On the Placement of an Obstacle or a Well so as to Optimize the Fundamental Eigenvalue

Source abstract

We investigate how to place an obstacle B within a domain $\Omega$ in Euclidean space so as to maximize or minimize the principal Dirichlet eigenvalue for the Laplacian on $\Omega \setminus B$. The shape of B is fixed 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 $\partial \Omega$, while the maximizing B is in the interior, typically at the center (supposing that the domain is sufficiently symmetric for this statement to be meaningful). Under special circumstances we can characterize the optimizing configurations 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 Schrödinger operator with this potential. Complementary facts are proved when the obstacle is replaced by a central nonpositive potential function.