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Explore the mathematical properties of minimizers in a variational model describing charged liquid droplet shapes. Delve into the competition between surface tension forces and repulsive effects between charged particles that determine droplet formation. Examine Lord Rayleigh's perturbative analysis of ball stability under charge thresholds and the experimental observations of Taylor's cones above critical charge values. Investigate the ill-posed nature of the optimal shape determination problem and the Debye-Hückel-type free energy model proposed by Muratov and Novaga. Learn about the partial regularity results for minimizers in this second model, providing insights into the complex behavior of charged droplets. This 59-minute lecture, part of the Hausdorff Trimester Program on Evolution of Interfaces, offers a deep dive into the mathematical analysis of charged droplet phenomena.
