The onset problem for a thin superconducting loop in a large magnetic field

We present a rigorous analysis of the eigenvalue problem associated with the onset of superconductivity for a thin domain in the presence of a large applied magnetic field. We prove the validity of the formal result of Richardson and Rubinstein (Proc. Roy. Soc. London A 455 (1999), 2549–2564) revealing that in this double limit of thin domain and large field, the appropriate Rayleigh quotient differs from the standard one for order 1 applied fields through the addition of a potential depending on the field.