On the integrodifferential equations of the nonlocal theory of superconductivity

The nonlocal model of superconductivity has, in recent years, received strong support on both microscopic theoretical and experimental grounds. In the present report two of the mathematical aspects of the theory are discussed. One concerns the question of well-posedness of the equations and boundary conditions. The other concerns some methods of obtaining approximate solutions. Existence and uniqueness theorems are given for the solutions of the integrodifferential equations governing the vector potential, under suitable restrictions on the kernels involved. Both diffuse and specular reflection types of boundary conditions are considered in both scalar (one-dimensional) and vector (three-dimensional) problems. An existence theorem is also given for the time-dependent equations. Approximate solutions for the problem of a film in a parallel magnetic field are given in two limiting cases. In one the small-coherence-length limit in the Pippard model under diffuse scattering boundary conditions is considered, and a boundary-layer-type solution is given. In the other a thin-film approximation to the diffuse problem is found. An exact series representation is also given for the solution of the problem of a circular cylinder in a parallel magnetic field with specular reflection boundary conditions.