New Perspectives. Functional Inequalities: and New Applications. Nassif Ghoussoub Amir Moradifam. Monographs. Surveys and
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1 Mathematical Surveys and Monographs Volume 187 Functional Inequalities: New Perspectives and New Applications Nassif Ghoussoub Amir Moradifam American Mathematical Society Providence, Rhode Island
2 Contents Preface Introduction xi xiii Part 1. Hardy Type Inequalities 1 Chapter 1. Bessel Pairs and Sturm's Oscillation Theory The class of Hardy improving potentials Sturm theory and integral criteria for Hi-potentials The class of Bessel pairs Further comments 17 Chapter 2. The Classical Hardy Inequality and Its Improvements One dimensional Poincare inequalities Hi-potentials and improved Hardy inequalities on balls Improved Hardy inequalities on domains with 0 in their interior Attainability of the best Hardy constant on domains with 0 in their interior Further comments 28 Chapter 3. Improved Hardy Inequality with Boundary Singularity Improved Hardy inequalities on conical domains with vertex at Attainability of the Hardy constants on domains having 0 on the boundary Best Hardy constant for domains contained in a half-space The Poisson equation on the punctured disc Further comments 42 Chapter 4. Weighted Hardy Inequalities Bessel pairs and weighted Hardy inequalities Improved weighted Hardy-type inequalities on bounded domains Weighted Hardy-type inequalities on K Hardy inequalities for functions in Further comments 57 Chapter 5. The Hardy Inequality and Second Order Nonlinear Eigenvalue Problems Second order nonlinear eigenvalue problems The role of dimensions in the regularity of extremal solutions Asymptotic behavior of stable solutions near the extremals The bifurcation diagram for small parameters 65 vii
3 viii 5.5. Further comments 67 Part 2. Hardy-Rellich Type Inequalities 69 Chapter 6. Improved Hardy-Rellich Inequalities on H$(Sl) General Hardy-Rellich inequalities for radial functions General Hardy-Rellich inequalities for non-radial functions Optimal Hardy-Rellich inequalities with power weights \x\m Higher order Rellich inequalities Calculations of best constants Further comments 90 Chapter 7. Weighted Hardy-Rellich Inequalities on H2 (ft) n Hi (ft) Inequalities between Hessian and Dirichlet energies on H2(Q) DHq(Q) Hardy-Rellich inequalities on H2(ft) f) Hi (ft) Further comments 107 Chapter 8. Critical Dimensions for 4th Order Nonlinear Eigenvalue Problems Fourth order nonlinear eigenvalue problems A Dirichlet boundary value problem with an exponential nonlinearity A Dirichlet boundary value problem with a MEMS nonlinearity A Navier boundary value problem with a MEMS nonlinearity Further comments 121 Part 3. Hardy Inequalities for General Elliptic Operators 123 Chapter 9. General Hardy Inequalities A general inequality involving interior and boundary weights Best pair of constants and eigenvalue estimates Weighted Hardy inequalities for general elliptic operators Non-quadratic general Hardy inequalities for elliptic operators Further comments 141 Chapter 10. Improved Hardy Inequalities For General Elliptic Operators General Hardy inequalities with improvements Characterization of improving potentials via ODE methods Hardy inequalities on Hl(ft) Hardy inequalities for exterior and annular domains Further comments 156 Chapter 11. Regularity and Stability of Solutions in Non-Self-Adjoint Problems Variational formulation of stability for non-self-adjoint eigenvalue problems Regularity of semi-stable solutions in non-self-adjoint boundary value problems Liouville type theorems for general equations in divergence form Further remarks 167 Part 4. Mass Transport and Optimal Geometric Inequalities 169
4 ix Chapter 12. A General Comparison Principle for Interacting Gases Mass transport with quadratic cost A comparison principle between configurations of interacting gases Further comments 179 Chapter 13. Optimal Euclidean Sobolev Inequalities A general Sobolev inequality Sobolev and Gagliardo-Nirenberg inequalities Euclidean Log-Sobolev inequalities A remarkable duality Further remarks and comments 189 Chapter 14. Geometric Inequalities Quadratic case of the comparison principle and the HWBI inequality Gaussian inequalities Trends to equilibrium in Fokker-Planck equations Further comments 197 Part 5. Hardy-Rellich-Sobolev Inequalities 199 Chapter 15. The Hardy-Sobolev Inequalities Interpolating between Hardy's and Sobolev inequalities Best constants and extremals when 0 is in the interior of the domain Symmetry of the extremals on half-space The Sobolev-Hardy-Rellich inequalities Further comments and remarks 211 Chapter 16. Domain Curvature and Best Constants in the Hardy-Sobolev Inequalities From the subcritical to the critical case in the Hardy-Sobolev inequalities Preliminary blow-up analysis Refined blow-up analysis and strong pointwise estimates Pohozaev identity and proof of attainability Appendix: Regularity of weak solutions Further comments 243 Part 6. Aubin-Moser-Onofri Inequalities 245 Chapter 17. on Log-Sobolev Inequalities the Real Line One-dimensional version of the Moser-Aubin inequality The Euler-Lagrange equation and the case a > The optimal bound in the one-dimensional Aubin-Moser-Onofri inequality Ghigi's inequality for convex bounded functions on the line Further comments 262 Chapter 18. Trudinger-Moser-Onofri Inequality on S The on Trudinger-Moser inequality The optimal Moser-Onofri inequality 267
5 x Conformal invariance of Ji and its applications Further comments 272 Chapter 19. Optimal Aubin-Moser-Onofri Inequality on S The Aubin inequality Towards an optimal Aubin-Moser-Onofri inequality on Bol's isoperimetric inequality Further comments 287 Bibliography 289
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