L∞ estimate for a limiting form of Ginzburg-Landau systems in convex domains

In this note, we study a semilinear system involving the curl operator in a bounded and convex domain in R3, which is a limiting form of the Ginzburg-Landau model for superconductors in three dimensions for a large value of the Ginzburg-Landau parameter. We show the existence and establish the L∞ estimate associated with the boundary data for the weak solution of this system. As an application, our result gives a lower bound of the strength of the applied field such that the superconducting state is locally stable.