# Thin domain limit and counterexamples to strong diamagnetism

## TL;DR

This paper investigates the magnetic properties of thin superconducting domains, providing counterexamples to strong diamagnetism and analyzing the transition behavior, with results aligning with experimental observations.

## Contribution

It introduces counterexamples to strong diamagnetism and characterizes the non-monotone transition in thin superconducting domains, extending understanding of magnetic responses.

## Key findings

- Counterexamples to strong diamagnetism in thin domains
- Non-monotone transition from superconducting to normal state
- Structure of the order parameter in nonlinear regimes

## Abstract

We study the magnetic Laplacian and the Ginzburg-Landau functional in a thin planar, smooth, tubular domain and with a uniform applied magnetic field. We provide counterexamples to strong diamagnetism, and as a consequence, we prove that the transition from the superconducting to the normal state is non-monotone. In some non-linear regime, we determine the structure of the order parameter and compute the super-current along the boundary of the sample. Our results are in agreement with what was observed in the Little-Parks experiment, for a thin cylindrical sample.