In this talk I shall discuss, within the phenomenological Ginzburg-Landau (GL) theory, the response of a superconductor to an applied magnetic field. I will present new results about the ground state of the GL functional for type II superconductors in magnetic fields varying between the second and third critical fields. In this regime, superconductivity is a surface phenomenon, restricted to a thin layer along the boundary of the sample. Our new results show that the Ginzburg-Landau energy is to subleading order entirely determined by the minimization of simplified 1D functionals. The leading order of the energy is given by a universal, sample-independent, problem, whereas corrections depend on the curvature of the sample. Refined estimates on the Ginzburg-Landau minimizer follow from these energy estimates. In particular we settle in the affirmative a conjecture of X. B. Pan about the uniform distribution of superconductivity along the boundary.
Joint work with Michele Correggi.
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