On families of optimal Hardy-weights for linear second-order elliptic operators

Abstract

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on (a, b), ∞ ≤ a < b ≤ ∞, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.2000 Mathematics Subject Classification. Primary 35B09; Secondary 35J08, 35J20, 47J20, 49J40.

Optimal Hardy-weights for elliptic operators with mixed boundary conditions

“…Theorem 1.10 is proved in [5] for the specific (classical) case ∂Ω Rob = ∅ and a = 0. Moreover, Theorem 1.10 with ∂Ω Rob = ∅ and a > 0 was proved in [19], and provides greater Hardy-weights than the optimal Hardy-weight W class given in [5], in the sense that…”

“…In the present paper, we utilize the approach developed in [5,19] to produce families of optimal Hardy-inequalities for a general linear, second-order, elliptic operator with degenerate mixed boundary conditions. Our approach relies on criticality theory of positive weak solutions for elliptic operators with mixed boundary conditions which has been recently established in [19].…”

“…In the present paper, we utilize the approach developed in [5,19] to produce families of optimal Hardy-inequalities for a general linear, second-order, elliptic operator with degenerate mixed boundary conditions. Our approach relies on criticality theory of positive weak solutions for elliptic operators with mixed boundary conditions which has been recently established in [19]. Though our methods are similar to those in [5,19], where (generalized) Dirichlet boundary conditions are considered, a more intricate treatment is required when mixed boundary conditions are considered.…”

“…Our approach relies on criticality theory of positive weak solutions for elliptic operators with mixed boundary conditions which has been recently established in [19]. Though our methods are similar to those in [5,19], where (generalized) Dirichlet boundary conditions are considered, a more intricate treatment is required when mixed boundary conditions are considered.…”

Optimal Hardy weights on the Euclidean lattice

“…[24,26,27,32], the novelity of the approach lies in the use of solutions with specific properties and the proof of optimality. For further recent work on optimal decay of Hardy weights in the continuum, see [8,18,42,52].…”