Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains

We study the Laplacian with zero magnetic field acting on complex functions of a planar domain Ω, with magnetic Neumann boundary conditions. If Ω is simply connected then the spectrum reduces to the spectrum of the usual Neumann Laplacian; therefore we focus on multiply connected domains bounded by convex curves and prove lower bounds for its ground state depending on the geometry and the topology of Ω. Besides the area, the perimeter and the diameter, the geometric invariants which play a crucial role in the estimates are the fluxes of the potential one-form around the inner holes and the distance between the boundary components of the domain; more precisely, the ratio between its minimal and maximal width. Then, we give a lower bound for doubly connected domains which is sharp in terms of this ratio, and a general lower bound for domains with an arbitrary number of holes. When the inner holes shrink to points, we obtain as a corollary a lower bound for the first eigenvalue of the so-called Aharonov-Bohm operators with an arbitrary number of poles

On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.Comment: 26 page

High orders of Weyl series for the heat content

This article concerns the Weyl series of spectral functions associated with the Dirichlet Laplacian in a $d$-dimensional domain with a smooth boundary. In the case of the heat kernel, Berry and Howls predicted the asymptotic form of the Weyl series characterized by a set of parameters. Here, we concentrate on another spectral function, the (normalized) heat content. We show on several exactly solvable examples that, for even $d$, the same asymptotic formula is valid with different values of the parameters. The considered domains are $d$-dimensional balls and two limiting cases of the elliptic domain with eccentricity $\epsilon$: A slightly deformed disk ($\epsilon\to 0$) and an extremely prolonged ellipse ($\epsilon\to 1$). These cases include 2D domains with circular symmetry and those with only one shortest periodic orbit for the classical billiard. We analyse also the heat content for the balls in odd dimensions $d$ for which the asymptotic form of the Weyl series changes significantly.Comment: 20 pages, 1 figur

Co-sintering of dense electrophoretically deposited YSZ films on porous NiO-YSZ substrates for SOFC applications

An original process for the preparation of YSZ dense films with a thickness lower than 10 μm over NiO-YSZ substrates is presented. This process involves the preparation of a green membrane of NiO-YSZ and subsequent electrophoretic deposition (EPD) of commercial YSZ powder on this polymer-rich membrane. A single thermal treatment allowed removal of the organic compounds, sintering of the layers and full densification of the electrolyte. © 2005 Materials Research Society

Electrochemical characterization of anode supported SOFC prepared by co-firing technique

One of the main problems in the fabrication of anode supported solid oxide fuel cells is related to the sintering of electrolyte layer on anodic substrate, because differential densification of the layers may result in cracks during thermal process. Co-firing approach consists of simultaneous sintering of both electrolyte and anode. In this way, shrinkage of porous layer is compatible with the densification of electrolyte film. In this work co-firing technique was used for the sintering of YSZ thick films deposited on green NiO-YSZ layers by electrophoretic deposition (EPD). EPD is a colloidal process based on the motion of charged particles in the electric field in the direction of the electrode with opposite charge, thus forming a compact layer. With respect to other techniques, EPD has several advantages: short formation times, little restriction in the shape of substrates, simple deposition apparatus, possibility to have a mass production, low cost, easy control of the thickness of the deposited film through simple regulation of applied potential and deposition time. The EPD/co-firing combined process allowed to obtain a dense, 10 μm thick, crack free electrolyte layer with a good bonding to the anode. A slurry was prepared starting from a commercial NiOYSZ anodic powder (Praxair), polyvinylidene fluoride (PVDF binder SOLEF 6020, Solvay), a nanometric carbon powder (super P, Carbon Belgium), dispersed in N-methyl-2-pyrrolidone. A green membrane was obtained after evaporation of the solvent. A suspension of YSZ powder was prepared starting from commercial YSZ (TZ8Y, Tosoh) in methanol and deposited by EPD on a green NiO-YSZ membrane using a planar EPD cell setup. Co-firing parameters were assessed from the results of TG-DTA analysis performed on green bodies. Figure 1 shows the results of Hg porosimetry performed on sintered anodes for the determination of residual porosity and surface area. Green and fired samples were characterized in terms of morphology by scanning electron microscopy (FE-SEM), as reported in Figure 2. EDS linescan performed on the cross section of the cell did not show nickel diffusion in the electrolyte layer. A cathode layer was deposited on fully sintered half cells via spray-powder technique, using a suspension of commercial LSFC powder (Nextech), followed by a low temperature sintering process. Electrochemical characterization was performed on button cells in hydrogen in the temperature range 600-800 degrees C. Data of the electrochemical characterization will be presented at the conference

Mode Confinement in Photonic Quasi-Crystal Point-Defect Cavities for Particle Accelerators

In this Letter, we present a study of the confinement properties of point-defect resonators in finite-size photonic-bandgap structures composed of aperiodic arrangements of dielectric rods, with special emphasis on their use for the design of cavities for particle accelerators. Specifically, for representative geometries, we study the properties of the fundamental mode (as a function of the filling fraction, structure size, and losses) via 2-D and 3-D full-wave numerical simulations, as well as microwave measurements at room temperature. Results indicate that, for reduced-size structures, aperiodic geometries exhibit superior confinement properties by comparison with periodic ones.Comment: 4 pages, 4 figures, accepted for publication in Applied Physics Letter

The Steklov spectrum of cuboids

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d ≥ 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d - 2) - dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids

Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions

Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.Comment: 7 page

Post-stroke depression increases disability more than 15% in ischemic stroke survivors: a case-control study

We performed a retrospective, case-control study in consecutive ischemic stroke patients admitted to our stroke rehabilitation unit. Patients were matched for severity of neurological impairment (evaluated with the Canadian Neurological Scale, CNS), age (difference within 1 year), and onset admission interval (difference within 3 days). Participants were divided into two subgroups according to the presence or absence of PSD. Aim was to assess the specific influence of post-stroke depression (PSD) and antidepressant treatment on both basal functional status and rehabilitation outcomes. All PSD patients were treated primarily with serotoninergic antidepressants (AD). The final sample included 280 patients with depression (out of 320 found in a whole case series of 993 ischemic patients, i.e., 32.25%) and 280 without depression. Forty patients with depression were excluded because they had a history of severe psychiatric illness or aphasia, with a severe comprehension deficit. On one hand, PSD patients obtained lower Barthel Index (BI) and Rivermead Mobility Index (RMI) scores at both admission and discharge, with minor effectiveness of rehabilitative treatment and longer length of stay; on the other hand, this group had a lower percentage of dropouts. Lastly, PSD patients showed a different functional outcome, based on their response to antidepressant therapy, that was significantly better in responders than in non-responders (13.13%). Our results confirm the unfavorable influence of PSD on functional outcome, despite pharmacological treatment

A Reilly formula and eigenvalue estimates for differential forms

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally we also obtain, as a by-product of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.Comment: 22 page