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On a nonlocal superconductivity problem

Damião J. Araújo, Aelson Sobral

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 Hölder regularity estimates for viscosity solutions.

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, 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 Hölder regularity estimates for viscosity solutions.

Paper Structure

This paper contains 8 sections, 15 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Challenges for the nonlocal setting
  3. Main results and consequences
  4. Preliminaries
  5. Nonlocal positivity argument
  6. Oscillation estimates
  7. Gradient regularity estimates
  8. Fully Nonlinear Operators

Key Result

Theorem 1.1

For $u \in C(B_1) \cap L^\infty(\mathbb{R}^n)$ and $\Omega \subset \mathbb{R}^n$ an open set, assume that $(u,\Omega)$ solves mainprob, for some $s\in (1/2,1)$. Then, $u$ is locally $C^{1,\alpha}$, for some universal $\alpha\in (0,1)$, depending only on $n$ and $s$. Furthermore, there exists $C$ dep

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 13 more