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We investigate Lamé systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it yields natural correctors that facilitate error estimates. A key ingredient is the asymptotic analysis for the rescaled cell problems, and this is studied by exploring the convergence of the periodic layer potentials for the Lamé system to those in the whole space when the period tends to infinity.

We address the topology of the set of singularities of a solution of a Hamilton-Jacobi equation. For this we will apply the idea of the first two authors (Generalized characteristic and Lax-Oleinik operators: global result, preprint, arXiv:1605.07581, 2016.) to use the positive Lax-Oleinik semi-group to propagate singularities. {\it To cite this article: P. Cannarsa, W. Cheng, A. Fathi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

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In this article, we study wave dynamics in the fractional nonlinear Schrödinger equation with a modulated honeycomb potential. This problem arises from recent research interests in the interplay between topological materials and nonlocal governing equations. Both are current focuses in scientific research fields. We first develop the Floquet-Bloch spectral theory of the linear fractional Schrödinger operator with a honeycomb potential. Especially, we prove the existence of conical degenerate points, i.e., Dirac points, at which two dispersion band functions intersect. We then investigate the dynamics of wave packets spectrally localized at a Dirac point and derive the leading effective envelope equation. It turns out the envelope can be described by a nonlinear Dirac equation with a varying mass. With rigorous error estimates, we demonstrate that the asymptotic solution based on the effective envelope equation approximates the true solution well in the weighted-H^s space.