Calculus of Variations and Geometric Measure Theory

P. Dondl - M. Novaga - S. Wojtowytsch - S. Wolff-Vorbeck

Connected Coulomb Columns: Analysis and Numerics

created by novaga on 01 Jul 2020
modified on 27 Jul 2021


Published Paper

Inserted: 1 jul 2020
Last Updated: 27 jul 2021

Journal: Nonlinearity
Volume: 34
Number: 9
Pages: 6120-6139
Year: 2021

ArXiv: 2007.00136 PDF


We consider a version of Gamow's liquid drop model with a short range attractive perimeter-penalizing potential and a long-range Coulomb interaction of a uniformly charged mass in $\mathbb R^3$. Here we constrain ourselves to minimizing among the class of shapes that are columnar, i.e., constant in one spatial direction. Using the standard perimeter in the energy would lead to non-existence for any prescribed cross-sectional area due to the infinite mass in the constant spatial direction. In order to heal this defect we use a connected perimeter instead. We prove existence of minimizers for this connected isoperimetric problem with long-range interaction and study the shapes of minimizers in the small and large cross section regimes. For an intermediate regime we use an Ohta-Kawasaki phase field model with connectedness constraint to study the shapes of minimizers numerically.