Hardy-type inequalities related to degenerate elliptic differential operators

Abstract

AbstractWe prove some Hardy type inequalities related to quasilinear second order de-generate elliptic differential operators L p u := −∇ ∗L (|∇ L u| p−2 ∇ L u). If φ is a positiveweight such that −L p φ ≥ 0, then the Hardy type inequalityc Z Ω |u| p φ p |∇ L φ| p dξ ≤ Z Ω |∇ L u| p dξ  u ∈ C 10 (Ω)  holds. We find an explicit value of the constant involved, which, in most cases, resultsoptimal. As particular case we derive Hardy inequalities for subelliptic operators onCarnot Groups. Mathematical Subject Classification(2000) 35H10, 22E30, 26D10, 46E35 1 Introduction An N-dimensional generalization of the classical Hardy inequality is the followingc Z Ω |u| p w −p dx≤ Z Ω |∇u| p dx, u∈ C 10 (Ω), (1.1)where p>1, Ω ⊂ R N and the weight wis, for instance, w:= |x| or w(x) := dist(x,∂Ω)(see for instance [5, 10, 23] and the references therein).A lot of efforts have been made to give explicit values of the constant c, and evenmore, to find its best value c n,p (see e.g. [5, 10, 23, 24, 31, 40, 41, 42]).