On the Placement of an Obstacle So As to Optimize the Dirichlet Heat Trace

We prove that among all domains of $\mathbb{R}^n$ 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 obstacle for any spectral property that can be obtained as a positivity-preserving or positivity-reversing transform of $Z(t)$, including the spectral zeta functi...

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