Conformal Killing Vector Fields and Rellich Type Identities on Riemannian Manifolds, II

Abstract

We propose a general Noetherian approach to Rellich integral identities. Using this method we obtain a higher order Rellich type identity involving the polyharmonic operator on Riemannian manifolds admitting homothetic transformations. Then we prove a biharmonic Rellich identity in a more general context. We also establish a nonexistence result for semilinear systems involving biharmonic operators.Mathematics Subject Classification (2010). 35J50, 35J20, 35J60.

A Caffarelli–Kohn–Nirenberg type inequality on Riemannian manifolds

“…If a = p − 1, b = 0 in (3), then we can get easily the sharp Hardy type inequality established in [3]. Namely: We observe that the Caffarelli-Kohn-Nirenberg Inequality (1) with b = 0 and a = −1 represents the classical Uncertainity Principle of quantum mechanics:…”

On Higher order Poincaré Inequalities with radial derivatives and Hardy improvements on the hyperbolic space

“…Furthermore, Carron [14] derived the classical Hardy inequality on Riemannian manifolds which open up new directions in the study of Hardy inequality on non-trivial geometry. Among all the recent work in these directions, we are bringing up only a few of them [11,15,17,23,24,30,37,25] without a claim of completeness. A large part of these works dealt with an improvement of inequalities with optimal Hardy weight.…”

Hardy inequalities on Riemannian manifolds and applications

“…Our principal result is a simple criterion to establish if there holds a Hardy inequality involving the weight ρ. Namely, if ρ is psuperharmonic in Ω, that is −∆ p ρ ≥ 0, then operators on Carnot groups. For this goal we shall mainly use a technique introduced by Mitidieri in [Mi] and developed in [Da1,Da2,Da3] and in [BM1,BM2]. The proof is based on the divergence theorem and on the careful choice of a vector field.…”

“…Let us point out some interesting outcomes of our approach. A first issue is that, since it is quite general, our approach includes Hardy inequalities already studied in [AS,BM1,Ca2,KO,LW] in the case p = 2 as well as their generalization for p > 1. Indeed, in all these cited papers, the authors assume extra conditions on the function ρ or on the manifold.…”