×

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 \(\beta >3\).

MSC:

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.