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

A defect energy J(beta), which measures jump discontinuities of a unit leng th gradient field, is studied. The number beta indicates the power of the j umps of the gradient fields that appear in the density of J(beta). It is sh own that J(beta) for beta = 3 is lower semicontinuous ton the space of unit gradient fields belonging to BV) in L-1-convergence of gradient fields. A similar result holds for the modified energy J(+)(beta), which measures onl y a particular type of defect. The result turns out to be very subtle, sinc e J(+)(beta) with beta > 3 is not lower semicontinuous, as is shown in this paper. The key idea behind semicontinuity is a duality representation for J(3) and J(+)(3). The duality representation is also important for obtainin g a lower bound by using J(3) for the relaxation limit of the Ginzburg-Land au type energy for gradient fields. The lower bound obtained here agrees wi th the conjectured value of the relaxation limit.