Unbounded planar domains whose second nodal line does not touch the boundary

Counting Nodal Lines Which Touch the Boundary of an Analytic Domain

We consider the zeros on the boundary $\partial Ω$ of a Neumann eigenfunction $ϕ_λ$ of a real analytic plane domain $Ω$. We prove that the number of its boundary zeros is $O (λ)$ where $-Δϕ_λ = λ^2 ϕ_λ$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(λ)$. It follows that the number of nodal lines of $ϕ_λ$ (components of the nodal set) which touch the boundary is of order $λ$. This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal …

A counterexample to Payne's nodal line conjecture with few holes

Payne conjectured in 1967 that the nodal line of the second Dirichlet eigenfunction must touch the boundary of the domain. In their 1997 breakthrough paper, Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili proved this to be false by constructing a counterexample in the plane with many holes and raised the question of the minimum number of holes a counterexample can have. In this paper we prove it is at most 6.

分数p-拉普拉斯在斯特拉赫对称域中的 Payne 节点集猜想

Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains

本文研究了分数p-拉普拉斯在斯特拉赫对称域中特征函数和节点解的边界接触性质，通过极化不等式分析证明了正负部分支持接触边界，并推导出节点集的相同性质。