论文标题
三角形上拉普拉斯特征值的形状优化及其在插值误差常数估计中的应用
Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error constant estimation
论文作者
论文摘要
针对直径约束下三角形域的拉普拉斯特征值最小化问题提出了计算机辅助证明。该证明利用了最近开发的为差异操作员的特征值和特征函数开发了保证的计算方法。该论文还提供了Hadamard形状衍生物的基本和简洁的证明，这有助于验证特征值相对于形状参数的单调性。除了模型均匀的Dirichlet特征值问题外，还考虑了与非均匀的Neumann边界条件相关的特征值问题，该问题与Crouzeix-Raviart插值误差常数有关。计算机辅助证明说，在带有单位直径的三角形中，等边三角形最大程度地减少了每个相关特征值问题的第一个特征值。
A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The proof utilizes recently developed guaranteed computation methods for both eigenvalues and eigenfunctions of differential operators. The paper also provides an elementary and concise proof of the Hadamard shape derivative, which helps to validate the monotonicity of eigenvalue with respect to shape parameters. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix--Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.
