Abstract:
This paper is devoted to the homogenization of Shr\"odinger type equations with periodically oscillating coefficients of the diffusion term, and a rapidly oscillating periodic time-dependent potential. One convergence theorem is proved and we derive the macroscopic homogenized model. Our approach is the well known two-scale convergence method.

Abstract:
The paper deals with homogenization problem for nonlinear elliptic and parabolic equations in a periodically perforated domain, a nonlinear Fourier boundary conditions being imposed on the perforation border. Under the assumptions that the studied differential equation satisfies monotonicity and 2-growth conditions and that the coefficient of the boundary operator is centered at each level set of unknown function, we show that the problem under consideration admits homogenization and derive the effective model.

Abstract:
We consider energies defined as the Dirichlet integral of curves taking values in fast-oscillating manifolds converging to a linear subspace. We model such manifolds as subsets of $R^{m+m'}$ described by a constraint $(x_{m+1},\ldots,x_{m'})= \delta \, \varphi(x_1/\epsilon,\ldots, x_m/\epsilon)$ where $\epsilon$ is the period of the oscillation, $\delta$ its amplitude and $\varphi$ its profile. The interesting case is $\epsilon<\!<\delta<\!<1$, in which the limit of the energies is described by a Finsler metric on $R^{m}$ which is defined by optimizing the contribution of oscillations on each level set $\{\varphi=c\}$. The formulas describing the limit mix homogenization and convexification processes, highlighting a multi-scale behaviour of optimal sequences. We apply these formulas to show that we may obtain all (homogeneous) symmetric Finsler metrics larger than the Euclidean metric as limits in the case of oscillating surfaces in $R^{3}$

Abstract:
We prove a homogenization result for integral functionals in domains with oscillating boundaries, showing that the limit is defined on a degenerate Sobolev space. We apply this result to the description of the asymptotic behaviour of thin films with fast-oscillating profile, proving that they can be described by first applying the homogenization result above and subsequently a dimension-reduction technique.

Abstract:
We study the homogenization of a Schrodinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory.

Abstract:
This note is a summary of the recent paper [9]. Here, we study the homogenization of elliptic systems with Dirichlet boundary condition, when both the coefficients and the boundary datum are oscillating. In particular, in the paper [9], we showed that, the solutions converge in L2 with a power rate, and we identified the homogenized limit system and the homogenized boundary data. Due to a boundary layer phenomenon, this homogenized system depends in a non trivial way on the boundary. The analysis in [9] answers a longstanding open problem, raised for instance in [4]

Abstract:
Rapidly oscillating Ap (roAp) stars have rarely been found in binary or higher order multiple systems. This might have implications for their origin. We intend to study the multiplicity of this type of chemically peculiar stars, looking for visual companions in the range of angular separation between 0.05" and 8". We carried out a survey of 28 roAp stars using diffraction-limited near-infrared imaging with NAOS-CONICA at the VLT. Additionally, we observed three non-oscillating magnetic Ap stars. We detected a total of six companion candidates with low chance projection probabilities. Four of these are new detections, the other two are confirmations. An additional 39 companion candidates are very likely chance projections. We also found one binary system among the non-oscillating magnetic Ap stars. The detected companion candidates have apparent K magnitudes between 6.8 and 19.5 and angular separations ranging from 0.23" to 8.9", corresponding to linear projected separations of 30-2400AU. While our study confirms that roAp stars are indeed not very often members of binary or multiple systems, we have found four new companion candidates that are likely physical companions. A confirmation of their status will help understanding the origin of the roAp stars.

Abstract:
A homogenization result for a family of integral energies is presented, where the fields are subjected to periodic first order oscillating differential constraints in divergence form. The work is based on the theory of A -quasiconvexity with variable coefficients and on two- scale convergence techniques.

Abstract:
We prove the homogenization of the Dirichlet problem for fully nonlinear elliptic operators with periodic oscillation in the operator and of the boundary condition for a general class of smooth bounded domains. This extends the previous results of Barles and Mironescu in half spaces. We show that homogenization holds despite a possible lack of continuity in the homogenized boundary data. The proof is based on a comparison principle with partial Dirichlet boundary data which is of independent interest.

Abstract:
This paper deals with the homogenization of fully nonlinear second order equation with an oscillating Dirichlet boundary data when the operator and boundary data are $\e$-periodic. We will show that the solution $u_\e$ converges to some function $\bar u(x)$ uniformly on every compact subset $K$ of the domain $D$. Moreover, $\bar u$ is a solution to some boundary value problem. For this result, we assume that the boundary of the domain has no (rational) flat spots and the ratio of elliptic constants $\Lambda / \lambda$ is sufficiently large.