Boundary perturbation of the Laplace eigenvalues and applications to electron bubbles and polygons

Abstract

We analyze the evolution of Laplace eigenvalues on a domain induced by the motion of the boundary. We apply our analysis to two problems: 1. We study the equilibrium and stability of electron bubbles. Electron bubbles are cavities formed around electrons injected into liquid helium. They can be treated as simple mathematical systems that minimize the energy with three terms: the energy of the electron proportional to a Laplace eigenvalue, the surface energy proportional to the surface area of the cavity, and the hydrostatic pressure proportional to its volume. This system possesses a surprising result: an instability of the 2S electron bubbles. 2. We compute the simple eigenvalues on a regular polygon with N sides. The polygon is treated as a perturbation of the unit circle and its eigenvalues are approximated by a Taylor series. The accuracy of our approach is measured by comparison with finite element estimates. For the lowest eigenvalue, the first Taylor term provides an estimate within 10-5 of the true value. The second term reduces the error to 10-7. We discuss how to utilize the available symmetry to obtain better finite element estimates. Finally, we briefly discuss the expansion of simple eigenvalues on regular polygons in powers of 1/N.