Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

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Aspects of topology: On dimension theory

A large number of polycrystalline materials, both manmade and natural, display preferred orientation of crystallites. Such alignment has a profound effect on anisotropy of physical properties. Preferred orientation or texture forms during growth or deformation and is modified during recrystallization or phase transformations and theories exist to predict its origin. Different methods are applied to characterize orientation patterns and determine the orientation distribution, most of them relying on diffraction. Conventionally x-ray polefigure goniometers are used. More recently single orientation measurements are performed with electron microscopes, both SEM and TEM. For special applications, particularly texture analysis at non-ambient conditions, neutron diffraction and synchrotron x-rays have distinct advantages. The review emphasizes such new possibilities. A second section surveys important texture types in a variety of materials with emphasis on technologically important systems and in rocks that contribute to anisotropy in the earth. In the former group are metals, structural ceramics and thin films. Seismic anisotropy is present in the crust (mainly due to phyllosilicate alignment), the upper mantle (olivine), the lower mantle (perovskite and magnesiowuestite) and the inner core (e-iron) and due to alignment by plastic deformation. There is new interest in the texturing of biological materials such as bones and shells. Preferred orientation is not restricted to inorganic substances but is also present in polymers that are not discussed in this review.

http://iopscience.iop.org/article/10.1088/0034-4885/67/8/R02/pdf

Texture and anisotropy

Multiple integrals in the calculus of variations and nonlinear elliptic systems

Linear operators part ii (spectral theory)

We analyse symmetry breaking for ground states of the Henon equation [7] in a ball. Asymptotic estimates of the transition are also given when p is close to either 2 or 2*.

Non-radial ground states for the hénon equation

We study a time-independent nonlinear Schrodinger equation with an attractive inverse square potential and a nonautonomous nonlinearity whose power is the critical Sobolev exponent. The problem shares a strong resemblance with the prescribed scalar curvature problem on the standard sphere. Particular attention is paid to the blow-up possibilities, i.e. the critical points at infinity of the corresponding variational problem. Due to the strong singularity in the potential, some new phenomenon appear. A complete existence result is obtained in dimension 4 using a detailed analysis of the gradient flow lines.

/pdf/nonlinear-schrodinger-equations-with-hardy-potential-and-2rkjzty74g.pdf

Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities

A short elementary proof based on polarizations yields a useful (new) rearrangement inequality for symmetrically weighted Dirichlet type functionals. It is then used to answer some symmetry related open questions in the literature. The non symmetry of the Henon equation ground states (previously proved in [19]) as well as their asymptotic behavior are analyzed more in depth. A special attention is also paid to the minimizers of the Caffarelli-Kohn-Nirenberg [8] inequalities.

Partial symmetry and asymptotic behavior for some elliptic variational problems

Solitary waves with prescribed speed on infinite lattices

. We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension N ≥ 3. We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg– Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3).

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=6&iss=1&rank=2

Didier Smets

Papers

Non-radial ground states for the hénon equation

Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities

Partial symmetry and asymptotic behavior for some elliptic variational problems

Solitary waves with prescribed speed on infinite lattices

Vortex rings for the Gross-Pitaevskii equation