Sharp Uncertainty Principle inequality for solenoidal fields

2021/04/06 by Hamamoto, Naoki
#26D10 (Primary) #26D15 (Secondary) #35A23 #81S07 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2104.02351

Abstract

This paper solves the L² version of Maz'ya's open problem (Integral Equations Operator Theory 2018) on the sharp uncertainty principle inequality ∫_ℝ^N|∇ \bf\it u|²dx∫_ℝ^N|\bf\it u|²|\bf\it x|²dx≥ C_N(∫_ℝ^N|\bf\it u|²dx)² for solenoidal (namely divergence-free) vector fields \bf\it u=\bf\it u(\bf\it x) on ℝ^N. The best value of the constant turns out to be C_N=(1)/(4)(√(N²-4(N-3))+2)² which exceeds the original value N²/4 for unconstrained fields. Moreover, we show the attainability of C_N 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 additionally exist extremal toroidal fields.

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