Convex approximations of functionals with curvature

Bellettini, Giovanni ; Paolini, Maurizio ; Verdi, Claudio

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 2 (1991) no. 4, pp. 297-306

Résumé

We address the numerical minimization of the functional \( \mathcal{F} (v) = \int_{\Omega} |Dv| + \int_{\partial \Omega} \mu v \, d\mathcal{H}^{n-1} - \int_{\Omega} x v \, dx \), for \( v \in BV(\Omega; \{-1,1\}) \). We note that \( \mathcal{F} \) can be equivalently minimized on the larger, convex, set \( BV(\Omega; \left[-1,1\right]) \) and that, on that space, \( \mathcal{F} \) may be regularized with a sequence \( \{ \mathcal{F}_{\epsilon}(v) = \int_{\Omega} \sqrt{ \epsilon^{2} + |Dv|^{2}} + \int_{\partial \Omega} \mu v \, d\mathcal{H}^{n-1} - \int_{\Omega} xv \, dx \}_{\epsilon} \)of regular functionals. Then both \( \mathcal{F} \) and \( \mathcal{F}_{\epsilon} \) can be discretized by continuous linear finite elements. The convexity of the functionals in \( BV(\Omega; \left[-1,1\right]) \) is useful for the numerical minimization of \( \mathcal{F} \). We prove the \( \Gamma - L^{1} (\Omega) \)-convergence of the discrete functionals to \( \mathcal{F} \) and present a few numerical examples.