$\mathrm{\Gamma }$-convergence of discrete approximations to interfaces with prescribed mean curvature

Bellettini, Giovanni, Paolini, Maurizio, and Verdi, Claudio. "\( \Gamma \)-convergence of discrete approximations to interfaces with prescribed mean curvature." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 1.4 (1990): 317-328. <http://eudml.org/doc/244180>.

@article{Bellettini1990,
abstract = {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.},
author = {Bellettini, Giovanni, Paolini, Maurizio, Verdi, Claudio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Calculus of variations; Surfaces with prescribed mean curvature; Finite elements; Convergence of discrete approximations; numerical approximation; prescribed mean curvature; - convergence},
language = {eng},
month = {12},
number = {4},
pages = {317-328},
publisher = {Accademia Nazionale dei Lincei},
title = {\( \Gamma \)-convergence of discrete approximations to interfaces with prescribed mean curvature},
url = {http://eudml.org/doc/244180},
volume = {1},
year = {1990},
}

TY - JOUR
AU - Bellettini, Giovanni
AU - Paolini, Maurizio
AU - Verdi, Claudio
TI - \( \Gamma \)-convergence of discrete approximations to interfaces with prescribed mean curvature
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1990/12//
PB - Accademia Nazionale dei Lincei
VL - 1
IS - 4
SP - 317
EP - 328
AB - 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.
LA - eng
KW - Calculus of variations; Surfaces with prescribed mean curvature; Finite elements; Convergence of discrete approximations; numerical approximation; prescribed mean curvature; - convergence
UR - http://eudml.org/doc/244180
ER -