Phase transitions with a minimal number of jumps in the singular limits of higher order theories

Plotnikov, P.I.; Toland, J.F.

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 655-691.

For a smooth $W:(0,\infty )\times {ℝ}^d\to ℝ$ and a family of L-periodic $W^{1,2}$-functions ${\vartheta }_{ϵ}:ℝ\to {ℝ}^d$ with ${\vartheta }_{ϵ}\rightharpoonup \vartheta$, the basic problem is to understand the weak* limit as $ϵ\to 0$ of L-periodic minimizers of

$$\begin{array}{cc}\underset{0}{\stackrel{L}{\int }}(\frac{ϵ}{2}{\varphi }^{\text{'}2}+W(\varphi ,{\vartheta }_{ϵ}))ds.& \text{(†)}\end{array}$$

It is assumed that $W(\phi ,\theta )\to \infty$ as $\phi \to 0,\infty$, and that $W(·,\theta )$, which has no more than three critical points counting multiplicity depending on $\theta \in {ℝ}^d$, is of a type that arises in the Cahn–Hilliard theory of phase separations where $d=1$. The limiting problem with $ϵ=0$ is to minimize, over bounded L-periodic measurable functions φ,

$$\begin{array}{cc}\underset{0}{\stackrel{L}{\int }}W(\varphi \left(s\right),\vartheta \left(s\right))ds.& \text{(‡)}\end{array}$$

Minimizers of (‡) need not be unique (there may be uncountably many), they may be discontinuous and minimizers with only simple jumps may coexist with minimizers with much more complicated discontinuities. Weak* limits of minimizers of (†) as $ϵ\to 0$ are minimizers of the relaxation of (‡). However it is shown that if, for a sequence of minimizers of (†),

$$\underset{k\to \infty }{\lim \sup }\sqrt{{ϵ}_k}\underset{0}{\stackrel{L}{\int }}|{\varphi }_{{ϵ}_k}^{\text{'}}\left(s\right){|}^{2}ds<\infty ,{ϵ}_k\to 0,$$

then the weak* limit of any subsequence of $\left\{{\varphi }_{{ϵ}_k}\right\}$ is an actual minimizer of (‡) which is continuous except at a finite number of simple jumps. Moreover, for sequences ${ϵ}_k\to 0$ from a set of positive Lebesgue density, it is shown that the weak* limit of L-periodic minimizers of (†) is a minimizer of (‡) with a finite number of simple jumps. Under additional hypotheses it is shown that, for sequences from a set of full Lebesgue density, the weak* limits of L-periodic minimizers of (†) are minimizers of (‡) with a minimal number of simple jumps.

[1] N.D. Alikakos, P.W. Bates, On the singular limit in phase field model of phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), 141-178 | EuDML | Numdam | MR | Zbl

[2] N.D. Alikakos, H.C. Simpson, A variational approach for a class of singular perturbation problems and applications, Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), 27-42 | MR | Zbl

[3] V.I. Arnold, Mathematical Methods of Classical Mechanics, Grad. Texts in Math. vol. 60, Springer, New York (1989) | MR

[4] A. Braides, Γ-Convergence for Beginners, Oxford Univ. Press, Oxford (2002) | MR

[5] J. Carr, M.E. Gurtin, M. Slemrod, Structured phase transitions on a finite interval, Arch. Ration. Mech. Anal. 86 (1984), 317-351 | MR | Zbl

[6] G. Buttazzo, G. Dal Maso, Singular perturbation problems in the calculus of variations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 11 (1984), 395-430 | EuDML | Numdam | MR

[7] J.L. Ericksen, Equilibrium of bars, J. Elasticity 5 (1975), 191-201 | MR | Zbl

[8] I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer, New York (2007) | MR | Zbl

[9] R.V. Kohn, S. Müller, Surface energy and microstructure in coherent phase transitions, Comm. Pure Appl. Math. 47 (1994), 405-435 | MR | Zbl

[10] M. Lilli, Qualitative behaviour of local minimizers of singular perturbed variational problems, J. Elasticity 87 (2007), 73-94 | MR | Zbl

[11] L. Modica, S. Mortola, Un esempio di Γ-convergenza, Boll. Unione Mat. Ital. (5) 14 (1977), 285-299 | MR | Zbl

[12] S. Müller, Singular perturbations as a selection criterion for periodic minimizing sequences, Calc. Var. Partial Differential Equations 1 (1993), 169-204 | MR | Zbl

[13] P.I. Plotnikov, J.F. Toland, Strain-gradient theory of hydroelastic travelling waves and their singular limits, University of Bath, Preprint, 2009

[14] M. Struwe, Variational Methods, Ergeb. Math. Grenzgeb. vol. 34, Springer-Verlag, Berlin (2008) | MR