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Singular perturbation and the energy of folds. (English) Zbl 0973.49009

The paper deals with the asymptotic behaviour as \(\varepsilon\downarrow 0\) of the energy functionals \[ E_\varepsilon(u):= \int_\Omega \varepsilon|\nabla\nabla u|^2+ {(1-|\nabla u|^2)^2\over\varepsilon} dx \tag{1} \] and contains a relevant progress towards the conjecture that the limiting energy is given by \[ {1\over 3}\int_D |[\nabla u]|^3 d{\mathcal H}^1 \tag{2} \] (here \(D\) is the discontinuity set of \(\nabla u\) and \([\nabla u]\) is its jump) on a suitable space of functions satisfying the eikonal equation \(|\nabla u|=1\) a.e. in \(\Omega\). The problem has several physical and mathematical motivations. Among the physical ones we mention that energies of this sort arise in the theory of smectic liquid crystals, in a model for blisters of compressed thin films and in the Cross-Newell theory of phase formation. Analogous models also appear in micromagnetics, where curl-free fields (gradients) are replaced by divergence free fields. On the mathematical side, the analysis of functionals (1) is intriguing because the zero-curl constraint leads to essential technical difficulties which have not been fully solved so far. Nowithstanding these difficulties, the paper contains several fundamental contributions to the problem:
(a) Using a new scheme for proving energy lower bounds (based on null lagrangians given by fields whose divergence can be controlled from above with the energy), the formula (2) for the limit energy is supported. Moreover, in the case when \(\Omega\) is an ellipse and the boundary conditions \(u=0\), \(u_\nu=-1\) on \(\partial\Omega\) hold, this scheme allows an explicit computation and confirms (2) as the limiting energy.
(b) An example of nonuniqueness for the limit problem is given.
(c) The basic ansatz of constancy of tangential derivative, leading to the form (2) of the limit energy, is shown to be false for an anisotropic version of the functionals (1).
(d) An example showing that the viscosity solution of the eikonal equation with boundary condition \(u=0\) on \(\partial\Omega\) is not a minimizer of the functional \(\int_D |[\nabla u]|^\beta d{\mathcal H}^1\) when \(\beta\neq 3\) is given.
Reviewer: L.Ambrosio (Pisa)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
35J60 Nonlinear elliptic equations
74G60 Bifurcation and buckling
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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