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By
Hopper, Christopher P.
3 Citations
We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Our approach uses the lifting of Sobolev mappings to the universal covering space, the connectedness of the covering space, an application of Ekeland’s variational principle and a certain tangential
$${\mathbb{A}}$$
harmonic approximation lemma obtained directly via a Lipschitz approximation argument. This allows regularity to be established directly on the level of the gradient. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium3 and nematic liquid crystals.
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By
Knüpfer, Hans; Muratov, Cyrill B.; Nolte, Florian
2 Citations
We investigate the scaling of the ground state energy and optimal domain patterns in thin ferromagnetic films with strong uniaxial anisotropy and the easy axis perpendicular to the film plane. Starting from the full threedimensional micromagnetic model, we identify the critical scaling for which the transition from single domain to multidomain ground states such as bubble or maze patterns occurs as the film thickness goes to zero and the lateral extent goes to infinity. Furthermore, we analyze the asymptotic behavior of the energy in these two asymptotic regimes. In the single domain regime, the energy Γconverges towards a much simpler twodimensional and local model. In the multidomain regime, we derive the scaling of the minimal energy and deduce a scaling law for the typical domain size.
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By
Pinaud, Olivier
3 Citations
This work is concerned with the semiclassical analysis of mixed state solutions to a Schrödinger–Position equation perturbed by a random potential with weak amplitude and fast oscillations in time and space. We show that the Wigner transform of the density matrix converges weakly and in probability to solutions of a Vlasov–Poisson–Boltzmann equation with a linear collision kernel.Aconsequence of this result is that a smooth nonlinearity such as the Poisson potential (repulsive or attractive) does not change the statistical stability property of the Wigner transform observed in linear problems.We obtain, in addition, that the local density and current are selfaveraging, which is of importance for some imaging problems in random media. The proof brings together the martingale method for stochastic equations with compactness techniques for nonlinear PDEs in a semiclassical regime. It relies partly on the derivation of an energy estimate that is straightforward in a deterministic setting but requires the use of a martingale formulation and wellchosen perturbed test functions in the random context.
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By
Ball, John M.; Zarnescu, Arghir
64 Citations
Uniaxial nematic liquid crystals are modelled in the Oseen–Frank theory through a unit vector field n. This theory has the apparent drawback that it does not respect the headtotail symmetry in which n should be equivalent to −n. This symmetry is preserved in the constrained Landau–de Gennes theory that works with the tensor
$${Q=s \left(n\otimes n\frac{1}{3} Id\right)}$$
. We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simplyconnected domains and in the natural energy class W^{1,2} the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains with holes and various boundary conditions, for the simplest form of the energy functional, we completely characterise the instances in which the predictions of the constrained Landau–de Gennes theory differ from those of the Oseen–Frank theory.
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By
Zhang, Tong; Zheng, Yuxi
17 Citations
Abstract.
We construct rigorously a three‐parameter family of self‐similar, globally bounded, and continuous weak solutions in two space dimensions for all positive time to the Euler equations with axisymmetry for polytropic gases with a quadratic pressure‐density law. We use the axisymmetry and self‐similarity assumptions to reduce the equations to a system of three ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. These solutions exhibit familiar structures seen in hurricanes and tornadoes. They all have finite local energy and vorticity with well‐defined initial and boundary values. These solutions include the one‐parameter family of explicit solutions reported in a recent article of ours.
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