Phase transitions in multiphase media taking into account the interface surface energy by an integral of higher derivatives of the displacement field
type: Item
instance of: scholarly article
zbMATH Open document ID: 0983.74046
author: A. S. Mikhajlov
publication date: 2002-02-24T00:00:00Z
Mathematics Subject Classification ID: 74N20
Mathematics Subject Classification ID: 74G65
zbMATH DE Number: 1710233
zbMATH Keywords: multiphase medium
zbMATH Keywords: phase transformation
zbMATH Keywords: minimization
zbMATH Keywords: energy functional
zbMATH Keywords: existence of equilibrium state
zbMATH Keywords: higher derivative of displacement field
zbMATH Keywords: interface surface energy
MaRDI profile type: Publication
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title: Phase transitions in multiphase media taking into account the interface surface energy by an integral of higher derivatives of the displacement field
published in: Problemy Matematicheskogo Analiza
review text: The author studies the minimization of energy functional NEWLINE\[NEWLINE I(u,\chi,\tau,\sigma)= \sigma\|u\|_H^p + \int_{\Omega}\chi^{\lambda}(x) F^{\lambda}\bigl(\dot{u}(x),u(x),x,\tau\bigr) dx NEWLINE\]NEWLINE which acts on the set of functions NEWLINE\[NEWLINE X = \bigl\{\{u,\chi\}: u\in H = W_2^2(\Omega,\mathbb R^m)\cap \overset\circ{W}^1_2(\Omega,\mathbb R^m),\;\chi = (\chi^+,\chi^-,\chi^0)\bigr\}, NEWLINE\]NEWLINE where \(0 < p <1\), \(\sigma > 0\), \(\Omega\subset\mathbb R^m\) is a bounded domain with Lipschitz boundary, \(\chi^{\lambda}\) are measurable characteristic functions, \(\lambda = +, -, 0\), and \(\chi^+(x) + \chi^-(x) + \chi^0(x) = 1\) for a.e. \(x\in\Omega\). The functional is minimized under the additional constraint NEWLINE\[NEWLINE \int_{\Omega}\bigl(\rho^+\chi^+(x) + \rho^-\chi^-(x) + \rho^0\chi^0(x)\bigr) dx = M, NEWLINE\]NEWLINE where \(|\Omega|\min\{\rho^+, \rho^-, \rho^0\} \leq M \leq |\Omega|\max\{\rho^+, \rho^-, \rho^0\}\).NEWLINENEWLINENEWLINEThe author proves existence of an equilibrium state for the considered functional and studies dependence of equilibrium states on parameters \(\sigma\) and \(\tau\). In addition, he compares two approaches for regularizing the energy functional which are based on using higher derivatives of the displacement field and the area of interface surface.
reviewed by: Vladimir N. Grebenev
description: scientific article; zbMATH DE number 1710233