Optimal hardy inequalities for general elliptic operators with improvements

Abstract

We establish Hardy inequalities of the form (Formula Presented) (1) where E is a positive function defined in Ω, -div(AΔE) is a nonnegative nonzero finite measure in Ω which we denote by μ and where A(x) is a n x n symmetric, uniformly positive definite matrix defined in Ω with |ξ|2A := A(x)ξ·ξ for ξ ∈ ℝn. We show that (1) is optimal if E = 0 on ∂Ω or E = ∞ on the support of μ and is not attained in either case. When E = 0 on ∂Ω we show (Formula Presented)(2) is optimal and not attained. Optimal weighted versions of these inequalities are also established. Optimal analogous versions of (1) and (2) are established for p ≠ 2 which, in the case that μ is a Dirac mass, answers a best constant question posed by Adimurthi and Sekar (see [1]). We examine improved versions of the above inequalities of the form (Formula Presented)(3) Necessary and sufficient conditions on V are obtained (in terms of the solvability of a linear pde) for (3) to hold. Analogous results involving improvements are obtained for the weighted versions. In addition we obtain various results concerning the above inequalities valid for functions μ which are nonzero on the boundary of Ω. We also examine the nonquadradic case ,ie. p ≠ 2 of the above inequalities.