众所周知，当球同心时，穿孔球的扭转刚度最小，穿孔具有球的形状，并且这种域的狄利克雷-拉普拉斯算子的第一特征值在这种情况下也是最大值。这些结果已获得Ashbaugh和Chatelain（私人通信），Harrell等。例如，”（《礼记·乐记·乐记·乐记》）本文将这些结果推广到了$1 < p < \infty$的$p$-Laplacian情形.为了证明这些结果，我们遵循与上述文章相同的思路，即利用动平面方法和比较原理研究形导数的符号。在此过程中，我们得到了一些有趣的新结果，如Dirichlet $p$ -Laplacian方程扭转刚度泛函的Hadamard摄动公式，非线性偏微分方程的存在唯一性结果，以及一些已知的非线性偏微分方程比较结果的推广.
英文摘要：
It is known that the torsional rigidity for a punctured ball, with the puncture having the shape of a ball, is minimum when the balls are concentric and the first eigenvalue for the Dirichlet Laplacian for such domains is also a maximum in this case. These results have been obtained by Ashbaugh and Chatelain (private communication), Harrell et. al., by Kesavan and, by Ramm and Shivakumar. In this paper we extend these results to the case of $p$-Laplacian for $1 < p < \infty$. For proving these results, we follow the same line of ideas as in the aforementioned articles, namely, study the sign of the shape derivative using the moving plane method and comparison principles. In the process, we obtain some interesting new side results such as the Hadamard perturbation formula for the torsional rigidity functional for the Dirichlet $p$-Laplacian, the existence and uniqueness result for a nonlinear pde and some extensions of known comparison results for nonlinear pdes.
An asymptotic shape optimization problem for Riesz means of Laplacian eigenvalues
Ludwig-Maximilians Universität München（慕尼黑大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与工程中心）；Caltech（加州理工学院）；Chalmers University of Technology（查尔姆斯理工大学）；University of Gothenburg（哥德堡大学）
Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains
Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；Caltech（加州理工学院）；Chalmers University of Technology（查尔姆斯理工大学）；the University of Gothenburg（哥德堡大学）
