Calculus of Variations and Geometric Measure Theory

L. Ambrosio - C. De Lellis - C. Mantegazza

Line Energies for Gradient Vector Fields in the Plane

created on 19 Jan 1999
modified by root on 03 Jun 2013


Published Paper

Inserted: 19 jan 1999
Last Updated: 3 jun 2013

Journal: Calc. Var.
Volume: 9
Pages: 327-355
Year: 1999


In this paper we study the singular perturbation of $\int (1-
\nabla u
^2)^2$ by $\epsilon^2
^2$. This problem, which could be thought as the natural second order version of the classical singular perturbation of the potential energy $\int (1-u^2)^2$ by $\epsilon^2
\nabla u
^2$, leads, as in the first order case, to energy concentration effects on hypersurfaces. In the two dimensional case we study the natural domain for the limiting energy and prove a compactness theorem in this class.