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**1 - 2**of**2**### WHERE TO PLACE A SPHERICAL OBSTACLE SO AS TO MAXIMIZE THE SECOND DIRICHLET EIGENVALUE

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### ON THE PLACEMENT OF AN OBSTACLE SO AS TO OPTIMIZE THE DIRICHLET HEAT TRACE

"... Abstract. We prove that among all doubly connected domains of Rn bounded by two spheres of given radii, Z(t), the trace of the heat kernel with Dirichlet boundary conditions, achieves its maximum when the spheres are concentric (i.e., for the spherical shell). The infimum is attained in the limiting ..."

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Abstract. We prove that among all doubly connected domains of Rn bounded by two spheres of given radii, Z(t), the trace of the heat kernel with Dirichlet boundary conditions, achieves its maximum when the spheres are concentric (i.e., for the spherical shell). The infimum is attained in the limiting situation where 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 domain so as to maximize or minimize the trace of the Dirichlet heat kernel. In this case, for each t the maximizing position of the center of the obstacle belongs to the “heart ” of the domain, while the minimizing 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), as well as for the spectral zeta function and the regularized determinant.