On the Geometry of Optimal Windows, with Special Focus on the Square - Burchard, Almut; Denzler, Jochen (SIAM journal on mathematical analysis, 2006)
For the Laplace operator with mixed (Dirichlet and Neumann) boundary conditions, the dependence of the principal eigenvalue on the placement of the Dirichlet part is investigated. An optimal window is a Dirichlet part of the boundary that minimizes the principal eigenvalue among all competitors of the same area. In the special case of a square, we provide both numerical evidence and rigorous partial results for the conjecture that optimal windows in a square are segments centered at either a corner or the midpoint of a side. In particular, we prove that the principal eigenvalue decreases as a window is shifted from a side-centered position towards the corner. An optimal window contained in two sides of the square is connected and contains a corner in its interior. Optimal windows whose length does not exceed the length of one side break the symmetry of the square. We also construct a star-shaped domain whose optimal window(s) must be disconnected. Finally we give, for general domains in $\R^d$, continuity results for the eigenvalue as a function of the window, and examples of discontinuity when crucial hypotheses are violated. We also give a variation formula that relates the eigenvalue to the singularities of the eigenfunction (stress intensity coefficient) near the boundary of the window. Methods are based on the variational problem and include rearrangement, Dirichlet--Neumann bracketing, capacity estimates, and deformation under a flow.