Geodesic
\( \Gamma \)-convergence of discrete approximations to interfaces with prescribed mean curvature
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 1 (1990) no. 4, pp. 317-328.

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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.
Si studia l'approssimazione numerica del seguente problema di minimo: \( \min_{A \subseteq \Omega} \tilde{\mathcal{F}}(A) \), ove \( \tilde{\mathcal{F}}(A) = P_{\Omega}(A) + \cos(\theta) \mathcal{H}^{n-1}(\partial A \cap \partial \Omega) - \int_{A} \kappa \), teso alla ricerca di un insieme \( A \subseteq \Omega \subset \mathbb{R}^{n} \) con curvatura media \( \kappa \) e angolo \( \theta \) all'intersezione di \( \partial A \) con \( \partial \Omega \). Il funzionale \( \tilde{\mathcal{F}} \) viene preliminarmente rilassato mediante una successione di funzionali non convessi definiti in \( H^{1}(\Omega) \), che sono successivamente discretizzati con elementi finiti. Si dimostrano la \( \Gamma \)-convergenza dei funzionali discreti al funzionale \( \tilde{\mathcal{F}} \) e la compattezza di qualunque successione di minimi assoluti dei funzionali discreti. 
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     author = {Bellettini, Giovanni and Paolini, Maurizio and Verdi, Claudio},
     title = {\( {\Gamma} \)-convergence of discrete approximations to interfaces with prescribed mean curvature},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
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     volume = {Ser. 9, 1},
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Bellettini, Giovanni; Paolini, Maurizio; Verdi, Claudio. \( \Gamma \)-convergence of discrete approximations to interfaces with prescribed mean curvature. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 1 (1990) no. 4, pp. 317-328. http://geodesic.mathdoc.fr/item/RLIN_1990_9_1_4_a6/

[1] L. Ambrosio, Variational problems in SBV. Acta Applicandae Matematicae, 17, 1989, 1-40. | DOI | MR | Zbl

[2] S. Baldo, Minimal interface criterion for phase transitions and mixture of a Cahn Hilliard fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. | fulltext mini-dml | MR | Zbl

[3] S. Baldo - G. Bellettini, \( \Gamma \)-convergence and numerical analysis: an application to the minimal partitions problem. Ricerche di Matematica, to appear. | MR | Zbl

[4] G. Bellettini - M. Paolini - C. Verdi, Numerical minimization of geometrical type problems related to calculus of variations. Calcolo, to appear. | DOI | MR | Zbl

[5] G. Bellettini - M. Paolini - C. Verdi, Front-tracking and variational methods to approximate interfaces with prescribed mean curvature. Proceedings of the Conference on Numerical Methods for Free Boundary Problems, Birkhäuser, Stuttgart, to appear. | MR | Zbl

[6] P. G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam 1978. | MR | Zbl

[7] G. Dal Maso - J. M. Morel - S. Solimini, A variational method in image segmentation: existence and approximation results. Preprint S.I.S.S.A., Trieste, 48 M, 1988, 1-79. | DOI | MR | Zbl

[8] E. De Giorgi, Free discontinuity problems in calculus of variations. Proceedings of the Meeting in honour of J. L. Lions (6-10/6/1988). North-Holland, Amsterdam, to appear. | MR | Zbl

[9] E. De Giorgi - L. Ambrosio, Su un nuovo tipo di funzionale del calcolo delle variazioni. Atti Acc. Lincei Rend. fis., s. 8, vol. 82, fasc. 2, 1988, 199-210. | MR | Zbl

[10] E. De Giorgi - M. Carriero - A. Leaci, Existence theorems for a minimum problems with free discontinuity sets. Arch. Rational Mech. Anal., 108, 1989, 195-218. | DOI | MR | Zbl

[11] E. De Giorgi - T. Franzoni, Su un tipo di convergenza variazionale. Atti Acc. Lincei Rend. fis., s. 8, vol. 58, 1975, 842-850. | MR | Zbl

[12] H. Federer, Geometric Measure Theory. Springer Verlag, Berlin 1968. | MR | Zbl

[13] R. Finn, Equilibrium Capillary Surfaces. Springer Verlag, Berlin 1986. | DOI | MR | Zbl

[14] L. Modica, The gradient theory of phase transitions and minimal interface criterion. Arch. Rational Mech. Anal., 98, 1987, 123-142. | DOI | MR | Zbl

[15] L. Modica - S. Mortola, Un esempio di \( \Gamma \)-convergenza. Boll. Un. Mat. Ital., 5, 14 B, 1977, 285-299. | MR | Zbl

[16] D. Mumford - J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Applied Math., 42, 1989, 577-685. | DOI | MR | Zbl

[17] N. C. Owen - J. Rubinstein - P. Sternberg, Minimizer and gradient flows for singularly perturbed bistable potentials with a Dirichlet condition. To appear. | Zbl

[18] W. P. Ziemer, Weakly Differentiable Functions. Springer Verlag, Berlin 1989. | DOI | MR | Zbl