A problem in analysis

摘要

Assume that D C R~2 is a bounded domain, diffeomorphic to a disc, star-shaped, with aC~1,λ boundary C, λ > 0, which can be represented in polar coordinates as r = f(Φ), where f > 0 is a smooth 2π -periodic function. Let Ψ±_n := Ψ±n(Φ) := e~(±inΦ) f(Φ). Theorem. Assume that f_0~(2π)Ψ±nf~2(Φ)dΦ=0 n = 1,2,… Then f = const.