热点猜想可能是错误的：一些数值例子
The hot spots conjecture can be false: Some numerical examples
摘要：
热点猜想只对特殊几何是正确的。数值结果表明，热点猜想可能不成立，从而易于构造有界单洞区域。用边界积分方程求解具有Neumann边界条件的Laplace方程的基本本征值问题，得到一个非线性本征值问题。通过边界元配置法结合Beyn的算法对其进行离散，由于超收敛性，对第一个非零特征值及其相应的特征函数都得到了高精度的结果。此外，可以用数值表示，域内的最大值/极小值与其边界上的最大值/极小值之间的比值可以大于$1+10^{-3}$。最后，数值例子，易于构造域与多达五个洞提供失败的热点猜想以及。
英文摘要：
The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than $1+10^{-3}$. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.
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