problems

Mass transport and optimal geometric inequalities

12. A general comparison principle for interacting gases

13. Optimal Euclidean Sobolev inequalities

14. Geometric inequalities

HardyRellichSobolev inequalities

15. The HardySobolev inequalities

16. Domain curvature and best constants in the HardySobolev inequalities

AubinMoserOnofri inequalities

17. LogSobolev inequalities on the real line

18. TrudingerMoserOnofri inequality on $\mathbb {S}^2$

19. Optimal AubinMoserOnofri inequality on $\mathbb {S}^2$