Browsing by Author "Barchiesi, Marco"

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    Harmonic Dipoles and the Relaxation of the Neo-Hookean Energy in 3D Elasticity
    (2023) Barchiesi, Marco; Henao, Duvan; Mora-Corral, Carlos; Rodiac, Remy
    We consider the problem of minimizing the neo-Hookean energy in 3D. The difficulty of this problem is that the space of maps without cavitation is not compact, as shown by Conti & De Lellis with a pathological example involving a dipole. In order to rule out this behaviour we consider the relaxation of the neo-Hookean energy in the space of axisymmetric maps without cavitation. We propose a minimization space and a new explicit energy penalizing the creation of dipoles. This new energy, which is a lower bound of the relaxation of the original energy, bears strong similarities with the relaxed energy of Bethuel-Brezis-Helein in the context of harmonic maps into the sphere.
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    On the lack of compactness in the axisymmetric neo-Hookean model
    (2024) Barchiesi, Marco; Henao, Duvan; Mora-Corral, Carlos; Rodiac, Remy
    We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti and De Lellis is generic in some sense. On this map, we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of $\mathbb {S}<^>2$ -valued harmonic maps.