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Functional Inequalities: New Perspectives and New Applications
 
Nassif Ghoussoub University of British Columbia, Vancouver, BC, Canada
Amir Moradifam Columbia University, New York, NY
Front Cover for Functional Inequalities: New Perspectives and New Applications
HardcoverISBN:  978-0-8218-9152-0
Product Code:  SURV/187
List Price: $104.00
MAA Member Price: $93.60
AMS Member Price: $83.20
eBookISBN:  978-1-4704-0947-0
Product Code:  SURV/187.E
List Price: $98.00
MAA Member Price: $88.20
AMS Member Price: $78.40
HardcoverISBN:  978-0-8218-9152-0
eBookISBN:  978-1-4704-0947-0
Product Code:  SURV/187.B
List Price: $202.00$153.00
MAA Member Price: $181.80$137.70
AMS Member Price: $161.60$122.40
Front Cover for Functional Inequalities: New Perspectives and New Applications
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  • Front Cover for Functional Inequalities: New Perspectives and New Applications
  • Back Cover for Functional Inequalities: New Perspectives and New Applications
Functional Inequalities: New Perspectives and New Applications
Nassif Ghoussoub University of British Columbia, Vancouver, BC, Canada
Amir Moradifam Columbia University, New York, NY
Hardcover ISBN:  978-0-8218-9152-0
Product Code:  SURV/187
List Price: $104.00
MAA Member Price: $93.60
AMS Member Price: $83.20
eBook ISBN:  978-1-4704-0947-0
Product Code:  SURV/187.E
List Price: $98.00
MAA Member Price: $88.20
AMS Member Price: $78.40
Hardcover ISBN:  978-0-8218-9152-0
eBookISBN:  978-1-4704-0947-0
Product Code:  SURV/187.B
List Price: $202.00$153.00
MAA Member Price: $181.80$137.70
AMS Member Price: $161.60$122.40
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 1872013; 299 pp
    MSC: 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 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
     
     
    • 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
    • Hardy-Rellich type inequalities
    • 6. Improved Hardy-Rellich inequalities on $H^2_0(\Omega )$
    • 7. Weighted Hardy-Rellich 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 non-self-adjoint problems
    • Mass transport and optimal geometric inequalities
    • 12. A general comparison principle for interacting gases
    • 13. Optimal Euclidean Sobolev inequalities
    • 14. Geometric inequalities
    • Hardy-Rellich-Sobolev inequalities
    • 15. The Hardy-Sobolev inequalities
    • 16. Domain curvature and best constants in the Hardy-Sobolev inequalities
    • Aubin-Moser-Onofri inequalities
    • 17. Log-Sobolev inequalities on the real line
    • 18. Trudinger-Moser-Onofri inequality on $\mathbb {S}^2$
    • 19. Optimal Aubin-Moser-Onofri inequality on $\mathbb {S}^2$
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1872013; 299 pp
MSC: 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 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.

  • 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
  • Hardy-Rellich type inequalities
  • 6. Improved Hardy-Rellich inequalities on $H^2_0(\Omega )$
  • 7. Weighted Hardy-Rellich 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 non-self-adjoint problems
  • Mass transport and optimal geometric inequalities
  • 12. A general comparison principle for interacting gases
  • 13. Optimal Euclidean Sobolev inequalities
  • 14. Geometric inequalities
  • Hardy-Rellich-Sobolev inequalities
  • 15. The Hardy-Sobolev inequalities
  • 16. Domain curvature and best constants in the Hardy-Sobolev inequalities
  • Aubin-Moser-Onofri inequalities
  • 17. Log-Sobolev inequalities on the real line
  • 18. Trudinger-Moser-Onofri inequality on $\mathbb {S}^2$
  • 19. Optimal Aubin-Moser-Onofri inequality on $\mathbb {S}^2$
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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