A problem in analysis
Abstract Assume that D ⊂ ℝ2 is a bounded domain, diffeomorphic to a disc, star-shaped, with a C1,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 ± inφfn(φ). Theorem. Assume that ∫02πψ ± n f2(φ)dφ = 0 n = 1, 2,⋯ Then f = const.