# On a nonlocal superconductivity problem

## TL;DR

This paper studies a nonlocal free boundary problem in superconductivity, extending classical local models, and establishes interior gradient Hölder regularity for viscosity solutions.

## Contribution

It introduces a nonlocal free boundary model in superconductivity and proves regularity results for viscosity solutions, extending prior local models.

## Key findings

- Established interior gradient Hölder regularity estimates.
- Extended local free boundary models to nonlocal settings.
- Provided a framework for analyzing nonlocal superconductivity problems.

## Abstract

This paper investigates degenerate nonlocal free boundary problems arising in the context of superconductivity, extending the nonlocal counterpart to the work of Caffarelli, Salazar, and Shahgholian \cite{CS02, CSS04} 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 H\"older regularity estimates for viscosity solutions.