A converse to generalized Runcorn's theorem
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 $$ \int_{\{ x\in\mathbb{R}^n : a \leq |x| \leq b \}} f(x) \nabla u(x)\cdot\nabla v(x)\,\mathrm{d}\mathrm{V}(x) = 0 $$ for every...