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A \(\Gamma\)-convergence result for the two-gradient theory of phase transitions. (English) Zbl 1029.49040

This 79-pages long paper deals with the asymptotic behaviour of some double-well singularly perturbed functionals for gradient vector fields.
Let \(W\colon {\mathbb R}^{d\times N}\to[0,+\infty[\) be a continuous double-well potential such that \(W(z)=0\) if and only if \(z\in\{A,B\}\), where \(A-B=a\otimes\nu\) for some \(a\in{\mathbb R}^d\backslash\{0\}\) and \(\nu\) in the unit sphere in \({\mathbb R}^N\), and let for every smooth bounded open set \(\Omega\), \(u\in W^{1,1}(\Omega;{\mathbb R}^d)\), and \(\varepsilon>0\), \[ I_\varepsilon(u,\Omega)=\begin{cases}\int_\Omega{1\over\varepsilon}W(\nabla u)+\varepsilon|\nabla^2u|^2dx & \text{ if } u\in W^{2,2}(\Omega;{\mathbb R}^d)\cr +\infty & \text{otherwise}.\end{cases} \] Then, under various sets of additional assumptions on \(W\), it is proved that as \(\varepsilon\to 0\) the functionals \(I_\varepsilon\) \(\Gamma\)-converge to \[ I(u,\Omega)=\begin{cases} K^*{\mathcal H}^{N-1}(S(\nabla u)\cap\Omega) & \text{if } u\in W^{2,2}(\Omega;{\mathbb R}^d), \nabla u\in \text{BV}(\Omega;\{A,B\})\cr +\infty & \text{ otherwise },\end{cases} \] where \(S(\nabla u)\) is the singular set of \(\nabla u\), \[ K^*=\inf\bigg\{\liminf_{n\to\infty}I_{\varepsilon_n}(u_n,Q_\nu) : \varepsilon_n\to 0^+,\;\{u_n\}\subseteq W^{2,2}(Q_\nu;{\mathbb R}^d),\;u_n\to u_0\text{ in }L^1(Q_\nu;{\mathbb R}^d)\bigg\}, \] \(Q_\nu\) is a unit cube in \({\mathbb R}^N\) centered at the origin and with two of its faces orthogonal to \(\nu\), and \[ \nabla u_0=\begin{cases} A & \text{if } x\cdot\nu>0\cr B & \text{ if } x\cdot\nu<0. \end{cases} \] Variants of the above result are proved depending on the different sets of additional assumptions on \(W\).
It is also proved that, under some assumptions on \(W\), the asymptotic problem has a one-dimensional character, and \(K^*\) reduces to the analogue of some constants existing in vector-valued phase transitions theory.
Some examples are also discussed.
An appendix, where the Poincaré inequality is generalized to Orlicz-Sobolev spaces, completes the paper.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
35B25 Singular perturbations in context of PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
26D10 Inequalities involving derivatives and differential and integral operators
49J10 Existence theories for free problems in two or more independent variables
74G65 Energy minimization in equilibrium problems in solid mechanics
74N15 Analysis of microstructure in solids
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