Mathematical Surveys and Monographs Volume 187 Functional Inequalities: New Perspectives and New Applications Nassif Ghoussoub Amir Moradifam American Mathematical Society Providence, Rhode Island
Contents Preface Introduction xi 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 Poincare 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 K 52 4.4. Hardy inequalities for functions in 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 5.5. Further comments 67 Part 2. Hardy-Rellich Type Inequalities 69 Chapter 6. Improved Hardy-Rellich Inequalities on H$(Sl) 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 Chapter 7. Weighted Hardy-Rellich Inequalities on H2 (ft) n Hi (ft) 93 7.1. Inequalities between Hessian and Dirichlet energies on H2(Q) DHq(Q) 93 7.2. Hardy-Rellich inequalities on H2(ft) f) Hi (ft) 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 Hl(ft) 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
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 inequality 191 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 domain 203 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. on Log-Sobolev Inequalities the Real Line 247 17.1. One-dimensional version of the Moser-Aubin inequality 247 17.2. The Euler-Lagrange equation and the case a > 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 on Trudinger-Moser inequality 2 263 18.2. The optimal Moser-Onofri inequality 267
x 18.3. Conformal invariance of Ji 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 2 277 19.3. Bol's isoperimetric inequality 283 19.4. Further comments 287 Bibliography 289