Optimal design problems for quasidisks and partially clamped drums: Existence, symmetrization, and numerical methods
It is shown that the class of quasidisks in the complex plane, with fixed quasicircle constant and area, is compact in both the Hausdorff metric and in the sense of Caratheodory convergence. Compactness for chord-arc domains with fixed chord-arc constant and area is shown as a result of the quasidisk compactness. Compactness is used to show that each eigenvalue of the Laplacian, subject to Dirichlet boundary conditions, attains its extrema over each of these classes. The design problem of extremizing the fundamental tone of a drum fastened only on a fraction of the boundary is considered. The special case of minimizing the fundamental frequency of a circular drum is solved using symmetrization. The gradient of the tone of the drum with respect to the design is considered and approximated appropriately. This approximate gradient is then used to compute examples numerically.
Uhlig, Paul Xavier. "Optimal design problems for quasidisks and partially clamped drums: Existence, symmetrization, and numerical methods." (1997) Diss., Rice University. https://hdl.handle.net/1911/19224.