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

Aviles, Patricio; Giga, Yoshikazu

Proc. R. Soc. Edinb., Sect. A, Math. 129, No. 1, 1-17 (1999).

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 \(\beta >3\).

Reviewer: Vicentiu D.Rădulescu (Craiova)

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

Keywords:

Ginzburg-Landau energy; jump discontinuity; Hausdorff measure; semicontinuity; defect energy

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