Singular perturbations as a selection criterion for periodic minimizing sequences
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 this result by considering transformation of medium carbon steel from austenite (body centred lattice) to martensite (face centred lattice particles), where martensitic phase appears in certain ratio. In the next section the author collects some estimates for the functional \[ J_ \ell (v) = \int^{+ 1/2}_{- 1/2}(\varepsilon^ 2 \ell^{-1} v^ 2_{xx} + \ell W(v_ x) + \ell^ 3 v^ 2)dx, \] including the study of the corresponding Euler-Lagrange equation, bounds for eigenvalues of the linearized Euler-Lagrange operator, and stability. There are several detailed results which cannot be listed in a short review. For example, the author proves that for the Dirichlet conditions the minimizer of \(I^ \varepsilon (u)\) is the same in \(H^ 2 \cap H^ 1_ 0\) as in \(H^ 2_ \#\). Some comments: This is an excellent article containing many interesting results and comments. The author mentions the \(\Gamma\)-convergence (i.e., convergence of the minimizers, rather than in Sobolev norms) and ignores the more difficult concept of \(G\)-convergence of Lurie and Cherkaev. There is a need for a monograph comparing these approaches with the ``classical'' perturbation theories, multiple scales, and the probabilistic theory proposed many years ago by Avner Friedman. This remark implies no criticism of this outstanding paper.
scientific article; zbMATH DE number 509964
scientific article