On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg–Landau type energy for gradient fields

A defect energy J β , which measures jump discontinuities of a unit length gradient field, is studied. The number β indicates the power of the jumps of the gradient fields that appear in the density of J β . It is shown that J β for β = 3 is lower semicontinuous (on the space of unit gradient fields belonging to BV) in L 1 -convergence of gradient fields. A similar result holds for the modified energy , which measures only a particular type of defect. The result turns out to be very subtle, since with β > 3 is not lower semicontinuous, as is shown in this paper. The key idea behind semicontinuity is a duality representation for J 3 and . The duality representation is also important for obtaining a lower bound by using J 3 + for the relaxation limit of the Ginzburg–Landau type energy for gradient fields. The lower bound obtained here agrees with the conjectured value of the relaxation limit.

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