On a nonlocal superconductivity problem

Abstract

This paper investigates degenerate nonlocal free boundary problems arising in the context of superconductivity, extending the nonlocal counterpart to the work of Caffarelli and Salazar (2002) and Caffarelli, Salazar and Shahgholian (2004) in the local setting. In these models, no partial differential equation governs the moving sets where the gradient vanishes, meaning that test functions are only required to have a nonzero gradient. Our main results provide interior gradient Hölder regularity estimates for viscosity solutions.