Singular perturbations as a selection criterion for periodic minimizing sequences
type: Item
instance of: scholarly article
title: Singular perturbations as a selection criterion for periodic minimizing sequences
zbMATH Open document ID: 0821.49015
DOI: 10.1007/BF01191616
published in: Calculus of Variations and Partial Differential Equations
publication date: 1995-09-18T00:00:00Z
review text: Mathematical models of solid-solid phase transformations and the simpler cases of liquid-vapour, and the simple problem of sailing into the wind, produce energy functionals that are not weakly lower semicontinuous in the setting of approximate Sobolev space. The simplest example is the minimization problem for the functional \[ \int^ 1_ 0 \bigl[ (u_ x^ 2 - 1)^ 2 + u^ 2 \bigr] dx, \] with periodic boundary conditions. Inclusion of surface energy leads to a perturbed functional: \[ \int^ 1_ 0 \bigl[ \varepsilon u^ 2_{xx} + (u^ 2_ x - 1)^ 2 + u^ 2 \bigr] dx. \] To improve generality of his results the author considers a double-well potential \(W\), with \(W(-1) = W(1) = 0\), \(W(z) = W(-z)\), \(W \in C^ 3\), \(W''(\pm 1) > 0\), \(W''(0) < 0\), \(\sigma = W'<0\) on \((- \infty, -1) \cup (0,1)\), \(\sigma > 0\) on \((-1,0) \cup (1, \infty)\), \(\lim_{z \to \pm \infty} \sigma (z) > 0\), and the functional \[ I^ \varepsilon = \int^ 1_ 0 \bigl[ \varepsilon^ 2 u^ 2_{xx} + W(u_ x) + u^ 2 \bigr] dx \] and \(A_ 0 = 2 \int^{+1}_{-1} W^{1/2} (z)dz\). He proves that there exists \(\varepsilon_ 0 > 0\) such that for any \(\varepsilon \in (0, \varepsilon_ 0)\) the following is true: If \(u^ \varepsilon\) minimizes \(I^ \varepsilon\) in the space of periodic functions \(H^ 2_ \#(0,1)\) then \(u^ \varepsilon\) is periodic with a minimal period of \(P^ \varepsilon : P^ \varepsilon = 2(6A_ 0 \varepsilon)^{1/3} + O(\varepsilon^{2/3})\). Moreover, \(u^ \varepsilon (x + {\varepsilon \over 2}) = - u^ \varepsilon (x)\) for all \(x \in (0,1)\), and \[ I^ \varepsilon (u^ \varepsilon) = {1 \over 4} (6A_ 0 \varepsilon)^{2/3} +O (\varepsilon^{4/3}). \] Therefore there are at most two distinct (up to translation) minimizers of \(I^ \varepsilon\), and there is a minimizer \(u^ \varepsilon\) such that \[ u^ \varepsilon (0) = u^ \varepsilon \left( {1 \over 2} \right) = 0, \quad u^ \varepsilon \left( {1 \over 2} - x \right) = - u^ \varepsilon \left( {1 \over 2} + x \right). \] The author interprets th…
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 35M10
zbMATH DE Number: 509964
zbMATH Keywords: solid-solid phase transition
zbMATH Keywords: singularly perturbed functionals
zbMATH Keywords: double- well potential
zbMATH Keywords: carbon steel
zbMATH Keywords: austenite
zbMATH Keywords: martensite
zbMATH Keywords: Euler-Lagrange equation
author: Stefan Müller
MaRDI profile type: Publication
cites work: Q3969238
cites work: Q5186236
cites work: Singularities and rank one properties of Hessian measures
cites work: Fine phase mixtures as minimizers of energy
cites work: Proposed experimental tests of a theory of fine microstructure and the two-well problem
cites work: Structured phase transitions on a finite interval
cites work: Metastable patterns in solutions of ut = ϵ2uxx − f(u)
cites work: Q4748693
cites work: Phase transitions of elastic solid materials
cites work: The gradient theory of phase transitions for systems with two potential wells
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Surface energy and microstructure in coherent phase transitions
cites work: Local minimisers and singular perturbations
cites work: One-dimensional infinite-horizon variational problems arising in continuum mechanics
cites work: Q3335096
cites work: The gradient theory of phase transitions and the minimal interface criterion
cites work: Nonconvex variational problems with anisotropic perturbations
cites work: Q4185363
cites work: Q3706042
full work available at URL: https://doi.org/10.1007/bf01191616
OpenAlex ID: W1970612242
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description: scientific article; zbMATH DE number 509964