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

被引:63
作者
Aviles, P [1 ]
Giga, Y
机构
[1] Hokkaido Univ, Dept Math, Sapporo, Hokkaido 060, Japan
[2] Univ Illinois, Champaign, IL 61801 USA
基金
日本学术振兴会;
关键词
D O I
10.1017/S0308210500027438
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
A defect energy J(beta), which measures jump discontinuities of a unit length gradient field, is studied. The number beta indicates the power of the jumps of the gradient fields that appear in the density of J(beta). It is shown 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 only a particular type of defect. The result turns out to be very subtle, since 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 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.
引用
收藏
页码:1 / 17
页数:17
相关论文
共 23 条
[1]   RANK ONE PROPERTY FOR DERIVATIVES OF FUNCTIONS WITH BOUNDED VARIATION [J].
ALBERTI, G .
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 1993, 123 :239-274
[2]   VARIATIONAL INTEGRALS ON MAPPINGS OF BOUNDED VARIATION AND THEIR LOWER SEMICONTINUITY [J].
AVILES, P ;
GIGA, Y .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 1991, 115 (03) :201-255
[3]   MINIMAL CURRENTS, GEODESICS, AND RELAXATION OF VARIATIONAL INTEGRALS ON MAPPINGS OF BOUNDED VARIATION [J].
AVILES, P ;
GIGA, Y .
DUKE MATHEMATICAL JOURNAL, 1992, 67 (03) :517-538
[4]   The distance function and defect energy [J].
Aviles, P ;
Giga, Y .
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 1996, 126 :923-938
[5]   SINGULARITIES AND RANK ONE PROPERTIES OF HESSIAN MEASURES [J].
AVILES, P ;
GIGA, Y .
DUKE MATHEMATICAL JOURNAL, 1989, 58 (02) :441-467
[6]  
Aviles P., 1987, Proc. Centre Math. Appl., V1987, P1
[7]  
DALMASO G, 1993, INTRO LINFINITY CONV
[8]  
De Giorgi E., 1955, Ricerche Mat., V4, P95
[9]  
DEGIORGI E, P ICM POL 1983 WARSZ, V2, P1175
[10]  
Federer H., 1969, GEOMETRIC MEASURE TH