\( \Gamma \)-convergence of discrete approximations to interfaces with prescribed mean curvature

Bellettini, Giovanni ; Paolini, Maurizio ; Verdi, Claudio

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 1 (1990) no. 4, pp. 317-328

Résumé

The numerical approximation of the minimum problem: \( \min_{A \subseteq \Omega} \tilde{\mathcal{F}}(A) \), is considered, where \( \tilde{\mathcal{F}}(A) = P_{\Omega}(A) + \cos(\theta) \mathcal{H}^{n-1}(\partial A \cap \partial \Omega) - \int_{A} \kappa \). The solution to this problem is a set \( A \subseteq \Omega \subset \mathbb{R}^{n} \) with prescribed mean curvature \( \kappa \) and contact angle \( \theta \) at the intersection of \( \partial A \) with \( \partial \Omega \). The functional \( \tilde{\mathcal{F}} \) is first relaxed with a sequence of nonconvex functionals defined in \( H^{1}(\Omega) \) which, in turn, are discretized by finite elements. The \( \Gamma \)-convergence of the discrete functionals to \( \tilde{\mathcal{F}} \) as well as the compactness of any sequence of discrete absolute minimizers are proven.