Joint work with Stefano Decio, Max Engelstein, Mario Michetti, and Svitlana Mayboroda. The Robin boundary condition is $\frac{1}{a} \frac{\partial u}{\partial n}+u=f$ on the boundary of a domain $U$, and we claim that for $0< a< +\infty$, the corresponding harmonic measure is mutually absolutely continuous with respect to surface measure. Here (we hope we will have finished checking that) we can consider any bounded domain $U$ in $\mathbb{R}^n$ whose boundary is Ahlfors regular of dimension $d$, $n-2< d< n$, with nontangential access. The Robin condition is then to be taken weakly, and surface measure becomes Hausdorff measure.[-]

Nonlocal interaction energies are continuum models for large systems of particles, where typically each particle interacts not only with its immediate neighbors, but also with particles that are far away. Examples of these energies arise in many different applications, such as biology (population dynamics), physics (Ginzburg-Landau vortices), and material science (dislocation theory). A fundamental question is understanding the optimal arrangement of particles at equilibrium, which are described, at least in average, by minimizers of the energy. In this talk I will focus on a class of nonlocal energies that are perturbations of the Coulomb energy and I will show how their minimizers can be explicitly characterized. This is based on joint works with J. Mateu, L. Rondi, L. Scardia, and J. Verdera.[-]