Mixed Boundary Value Problems in Singularly Perturbed Two-Dimensional Domains with the Steklov Spectral Condition

We study the asymptotic behavior of the spectrum of the Laplace equation with the Steklov, Dirichlet, Neumann boundary conditions or their combination in a twodimensional domain with small holes of diameter O(ε) as ε → +0. We derive and justify asymptotic expansions of eigenvalues and eigenfunctions of two types: series in ʓ= | ln ε|−1 and power series with rational and holomorphic terms in ʓ respectively. For the overall Steklov problem we obtain asymptotic expansions in the low and middle frequency ranges of the spectrum. Bibliography: 18 titles

Steklov spectral problems in a set with a thin toroidal hole

The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity $Gamma_e$ having the shape of a thin toroidal set, with a constant cross-section of diameter $ell 1$. We construct the main terms of the asymptotic expansion of the eigenvalues in terms of real-analytic functions of the variable $|lne|^{-1}$, and we prove that the relative asymptotic error is of much smaller order $O(e|ln e|)$ as $e o 0^+$. The asymptotic analysis involves eigenvalues and eigenfunctions of a certain integral operator on the smooth curve $Gamma$, the axis of the cavity $Gamma_e$

The stiff Neumann problem: Asymptotic specialty and "kissing" domains

We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain Omega subset of R-d which is divided into two subdomains: an annulus Omega(1) and a core Omega(0). The density and the stiffness constants are of order epsilon(-2m) and epsilon(-1) in Omega(0), while they are of order 1 in( )Omega(1). Here m is an element of R is fixed and epsilon > 0 is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as epsilon -> 0 for any m. In dimension 2 the case when Omega(0) touches the exterior boundary partial derivative Omega S and Omega(1) gets two cusps at a point O is included into consideration. The possibility to apply the same asymptotic procedure as in the "smooth" case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as x -> O for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given

Correctors for some nonlinear monotone operators

In this paper we study homogenization of quasi-linear partial differential equations of the form -\mbox{div}\left( a\left( x,x/\varepsilon _h,Du_h\right) \right) =f_h on $\Omega$ with Dirichlet boundary conditions. Here the sequence $\left( \varepsilon _h\right)$ tends to $0$ as $h\rightarrow \infty$ and the map $a\left( x,y,\xi \right)$ is periodic in $y,$ monotone in $\xi$ and satisfies suitable continuity conditions. We prove that $u_h\rightarrow u$ weakly in $W_0^{1,p}\left( \Omega \right)$ as $h\rightarrow \infty ,$ where $u$ is the solution of a homogenized problem of the form -\mbox{div}\left( b\left( x,Du\right) \right) =f on $\Omega .$ We also derive an explicit expression for the homogenized operator $b$ and prove some corrector results, i.e. we find $\left( P_h\right)$ such that $Du_h-P_h\left( Du\right) \rightarrow 0$ in $L^p\left( \Omega, \mathbf{R}^n\right)$

Some homogenization and corrector results for nonlinear monotone operators

This paper deals with the limit behaviour of the solutions of quasi-linear equations of the form \ \ds -\limfunc{div}\left(a\left(x, x/{\varepsilon _h},Du_h\right)\right)=f_h on $\Omega$ with Dirichlet boundary conditions. The sequence $(\varepsilon _h)$ tends to $0$ and the map $a(x,y,\xi )$ is periodic in $y$, monotone in $\xi$ and satisfies suitable continuity conditions. It is proved that $u_h\rightarrow u$ weakly in $H_0^{1,2}(\Omega )$, where $u$ is the solution of a homogenized problem \ -\limfunc{div}(b(x,Du))=f on $\Omega$. We also prove some corrector results, i.e. we find $(P_h)$ such that $Du_h-P_h(Du)\rightarrow 0$ in $L^2(\Omega ,R^n)$

La tournée de Forêt Méditerranéenne - Récit d’une 11e tournée à la découverte des Parcs naturels d’Andalousie

Cette année, l’association Forêt Méditerranéenne a choisi d’organiser sa tournée forestière annuelle dans le sud de l’Espagne, en Andalousie. Du 5 au 8 mai 2016, plus de vingt participants sont ainsi partis à la découverte des plus grands Parcs et espaces naturels d’Espagne. Cet article est le compte rendu de ce voyage

On weak convergence of locally periodic functions

We prove a generalization of the fact that periodic functions converge weakly to the mean value as the oscillation increases. Some convergence questions connected to locally periodic nonlinear boundary value problems are also considered.Comment: arxiv version is already officia

The High Frequency Instrument of Planck: Requirements and Design

The Planck satellite is a project of the European Space Agency based on a wide international collaboration, including United States and Canadian laboratories. It is dedicated to the measurement of the anisotropy of the Cosmic Microwave Background (CMB) with unprecedented sensitivity and angular resolution. The detectors of its High frequency Instrument (HFI) are bolometers cooled down to 100 mK. Their sensitivity will be limited by the photon noise of the CMB itself at low frequencies, and of the instrument background at high frequencies. The requirements on the measurement chain are directly related to the strategy of observation used for the satellite. Due to the scanning on the sky, time features of the measurement chain are directly transformed into angular features in the sky maps. This impacts the bolometer design as well as other elements: For example, the cooling system must present outstanding temperature stability, and the amplification chain must show, down to very low frequencies, a flat noise spectrum

Use of High Sensitivity Bolometers for Astronomy: Planck High Frequency Instrument

The Planck satellite is dedicated to the measurement of the anisotropy of the Cosmic Microwave Background (CMB) with unprecedented sensitivity and angular resolution. It is a project of the European Space Agency based on a wide international collaboration, including United States and Canadian laboratories. The detectors of its High Frequency Instrument (HFI) are bolometers cooled down to 100 mK. Their sensitivity will be limited by the photon noise of the CMB itself at low frequencies, and of the instrument background at high frequencies. The requirements on the measurement chain are directly related to the strategy of observation used for the satellite. This impacts the bolometer design as well as other elements: The cooling system must present outstanding temperature stability, and the amplification chain must show a flat noise spectrum down to very low frequencies