## Functional Inequalities: New Perspectives and New Applications: New Perspectives and New ApplicationsThe 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 |

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### Contents

Bessel Pairs and Sturms Oscillation Theory | 3 |

The Classical Hardy Inequality and Its Improvements | 19 |

Improved Hardy Inequality with Boundary Singularity | 31 |

Weighted Hardy Inequalities | 45 |

The Hardy Inequality and Second Order Nonlinear Eigenvalue | 59 |

Improved HardyRellich Inequalities on H20 Ω | 71 |

Weighted HardyRellich Inequalities on H2Ω H10 Ω | 93 |

Critical Dimensions for 4th Order Nonlinear Eigenvalue Problems109 8 1 Fourth order nonlinear eigenvalue problems | 109 |

Optimal Euclidean Sobolev Inequalities | 181 |

Geometric Inequalities | 191 |

The HardySobolev Inequalities | 201 |

Domain Curvature and Best Constants in the HardySobolev | 213 |

9 | 247 |

21 | 253 |

52 | 259 |

TrudingerMoserOnofri Inequality on S2 | 263 |

General Hardy Inequalities | 125 |

Improved Hardy Inequalities For General Elliptic Operators | 143 |

Regularity and Stability of Solutions in NonSelfAdjoint | 157 |

A General Comparison Principle for Interacting Gases | 171 |

Optimal AubinMoserOnofri Inequality on S2 | 275 |

289 | |

### Common terms and phrases

assume attained ball of radius barycenters best constant boundary conditions boundary weight change of variable chapter consider contradiction convex function COROLLARY deﬁned deﬁnition denote dimension Dirichlet Dirichlet boundary conditions domain Q eigenvalue problems elliptic operators entropy Euler-Lagrange equation exists extremal solution ﬁrst following holds following inequality holds follows from standard G CEO G H3 G H6 g n g Hardy inequality Hence HI-potential improved Hardy-Rellich inequalities improving potential integrate interior weight Legendre transform Lemma Let Q maximum principle mean curvature Moreover n-dimensional Bessel pair non-negative nonlinear Note obtain Open problem optimal positive solution proof of Theorem Proposition prove radial functions resp result satisﬁes singular smooth bounded domain Sobolev inequality spherical harmonics standard elliptic theory sub-solution sufﬁces Suppose unit ball weight on Q zero