Abstract
We are dealing with the Ginzburg-Landau equation with magnetic effect in a 3-dimensional thin domain, where the thinness of the domain is controlled by a small positive parameter, epsilon, and the variable thickness is represented by the graph of a smooth function. Consider the case that the platform of the thin domain consists of a family of disjoint subdomains with narrow channels. This gives an example of a weak link, called an S-c-S junction in superconductivity. We prove that if the reduced equation, obtained in the limit as the thinness vanishes, has a nondegenerate stable solution in each subdomain and if the volume of the channels is sufficiently small, then there exists a stable solution to the original equation in the thin domain. Using the same argument, we can also prove the existence of non-trivial stable solutions for the case that the variable surface of a thin domain has deep wells in the scale by epsilon.
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