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Functional Inequalities: New Perspectives and New Applications Mathematical Surveys and Monographs Volume 187 Functional Inequalities: New Perspectives and New Applications Nassif Ghoussoub Amir Moradifam American Mathematical Society http://dx.doi.org/10.1090/surv/187 Functional Inequalities: New Perspectives and New Applications Mathematical Surveys and Monographs Volume 187 Functional Inequalities: New Perspectives and New Applications Nassif Ghoussoub Amir Moradifam American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov MichaelA.Singer MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 42B25, 35A23, 26D10, 35A15, 46E35. For additional information and updates on this book, visit www.ams.org/bookpages/surv-187 Library of Congress Cataloging-in-Publication Data Library of Congress Cataloging-in-Publication Data has been applied for by the AMS. See www.loc.gov/publish/cip. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 181716151413 To my son Joseph Ghoussoub To my parents Saed and Iran Moradifam Contents Preface xi Introduction xiii Part 1. Hardy Type Inequalities 1 Chapter 1. Bessel Pairs and Sturm’s Oscillation Theory 3 1.1. The class of Hardy improving potentials 3 1.2. Sturm theory and integral criteria for HI-potentials 9 1.3. The class of Bessel pairs 14 1.4. Further comments 17 Chapter 2. The Classical Hardy Inequality and Its Improvements 19 2.1. One dimensional Poincar´e inequalities 19 2.2. HI-potentials and improved Hardy inequalities on balls 21 2.3. Improved Hardy inequalities on domains with 0 in their interior 24 2.4. Attainability of the best Hardy constant on domains with 0 in their interior 26 2.5. Further comments 28 Chapter 3. Improved Hardy Inequality with Boundary Singularity 31 3.1. Improved Hardy inequalities on conical domains with vertex at 0 31 3.2. Attainability of the Hardy constants on domains having 0 on the boundary 34 3.3. Best Hardy constant for domains contained in a half-space 38 3.4. The Poisson equation on the punctured disc 41 3.5. Further comments 42 Chapter 4. Weighted Hardy Inequalities 45 4.1. Bessel pairs and weighted Hardy inequalities 45 4.2. Improved weighted Hardy-type inequalities on bounded domains 49 4.3. Weighted Hardy-type inequalities on Rn 52 4.4. Hardy inequalities for functions in H1(Ω) 54 4.5. Further comments 57 Chapter 5. The Hardy Inequality and Second Order Nonlinear Eigenvalue Problems 59 5.1. Second order nonlinear eigenvalue problems 59 5.2. The role of dimensions in the regularity of extremal solutions 61 5.3. Asymptotic behavior of stable solutions near the extremals 62 5.4. The bifurcation diagram for small parameters 65 vii viii CONTENTS 5.5. Further comments 67 Part 2. Hardy-Rellich Type Inequalities 69 2 Chapter 6. Improved Hardy-Rellich Inequalities on H0 (Ω) 71 6.1. General Hardy-Rellich inequalities for radial functions 71 6.2. General Hardy-Rellich inequalities for non-radial functions 74 6.3. Optimal Hardy-Rellich inequalities with power weights |x|m 78 6.4. Higher order Rellich inequalities 83 6.5. Calculations of best constants 85 6.6. Further comments 90 2 ∩ 1 Chapter 7. Weighted Hardy-Rellich Inequalities on H (Ω) H0 (Ω) 93 2 ∩ 1 7.1. Inequalities between Hessian and Dirichlet energies on H (Ω) H0 (Ω) 93 2 ∩ 1 7.2. Hardy-Rellich inequalities on H (Ω) H0 (Ω) 101 7.3. Further comments 107 Chapter 8. Critical Dimensions for 4th Order Nonlinear Eigenvalue Problems 109 8.1. Fourth order nonlinear eigenvalue problems 109 8.2. A Dirichlet boundary value problem with an exponential nonlinearity 110 8.3. A Dirichlet boundary value problem with a MEMS nonlinearity 113 8.4. A Navier boundary value problem with a MEMS nonlinearity 118 8.5. Further comments 121 Part 3. Hardy Inequalities for General Elliptic Operators 123 Chapter 9. General Hardy Inequalities 125 9.1. A general inequality involving interior and boundary weights 125 9.2. Best pair of constants and eigenvalue estimates 132 9.3. Weighted Hardy inequalities for general elliptic operators 134 9.4. Non-quadratic general Hardy inequalities for elliptic operators 137 9.5. Further comments 141 Chapter 10. Improved Hardy Inequalities For