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

Abstract

A defect energy Jβ, 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 Jβ. It is shown that Jβ for β = 3 is lower semicontinuous (on the space of unit gradient fields belonging to BV) in L1-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 J3 and . The duality representation is also important for obtaining a lower bound by using J3+ 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.