Line energies for gradient vector fields in the plane (Q1970224)

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.