We introduce an analogue of Payne's nodal line conjecture, which asserts thatthe nodal (zero) set of any eigenfunction associated with the second eigenvalueof the Dirichlet Laplacian on a bounded planar domain should reach the boundaryof the domain. The assertion here is that any eigenfunction associated with thefirst nontrivial eigenvalue of the Neumann Laplacian on a domain $\Omega$ withrotational symmetry of order two (i.e., $x\in\Omega$ iff $-x\in\Omega$) "shouldnormally" be rotationally antisymmetric. We give both positive and negativeresults which highlight the heuristic similarity of this assertion to the nodalline conjecture, while demonstrating that the extra structure of the problemmakes it easier to obtain stronger statements: it is true for all simplyconnected planar domains, while there is a counterexample domain homeomorphicto a disk with two holes.
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机译：我们引入一个Payne的节点线猜想的类似物，该断言认为与有界平面域上的Dirichlet Laplacian的第二个本征值相关的任何本征函数的节点（零）集都应到达该域的边界。这里的断言是，与域$ \ Omega $上具有第二阶旋转对称性（即$ x \ in \ Omega $ iff $ -x \ in \ Omega $）上的与Neumann Laplacian的第一个非平凡特征值相关的任何本征函数“旋转不对称。我们给出的正面和负面结果都突出了该断言与节点线猜想的启发式相似性，同时证明了问题的额外结构使其更易于获得更强的陈述：对于所有简单连接的平面域都是如此，而存在反例域同胚为具有两个孔的磁盘。
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