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

Bellettini, Giovanni; Paolini, Maurizio; Verdi, Claudio

Γ

-convergence of discrete approximations to interfaces with prescribed mean curvature

Bellettini, Giovanni ; Paolini, Maurizio ; Verdi, Claudio

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 1 (1990), p. 317-328 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

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

Si studia l'approssimazione numerica del seguente problema di minimo: ${min}_{A \subseteq Ω} \tilde{F} (A)$ , ove $\tilde{F} (A) = P_{Ω} (A) + \cos (θ) H^{n - 1} (\partial A \cap \partial Ω) - \int_{A} κ$ , teso alla ricerca di un insieme $A \subseteq Ω \subset R^{n}$ con curvatura media $κ$ e angolo $θ$ all'intersezione di $\partial A$ con $\partial Ω$ . Il funzionale $\tilde{F}$ viene preliminarmente rilassato mediante una successione di funzionali non convessi definiti in $H^{1} (Ω)$ , che sono successivamente discretizzati con elementi finiti. Si dimostrano la $Γ$ -convergenza dei funzionali discreti al funzionale $\tilde{F}$ e la compattezza di qualunque successione di minimi assoluti dei funzionali discreti.

Publié le : 1990-12-01

MR 1096825 | Zbl 0721.49038

@article{RLIN_1990_9_1_4_317_0,
     author = {Giovanni Bellettini and Maurizio Paolini and Claudio Verdi},
     title = {\( \Gamma \)-convergence of discrete approximations to interfaces with prescribed mean curvature},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {1},
     year = {1990},
     pages = {317-328},
     zbl = {0721.49038},
     mrnumber = {1096825},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1990_9_1_4_317_0}
}

Bellettini, Giovanni; Paolini, Maurizio; 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, Tome 1 (1990) pp. 317-328. http://gdmltest.u-ga.fr/item/RLIN_1990_9_1_4_317_0/

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