Semicontinuity of a functional depending on curvature

Abstract.

In this paper we prove a lower semicontinuity result for a functional ${\vec F}(E)$, defined on a class of bounded subsets of ${\mathbb R}^2$ with a piecewise ${\mathcal C}^2$ boundary, with respect to the $L^1$-convergence of the sets. The functional ${\vec F}(E)$ depends on the curvature of $\partial E$ in a linear way and contains a penalizing term which prevents the appearance of thin sets in the symmetric difference $E\vartriangle E_h$, where $\{E_h\}$ in an $L^1$-approximating sequence of $E$.