Shape Optimization and Electron Bubbles

Abstract

We present an analytical treatment of the shape optimization problem that arises in the study of electron bubbles. The problem is to minimize a weighted sum of a Laplace eigenvalue, volume, and surface area with respect to the shape of the domain. The analysis employs the calculus of moving surfaces and yields surprising conclusions regarding the stability of equilibrium spherical configurations. Namely, all but the lowest eigenvalue result in unstable configurations and certain combinations of parameters, near-spherical equilibrium stable configurations exist. Two-dimensional and three-dimensional problems are considered and numerical results are presented for the two-dimensional case.

Keywords:

• Calculus of moving surfaces
• Laplace eigenvalues
• Shape optimization

AMS Subject Classification:

• 49R50

Notes

∗The negative values, indicating instability against a particular harmonic, are emphasized in bold.

∗The negative values, indicating instability against a particular harmonic, are emphasized in bold.