关于障碍物的放置以优化Dirichlet热迹
On the placement of an obstacle so as to optimize the Dirichlet heat trace
摘要：
我们证明了在由两个给定半径的球面包围的$Z(t)$的所有双连通域中，当球面同心时（即，球壳）。当内球与外球相接触时，上确界达到，这是描述球形障碍物在凸区域内的最优位置以最大化或最小化Dirichlet热核迹的更一般定理的特殊情况.在这种情况下，障碍物中心的最小化位置属于域的“心脏”，而最大化情况发生在心脏的内部或障碍物与外边界接触的点处。类似的陈述适用于任何光谱特性的障碍物的最佳位置，这些光谱特性可以作为$Z(t)$的正性保持或正性反转变换获得，包括光谱zeta函数以及通过它的正则化行列式。
英文摘要：
We prove that among all doubly connected domains of $\R^n$ bounded by two spheres of given radii, $Z(t)$, the trace of the heat kernel with Dirichlet boundary conditions, achieves its minimum when the spheres are concentric (i.e., for the spherical shell). The supremum is attained when the interior sphere is in contact with the outer sphere.This is shown to be a special case of a more general theorem characterizing the optimal placement of a spherical obstacle inside a convex domain so as to maximize or minimize the trace of the Dirichlet heat kernel. In this case the minimizing position of the center of the obstacle belongs to the "heart" of the domain, while the maximizing situation occurs either in the interior of the heart or at a point where the obstacle is in contact with the outer boundary. Similar statements hold for the optimal positions of the obstaclefor any spectral property that can be obtained as a positivity-preserving or positivity-reversing transform of $Z(t)$,including the spectral zeta function and, through it, the regularized determinant.
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