A converse to generalized Runcorn's theorem

2026/07/20 by Vjekoslav Kovač, Ivica Smolić
#math-ph #math.MP

paper · pdf

Abstract

We study an inverse problem for generalized Runcorn's theorem motivated by an apparent paradox of lunar magnetism. In mathematical terms, we characterize square-integrable complex functions f such that ∫_\ x∈ℝⁿ : a ≤ |x| ≤ b \ f(x) ∇ u(x)⋅∇ v(x) dV(x) = 0 for every complex harmonic function u in the inner ball |x|<r₊ and every complex harmonic function v in the exterior region |x|>r_- that vanishes at infinity. The solution space depends on whether the radii a and b such that r_-<a<b<r₊ are regarded as varying or fixed. The solutions are described through the expansion of f into spherical harmonics, and explicit representations are provided.

Related

• Zero sets of holomorphic functions in the unit ball: non-radial growth characteristics
• A proof of the generalized Khavinson conjecture
• (L^r, L^s) Resolvent Estimate for the Sphere off the Line (1)/(r)-(1)/(s)=(2)/(n)
• On the harmonic characterization of the spheres: a sharp stability inequality and some of its consequences
• Higher Dimensional Versions of the Douglas-Ahlfors Identities