A converse to generalized Runcorn's theorem

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 $${\int }_{\{x\in {\mathbb{R}}^n:a\le |x|\le b\}}f(x)\nabla u(x)\cdot \nabla v(x)\mathrm{dV}(x)=0$$ 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.

16 pages