General Elliptic Operators 143 10.1. General Hardy inequalities with improvements 143 10.2. Characterization of improving potentials via ODE methods 147 10.3. Hardy inequalities on H1(Ω) 151 10.4. Hardy inequalities for exterior and annular domains 154 10.5. Further comments 156 Chapter 11. Regularity and Stability of Solutions in Non-Self-Adjoint Problems 157 11.1. Variational formulation of stability for non-self-adjoint eigenvalue problems 157 11.2. Regularity of semi-stable solutions in non-self-adjoint boundary value problems 159 11.3. Liouville type theorems for general equations in divergence form 161 11.4. Further remarks 167 Part 4. Mass Transport and Optimal Geometric Inequalities 169 CONTENTS ix Chapter 12. A General Comparison Principle for Interacting Gases 171 12.1. Mass transport with quadratic cost 171 12.2. A comparison principle between configurations of interacting gases 173 12.3. Further comments 179 Chapter 13. Optimal Euclidean Sobolev Inequalities 181 13.1. A general Sobolev inequality 181 13.2. Sobolev and Gagliardo-Nirenberg inequalities 182 13.3. Euclidean Log-Sobolev inequalities 183 13.4. A remarkable duality 185 13.5. Further remarks and comments 189 Chapter 14. Geometric Inequalities 191 14.1. Quadratic case of the comparison principle and the HWBI inequality191 14.2. Gaussian inequalities 193 14.3. Trends to equilibrium in Fokker-Planck equations 196 14.4. Further comments 197 Part 5. Hardy-Rellich-Sobolev Inequalities 199 Chapter 15. The Hardy-Sobolev Inequalities 201 15.1. Interpolating between Hardy’s and Sobolev inequalities 201 15.2. Best constants and extremals when 0 is in the interior of the domain203 15.3. Symmetry of the extremals on half-space 206 15.4. The Sobolev-Hardy-Rellich inequalities 208 15.5. Further comments and remarks 211 Chapter 16. Domain Curvature and Best Constants in the Hardy-Sobolev Inequalities 213 16.1. From the subcritical to the critical case in the Hardy-Sobolev inequalities 213 16.2. Preliminary blow-up analysis 219 16.3. Refined blow-up analysis and strong pointwise estimates 227 16.4. Pohozaev identity and proof of attainability 236 16.5. Appendix: Regularity of weak solutions 240 16.6. Further comments 243 Part 6. Aubin-Moser-Onofri Inequalities 245 Chapter 17. Log-Sobolev Inequalities on the Real Line 247 17.1. One-dimensional version of the Moser-Aubin inequality 247 ≥ 2 17.2. The Euler-Lagrange equation and the case α 3 250 17.3. The optimal bound in the one-dimensional Aubin-Moser-Onofri inequality 252 17.4. Ghigi’s inequality for convex bounded functions on the line 258 17.5. Further comments 262 Chapter 18. Trudinger-Moser-Onofri Inequality on S2 263 18.1. The Trudinger-Moser inequality on S2 263 18.2. The optimal Moser-Onofri inequality 267 xCONTENTS 18.3. Conformal invariance of J1 and its applications 270 18.4. Further comments 272 Chapter 19. Optimal Aubin-Moser-Onofri Inequality on S2 275 19.1. The Aubin inequality 275 19.2. Towards an optimal Aubin-Moser-Onofri inequality on S2 277 19.3. Bol’s isoperimetric inequality 283 19.4. Further comments 287 Bibliography 289 Preface This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. It is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to “systematic” approaches for proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand. Our aim is therefore to describe how a few general principles are behind the validity of large classes – and often “equivalence classes” – of functional inequal- ities, old and new. As such, 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. Similarly, allowable non-radial weights in Hardy-type inequalities for more general uniformly elliptic operators are closely related to the resolution of certain linear PDEs in divergence form with either a prescribed boundary condition or with prescribed singularities in the interior of the domain. On the other hand, most geometric inequalities including those of Sobolev and Log-Sobolev type, are simply expressions of the convexity of certain free en- ergy functionals along the geodesics of the space of probability measures equipped with the optimal mass transport (Wasserstein) metric.
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