Convex approximation of an inhomogeneous anisotropic functional

Bellettini, Giovanni, and Paolini, Maurizio. "Convex approximation of an inhomogeneous anisotropic functional." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.2 (1994): 177-187. <http://eudml.org/doc/244308>.

@article{Bellettini1994,
abstract = {The numerical minimization of the functional \( \mathcal\{F\} (u) = \int\_\{\Omega\} \phi (x,\nu\_\{u\}) |Du| + \int\_\{\partial \Omega\} \mu u \, d\mathcal\{H\}^\{n-1\} - \int\_\{\Omega\} \kappa u \, dx \), \( u \in BV(\Omega; \\{-1, 1\\}) \) is addressed. The function \( \phi \) is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that \( \mathcal\{F\} \) can be equivalently minimized on the convex set \( BV(\Omega; \left[-1, 1\right]) \) and then regularized with a sequence \( \\{\mathcal\{F\}\_\{\epsilon\}(u)\\}\_\{\epsilon\} \), of stricdy convex functionals defined on \( BV(\Omega; \left[-1, 1\right]) \). Then both \( \mathcal\{F\} \) and \( \mathcal\{F\}\_\{\epsilon\} \), can be discretized by continuous linear finite elements. The convexity property of the functionals on \( BV(\Omega; \left[-1, 1\right]) \) is useful in the numerical minimization of \( \mathcal\{F\} \). The \( \Gamma — L\_\{1\} (\Omega) \)-convergence of the discrete functionals \( \\{ \mathcal\{F\}\_\{h\} \\}\_\{h\} \) and \( \\{ \mathcal\{F\}\_\{\epsilon,h\} \\}\_\{\epsilon,h\} \) to \( \mathcal\{F\} \), as well as the compactness of any sequence of discrete absolute minimizers, are proven.},
author = {Bellettini, Giovanni, Paolini, Maurizio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Calculus of variations; Anisotropic surface energy; Finite elements; Convergence of discrete approximations; convex approximation; anisotropic functional; phase transitions; crystal growth; finite element; uniform convergence},
language = {eng},
month = {6},
number = {2},
pages = {177-187},
publisher = {Accademia Nazionale dei Lincei},
title = {Convex approximation of an inhomogeneous anisotropic functional},
url = {http://eudml.org/doc/244308},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Bellettini, Giovanni
AU - Paolini, Maurizio
TI - Convex approximation of an inhomogeneous anisotropic functional
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/6//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 2
SP - 177
EP - 187
AB - The numerical minimization of the functional \( \mathcal{F} (u) = \int_{\Omega} \phi (x,\nu_{u}) |Du| + \int_{\partial \Omega} \mu u \, d\mathcal{H}^{n-1} - \int_{\Omega} \kappa u \, dx \), \( u \in BV(\Omega; \{-1, 1\}) \) is addressed. The function \( \phi \) is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that \( \mathcal{F} \) can be equivalently minimized on the convex set \( BV(\Omega; \left[-1, 1\right]) \) and then regularized with a sequence \( \{\mathcal{F}_{\epsilon}(u)\}_{\epsilon} \), of stricdy convex functionals defined on \( BV(\Omega; \left[-1, 1\right]) \). Then both \( \mathcal{F} \) and \( \mathcal{F}_{\epsilon} \), can be discretized by continuous linear finite elements. The convexity property of the functionals on \( BV(\Omega; \left[-1, 1\right]) \) is useful in the numerical minimization of \( \mathcal{F} \). The \( \Gamma — L_{1} (\Omega) \)-convergence of the discrete functionals \( \{ \mathcal{F}_{h} \}_{h} \) and \( \{ \mathcal{F}_{\epsilon,h} \}_{\epsilon,h} \) to \( \mathcal{F} \), as well as the compactness of any sequence of discrete absolute minimizers, are proven.
LA - eng
KW - Calculus of variations; Anisotropic surface energy; Finite elements; Convergence of discrete approximations; convex approximation; anisotropic functional; phase transitions; crystal growth; finite element; uniform convergence
UR - http://eudml.org/doc/244308
ER -