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

Ambrosio, Luigi; De Lellis, Camillo; Mantegazza, Carlo

Calc. Var. Partial Differ. Equ. 9, No. 4, 327-355 (1999).

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.

Reviewer: Riccardo De Arcangelis (Napoli)

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

Keywords:

singular perturbation problems; energy concentration effects; eikonal equation; integral functional