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

Spectral gaps for the linear water-wave problem in a channel with thin structures

Peer reviewe

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 $L^2(\Omega ,R^n)$

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 $L^p\left( \Omega, \mathbf{R}^n\right)$

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

Electrospun nanofibers: From food to energy by engineered electrodes in microbial fuel cells

Microbial fuel cells (MFCs) are bio-electrochemical devices able to directly transduce chemical energy, entrapped in an organic mass named fuel, into electrical energy through the metabolic activity of specific bacteria. During the last years, the employment of bio-electrochemical devices to study the wastewater derived from the food industry has attracted great interest from the scientific community. In the present work, we demonstrate the capability of exoelectrogenic bacteria used in MFCs to catalyze the oxidation reaction of honey, employed as a fuel. With the main aim to increase the proliferation of microorganisms onto the anode, engineered electrodes are proposed. Polymeric nanofibers, based on polyethylene oxide (PEO-NFs), were directly electrospun onto carbon-based material (carbon paper, CP) to obtain an optimized composite anode. The crucial role played by the CP/PEO-NFs anodes was confirmed by the increased proliferation of microorganisms compared to that reached on bare CP anodes, used as a reference material. A parameter named recovered energy (Erec) was introduced to determine the capability of bacteria to oxidize honey and was compared with the Erec obtained when sodium acetate was used as a fuel. CP/PEO-NFs anodes allowed achieving an Erec three times higher than the one reached with a bare carbon-based anode