AI 中文总结
研究广义朗科恩定理逆问题，刻画满足特定积分等式的平方可积复函数\(f\)，通过将\(f\)展开为球谐函数描述解，依半径情况分述解空间，给出显式表示。
AI 中文摘要
我们研究了由月球磁学中一个明显的悖论引发的广义朗科恩定理的逆问题。从数学角度，我们刻画了平方可积复函数\(f\)，使得对于内球\(|x|<r_+\)中的每个复调和函数\(u\)以及外区域\(|x|>r_-\)中在无穷远处消失的每个复调和函数\(v\)，有\(\int_{\{ x\in\mathbb{R}^n: a \leq |x| \leq b \}} f(x) \nabla u(x)\cdot\nabla v(x)\,\mathrm{d}\mathrm{V}(x) = 0\)。解空间取决于\(r_-<a<b<r_+\)时半径\(a\)和\(b\)是视为变化还是固定。通过将\(f\)展开为球谐函数来描述解，并给出了显式表示。
英文摘要
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 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.
Comments16 pages
