This paper concerns the asymptotic behavior of the solutions of a singularly perturbed quasilinear system involving a small parameter λ and a quadratic form gM(curl H) = 〈Mcurl H, curl H〉. This system arises in the mathematical theory of nucleation of instability of the Meissner states of anisotropic superconductors, and the location of the maximum points of the quadratic form suggests the location where instability begins to nucleate. The existence and regularity of the solutions of this system for fixed λ have been established by Pan [“A quasilinear system involving the operator curl,” Calculus Var. Partial Differ. Equ. 36, 317 (2009)]. In this paper we find an optimal bound of the boundary data for the solvability for all small λ and find the asymptotic behavior of the solutions as λ goes to zero. In particular it is proved that the maximum points of the quadratic form approach a point on the domain boundary where the tangential component of the boundary datum is the maximal. In the special case where the applied magnetic field has constant direction and constant magnitude, the maximum points of the quadratic form approach a point on the domain boundary where the applied field is tangential to the surface.
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2011
American Institute of Physics
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