## 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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