Sharp Uncertainty Principle inequality for solenoidal fields

Abstract

This paper solves the L2 version of Maz'ya's open problem [1, Section 3.9] on the sharp uncertainty principle inequality∫RN|∇u|2dx∫RN|u|2|x|2dx≥CN(∫RN|u|2dx)2 for solenoidal (namely divergence-free) vector fields u=u(x) on RN. The best value of the constant for N≥3 turns out to be CN=14(N2−4(N−3)+2)2 which exceeds the original value N2/4 for unconstrained fields. Moreover, we show the attainability of CN and specify the profiles of the extremal solenoidal fields: for N≥4, the extremals are proportional to a poloidal field that is axisymmetric and unique up to the axis of symmetry; for N=3, there exist extremal toroidal fields, in addition to extremal poloidal fields; for N=2, the extremal fields are all toroidal.