A boundary piecewise constant level set method for boundary control of eigenvalue optimization problems

This paper investigates optimization of the least eigenvalue of −Δ with the constraint of one-dimension Hausdorff measure of Dirichlet boundary. We propose the boundary piecewise constant level set (BPCLS) method based on the regularity technique to combine two types of boundary conditions into a single Robin boundary condition. We derive the first variation of the least eigenvalue w.r.t. the BPCLS function and propose a penalty BPCLS algorithm and an augmented Lagrangian BPCLS algorithm. Numerical results are reported for experiments on ellipse and L-shape domains.