\(\Gamma\) -Convergence of Some Nonlocal Perimeters in Bounded Domains with General Boundary Conditions
Antoine Mellet, Yijing Wu · SIAM Journal on Mathematical Analysis · 2023
Abstract. We establish the [Formula: see text]-convergence of some energy functionals describing nonlocal attractive interactions in bounded domains. The interaction potential solves an elliptic equation (local or nonlocal) in the bounded domain and the primary interest of our results is to identify the effects that the boundary conditions imposed on the potential have on the limiting functional. We consider general Robin boundary conditions, which include Dirichlet and Neumann conditions as particular cases. Depending on the order of the elliptic operator the limiting functional involves the usual perimeter or some fractional perimeter. We also consider the [Formula: see text]-convergence of a related energy functional combining the usual perimeter functional and a nonlocal repulsive interaction energy.