Convex approximations of functionals with curvature
For a set \(\Omega\) in \(\mathbb{R}^ n\) with Lipschitz boundary \(\partial\Omega\), \(dH\) the \((n-1)\) dimensional Hausdorff measure in \(\mathbb{R}^ n\) and bounded functions \(\mu\) on \(\partial \Omega\), \(\kappa\) on \(\Omega\), define \(BV(\Omega)\) as space of functions of bounded variation on \(\Omega\), with \(\int| Dv|\) the total variation, \(\int(| 1+| Dv|^ 2)^{1/2}\) the area of \(v\). The authors study the numerical minimization of the functional \(\int_ \Omega| Dv|+\int_ \Omega\mu vdH-\int_ \Omega \kappa vdx\). They first turn the functional into a strictly convex one by replacing \(| Dv |\) by (\(\varepsilon^ 2+| Dv|^ 2)^{1/2}\) and then minimize the new functional by finite elements. They prove that the solution of the modified functional tends to the solution of the original one for \(\varepsilon\to 0\) and that the computed solutions tend to the true solution for the meshsize \(\to 0\) and \(\varepsilon\to 0\).
scientific article; zbMATH DE number 61697