平面Dirichlet Laplacian第三本征函数的闭节线猜想的一个反例
A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian
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中文总结 AI 辅助
针对平面Dirichlet Laplacian第三本征函数的闭节线猜想，构造出一个第三Dirichlet特征值为单重、节集含两条不相交闭曲线的反例，完成猜想的否定。
中文摘要 AI 辅助
Deng、Levitin和Yagudin在《LMS J. Comput. Math.》2003年第6卷的猜想4.3中提出猜想：若连通平面区域的第三Dirichlet特征值是单重的，则对应本征函数的所有节线不可能都是闭的。我们构造了一个反例：一个有界连通平面区域，其边界光滑，第三Dirichlet特征值是单重的，且对应每个非零实本征函数的整个内部节集都紧包含在该区域内。实际上，该节集恰好由两条不相交的实解析简单闭曲线组成。我们从Dahne、Gomez-Serrano和Hou的Payne猜想反例出发，通过一个对称细颈连接两个反射副本。反射将谱分裂为半区域上的混合Dirichlet-Neumann谱和Dirichlet谱。横向Poincaré不等式和极小极大原理使得两个谱都收敛到基区域谱，而边界唯一延拓在等指标处给出严格分离，并将偶次第二分支置于第三谱位置。局部本征函数收敛保持严格符号不等式；Courant定理限制了节集；欧拉计数确定其拓扑。光滑内穷竭完成了构造。
英文摘要
A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian Zikang Deng Levitin and Yagudin conjectured in Conjecture 4.3 of LMS J. Comput. Math. 6 (2003) that, if the third Dirichlet eigenvalue of a connected planar domain is simple, then not all nodal lines of a corresponding eigenfunction can be closed. We construct a counterexample: a bounded connected planar domain with smooth boundary whose third Dirichlet eigenvalue is simple and for which the entire interior nodal set of every corresponding nonzero real eigenfunction is compactly contained in the domain. In fact, the nodal set consists of exactly two disjoint real-analytic simple closed curves. Starting from the Payne-conjecture counterexample of Dahne, Gomez-Serrano, and Hou, we connect two reflected copies by a symmetric thin neck. Reflection splits the spectrum into mixed Dirichlet-Neumann and Dirichlet spectra on a half-domain. A transverse Poincare inequality and the min-max principle yield convergence of both spectra to the base-domain spectrum, while boundary unique continuation gives strict separation at equal indices and places the even second branch at the third spectral position. Local eigenfunction convergence preserves strict sign inequalities; Courant's theorem confines the nodal set; and an Euler count determines its topology. A smooth inner exhaustion completes the construction.
