An optimality criterion for shape optimization in eigenfrequency problems

For a broad class of static problems an optimality criterion of constant energy density at the designed boundary is known. In the present paper we prove a similar criterion for eigenfrequency problems. This optimality criterion serves as the tool for more basic understanding and for idealized reference cases as well as the basis for recursive procedures. Eigenfrequencies for in-plane vibrations as well as for out-of-plane vibrations of plates are optimized. The focus is on simplicity and multiple eigenfrequencies are not considered.