发表机构
University of Cambridge; University of Reading（剑桥大学; 雷丁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
构造有界单连通平面Lipschitz域，其双层算子本质谱半径大于1/2，从而反驳Kenig关于双层算子的谱半径猜想，证明基于计算机辅助的矩阵不等式与Floquet变换。
AI 中文摘要
层势为Lipschitz域上拉普拉斯方程的边值问题提供了经典方法。Kenig在1994年关于双层算子的谱半径猜想，在边界连通时，将确保相关Neumann级数在零均值$L^2$密度上的算子范数收敛。我们通过构造一个有界单连通平面Lipschitz域来反驳这一猜想，该域在弧长$L^2$上的双层算子的本质谱半径严格大于$1/2$。更精确地，对于每个充分接近$1/2$的$t>1/2$，我们得到这样的域，其Fredholm本质谱中包含$\pm i t$。该构造从形状在平移下重复的光滑图形开始。在尺度分离的极限下，细化使伴随预解方程的解在固定强迫下增长。图形斜率保持一致有界。一个计算机辅助证明通过Hermitian $2\times2$矩阵的不等式证明了这种增长。其在$- i/2$处的严格余量在邻近谱参数处持续存在。归一化和Floquet变换随后给出在全图形上具有小残差的紧支撑密度。我们将连续图形的缩放段插入一个有界边界中，其中这些密度形成弱零序列的近似特征向量。相同的谱结论在单个周期Lipschitz图形上成立。该证明结合了连续估计、精确有理算术和严格的区间包含。
英文摘要
Layer potentials provide a classical approach to boundary value problems for Laplace's equation on Lipschitz domains. Kenig's 1994 spectral-radius conjecture for the double-layer operator would ensure operator-norm convergence of the associated Neumann series on mean-zero $L^2$ densities when the boundary is connected. We disprove this conjecture by constructing a bounded simply connected planar Lipschitz domain whose double-layer operator on arclength $L^2$ has essential spectral radius strictly greater than $1/2$. More precisely, for every $t>1/2$ sufficiently close to $1/2$, we obtain such a domain with $\pm i t$ in its Fredholm essential spectrum. The construction starts from smooth graphs whose shapes repeat under translation. In the limit of separated scales, refinement makes solutions of adjoint resolvent equations grow with fixed forcing. The graph slopes remain uniformly bounded. A computer-assisted certificate proves this growth through an inequality for Hermitian $2\times2$ matrices. Its strict margin at $- i/2$ persists at nearby spectral parameters. Normalisation and a Floquet transform then give compactly supported densities with small residuals on the full graphs. We insert rescaled segments of successive graphs into one bounded boundary, where these densities form a weakly null sequence of approximate eigenvectors. The same spectral conclusion holds on a single periodic Lipschitz graph. The certificate combines continuous estimates, exact rational arithmetic and rigorous interval enclosures.
