Title: A geometric stability result for the Coulomb energy
Abstract: I will discuss work with Greg Chambers on the Coulomb energy in the context of recent geometric stability results (due to Christ, Figalli, Jerison, Carlen, Maggi, and others) for functionals that describe non-local interactions.
It is known that the Coulomb energy of a positive charge distribution increases under symmetrization: The physical reason is that the interaction energy between the charges grows as the typical distance between them shrinks. The energy increases strictly, unless the charge distribution is already radially decreasing about some point. Is this characterization of equality cases "stable"? In other words, must near-maximizers be close to radially decreasing? (How close?) Greg and I answer this question for charge distributions that are uniform on a set of finite positive volume.