Calculus of Variations and Geometric Measure Theory

G. Bellettini - L. Mugnai

Characterization and representation of the lower semicontinuous envelope of the elastica functional

created on 09 May 2003
modified by belletti on 13 Dec 2004


Published Paper

Inserted: 9 may 2003
Last Updated: 13 dec 2004

Journal: Ann. Inst. H. Poincare`, Anal. Non Lin.
Volume: 6
Number: 21
Pages: 839-880
Year: 2004


We characterize the lower semicontinuous envelope $\overline \F$ of the functional $\F(E) := \int_{\partial E} [1 + \vert \kappa_{\partial E}^{} \vert^p] d \HH^1$, defined on boundaries of sets $E \subset \R^2$, where $\kappa_{\partial E}^{}$ denotes the curvature of $\partial E$ and $p>1$. Through a desingularization procedure, we find the domain of $\overline \F$ and its expression, by means of different representation formulas.