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 $-\operatorname{div}(A \nabla u )=0$ 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].
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | 35B45, 35B65, 35J70 |
| Cite as: | arXiv:2605.23666 [math.AP] |
| (or arXiv:2605.23666v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.23666 arXiv-issued DOI via DataCite (pending registration) |
From: Gabriele Fioravanti [view email]
[v1]
Fri, 22 May 2026 14:14:43 UTC (18 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。