Where Best to Hold a Drum Fast
Steven James Cox, Paul X. Uhlig · SIAM Journal on Optimization · 1999
If we are allowed to fasten, say, one half of a drum's boundary, which half produces the lowest or highest bass note? The answer is a natural limit of solutions to a family of extremal Robin problems for the least eigenvalue of the Laplacian. We produce explicit extremizers when the drum is a disk while for general shapes we establish existence and necessary conditions, and we build and test a pair of numerical methods.