Browsing by Author "Rodiac, Remy"

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    Boundary regularity of weakly anchored harmonic maps
    (2015) Contreras, Andres; Lamy, Xavier; Rodiac, Remy
    In this note, we study the boundary regularity of the minimizers of a family of weak anchoring energies that model the states of liquid crystals. We establish optimal boundary regularity in all dimensions n >= 3. In dimension n = 3, this yields full regularity at the boundary, which stands in sharp contrast with the observation of boundary defects in physics works. We also show that, in the cases of weak and strong anchoring, the regularity of the minimizers is inherited from that of their corresponding limit problems. (C) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
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    Existence and non-existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees
    (WORLD SCIENTIFIC PUBL CO PTE LTD, 2016) Dos Santos, Mickael; Rodiac, Remy
    Let D = Omega\(omega) over bar subset of R-2 be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy E-epsilon (u) = 1/2 integral (D) vertical bar del u vertical bar(2) + 1/4s(2) integral (D) (1 - vertical bar u vertical bar(2))(2) where u : D -> C, and look for minimizers of E-epsilon with prescribed degrees deg (u, partial derivative Omega) = p, deg (u, partial derivative omega) = q on the boundaries of the domain. For large epsilon and for balanced degrees (i.e. p = q), we obtain existence of minimizers for domains with large capacity ( corresponding to thin annulus). We also prove non-existence of minimizers of E-epsilon, for large epsilon, if p not equal q, pq > 0 and if D is a circular annulus with large capacity. Our approach relies on similar results obtained for the Dirichlet energy E-infinity (u) = 1/2 integral (D) vertical bar del u vertical bar(2), on a previous existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.
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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 Convergence of Minimizers of Singular Perturbation Functionals
    (2018) Contreras, Andres; Lamy, Xavier; Rodiac, Remy
    The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the geometric-driven profile of ground states. In this work, we study, under very general assumptions, the convergence of minimizers towards harmonic maps. We show that the convergence is locally uniform up to the boundary, away from the lower-dimensional singular set. Our results generalize related findings, most notably in the theory of liquid-crystals, to all dimensions n >= 3, and to general nonlinearities. Our proof follows a well-known scheme, relying on a small energy estimate and a monotonicity formula. It departs substantially from previous studies in the treatment of the small energy estimate at the boundary, since we do not rely on the specific form of the potential. In particular, this extends existing results in three-dimensional settings. In higher dimensions, we also deal with additional difficulties concerning the boundary monotonicity formula.
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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.