On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields. (English) Zbl 0923.49008

The paper deals with the study of the defect energy \[ J^\beta (\nabla u)=\int_\Sigma | [\nabla u]| ^\beta d{\mathcal H}^{n-1}, \] where \(\nabla u\) is the gradient vector field in a bounded domain \(\Omega\) in R\(^n\) and \(\Sigma\) denotes the jump discontinuity of \(\nabla u\). The positive number \(\beta\) indicates the power of the jumps of the gradient fields that appear in the density of \(J^\beta\). The authors show that \(J^3\) is lower semicontinuous in the \(L^1\)-topology of gradient fields. The same property holds for the modified energy \(J^3_+\), which measures only a particular type of defect. It is also shown in the paper that this lower semicontinuity property fails for \(\beta >3\).


49J45 Methods involving semicontinuity and convergence; relaxation
49K20 Optimality conditions for problems involving partial differential equations
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
81T13 Yang-Mills and other gauge theories in quantum field theory
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[1] DOI: 10.1063/1.337352
[2] DOI: 10.1063/1.339533
[3] DOI: 10.1007/BF00253122 · Zbl 0647.49021
[4] De Giorgi, Proc. ICM Poland 1983, Warszawa 2 pp 1175–
[5] Giorgi, Ricerche Mat. 4 pp 95– (1995)
[6] DOI: 10.1007/978-1-4612-0327-8
[7] Aviles, Proc. R. Soc. Edinb. A 126 pp 923– (1996) · Zbl 0878.49017
[8] DOI: 10.1215/S0012-7094-92-06720-2 · Zbl 0769.49010
[9] DOI: 10.1007/BF00380769 · Zbl 0737.49011
[10] DOI: 10.1215/S0012-7094-89-05820-1 · Zbl 0711.49062
[11] Aviles, Proc. Centre Math. Analysis Austral. Natn. Univ. 12 pp 1– (1987)
[12] Alberti, Proc. R. Soc. Edinb. A 123 pp 239– (1993) · Zbl 0791.26008
[13] Simon, Lectures on geometric measure theory. Proc. Centre Math. Analysis Austral. Natn. Univ. 3 (1983) · Zbl 0546.49019
[14] DOI: 10.1103/PhysRevA.26.3037
[15] DOI: 10.1016/0022-5096(94)90030-2 · Zbl 0832.73051
[16] DOI: 10.1007/BF00251230 · Zbl 0616.76004
[17] Lions, Generalized solutions of Hamilton–Jacobi equations (1982)
[18] DOI: 10.1002/cpa.3160470402 · Zbl 0803.49007
[19] DOI: 10.1007/BF02925437 · Zbl 0819.49014
[20] Giusti, Minimal surfaces and functions of bounded variations (1984) · Zbl 0545.49018
[21] Federer, Geometric measure theory (1969) · Zbl 0176.00801
[22] DOI: 10.1070/SM1967v002n02ABEH002340 · Zbl 0168.07402
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