Sharp Rellich–Leray inequality with any radial power weight for solenoidal fields

Abstract

In the previous work Hamamoto (Calc Var Partial Differ Equ 58(4):23, 2019), following from an idea of Costin–Maz’ya (Costin and Maz’ya in Calc Var Partial Differ Equ 32(4):523–532, 2008), we computed the best constant in Rellich–Leray inequality for axisymmetric solenoidal fields, including any radial power weight. In the present paper, we recompute it without such a symmetry assumption. As a result, it turns out that the best constant in the same inequality for solenoidal fields is distinct from the one for unconstrained fields, only when the weight exponent stays within a bounded range.