Convex approximation of an inhomogeneous anisotropic functional

Bellettini, Giovanni ; Paolini, Maurizio

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 5 (1994) no. 2, pp. 177-187

Résumé

The numerical minimization of the functional \( \mathcal{F} (u) = \int_{\Omega} \phi (x,\nu_{u}) |Du| + \int_{\partial \Omega} \mu u \, d\mathcal{H}^{n-1} - \int_{\Omega} \kappa u \, dx \), \( u \in BV(\Omega; \{-1, 1\}) \) is addressed. The function \( \phi \) is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that \( \mathcal{F} \) can be equivalently minimized on the convex set \( BV(\Omega; \left[-1, 1\right]) \) and then regularized with a sequence \( \{\mathcal{F}_{\epsilon}(u)\}_{\epsilon} \), of stricdy convex functionals defined on \( BV(\Omega; \left[-1, 1\right]) \). Then both \( \mathcal{F} \) and \( \mathcal{F}_{\epsilon} \), can be discretized by continuous linear finite elements. The convexity property of the functionals on \( BV(\Omega; \left[-1, 1\right]) \) is useful in the numerical minimization of \( \mathcal{F} \). The \( \Gamma — L_{1} (\Omega) \)-convergence of the discrete functionals \( \{ \mathcal{F}_{h} \}_{h} \) and \( \{ \mathcal{F}_{\epsilon,h} \}_{\epsilon,h} \) to \( \mathcal{F} \), as well as the compactness of any sequence of discrete absolute minimizers, are proven.