On the Placement of an Obstacle or a Well so as to Optimize the Fundamental Eigenvalue Evans M. Harrell II
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چکیده
We investigate how to place an obstacle B within a domain in Euclidean space so as to maximize or minimize the principal Dirichlet eigenvalue for the Laplacian on nB. The shape of B is xed a priori (usually as a ball), and only its position varies. We establish that for a certain class of domains the minimizing B is in contact with @ , while the maximizing B is in the interior, typically at the center (supposing that the domain is su ciently symmetric for this statement to be meaningful). Under special circumstances we can characterize the optimizing con gurations with multiple obstacles. Our method relies on the Hadamard perturbation formula and a moving plane analysis. Similar facts are proved when the hard obstacle is replaced by a central nonnegative potential function supported in B, and we consider the Schrodinger operator with this potential. Complementary facts are proved when the obstacle is replaced by a central nonpositive potential function.
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تاریخ انتشار 2000