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Line energies for gradient vector fields in the plane. (English) Zbl 0960.49013

The paper is devoted to the study of the asymptotic behaviour, as \(\varepsilon\) decreases to 0, of the functionals \[ F_\varepsilon(u) ={1\over 2}\int_\Omega\left(\varepsilon|\nabla^2u|^2+{(1-|\nabla u|^2)^2\over\varepsilon}\right)dx, \] where \(\Omega\) is an open subset of \({\mathbb R}^n\). In the two dimensional case, a space of functions which seems to be the natural domain for the limiting energy is defined, and the equicoerciveness of \(F_\varepsilon\) in this space is proved. The functionals \(F_\varepsilon\) appear in connection with the theory of smetic liquid crystals, and in the modeling of the energy deformation of thin film blister undergoing a biaxial compression.
A discussion on the form of the limit of the family \(F_\varepsilon\) is also carried out.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables
49Q20 Variational problems in a geometric measure-theoretic setting
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