A Priori Regularity Estimates for Ratio of Solutions to Elliptic Equations with a Product Structure of Two-Dimensional Nodal Sets

Abstract

In this paper, we establish optimal a priori $C^{1,\alpha }$ regularity estimates for the ratio $w=v/u$ of two solutions to the same elliptic equation $-\mathrm{div}(A\nabla u)=0$ with Lipschitz coefficients $A$, under the assumption that their nodal sets satisfy $Z(u)\subseteq Z(v)$. We specifically address the case where the zero set $Z(u)$ exhibits a product structure of $2$-dimensional nodal sets, namely $Z(u)=Z(u_1)\times \cdots \times Z(u_m)$, where the $u_i$ are $2$-dimensional functions. This result extends the regularity estimates previously proved in dimension $2$ by [Logunov and Malinnikova, 2016] and by [Terracini, Tortone, and Vita, 2026].