Multiple boundary peak solutions for some singularly perturbed Neumann problems

We consider the problem \left \{ \begin{array}{rcl} \varepsilon^2 \Delta u - u + f(u) = 0 & \mbox{ in }& \ \Omega\\ u > 0 \ \mbox{ in} \ \Omega, \ \frac{\partial u}{\partial \nu} = 0 & \mbox{ on }& \ \partial\Omega, \end{array} \right. where \Omega is a bounded smooth domain in R^N, \varepsilon>$is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as \varepsilon approaches zero, at a critical point of the mean curvature function H(P), P \in \partial \Omega . It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P) or multiple local maximum points of H(P). In this paper, we prove that for any fixed positive integer$K$there exist boundary$K-peak$solutions at a local minimum point of$H(P)$. This implies that for any smooth and bounded domain there always exist boundary$K-peak$ solutions. We first use the Liapunov-Schmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes

Spontaneous direct bonding of thick silicon nitride

Wafers with LPCVD silicon-rich nitride layers have been successfully direct bonded to silicon-rich nitride and boron-doped silicon surfaces. A chemical - mechanical polishing treatment was necessary to reduce the surface roughness of the nitride before bonding. The measured surface energies of the room-temperature bond were comparable to values found for Si - Si hydrophilic bonding. A mechanism similar to this bonding is suggested for silicon nitride bonding

The surface adhesion parameter: a measure for wafer bondability

A theory is presented which describes the initial direct wafer bonding process. The effect of surface microroughness on the bondability is studied on the basis of the theory of contact and adhesion of elastic solids. An effective bonding energy, the maximum of which is the specific surface energy of adhesion, is proposed to describe the real binding energy of the bonding interface including the influence of the wafer surface microroughness. Both the effective bonding energy and the real area of contact between rough surfaces depend on a dimensionless surface adhesion parameter, &thetas;. Using the adhesion parameter as a measure, three kinds of wafer contact interfaces can be identified with respect to their bondability; viz. the non-bonding regime (&thetas;>&ap;12), the bonding regime (&thetas;<&ap;1), and the adherence regime (1<&thetas;<12). Experimental data are in agreement with this theor

An omnidirectional retroreflector based on the transmutation of dielectric singularities

In the field of transformation optics, metamaterials mimic the effect of coordinate transformations on electromagnetic waves, creating the illusion that the waves are propagating through a virtual space. Transforming space by appropriately designed materials makes devices possible that have been deemed impossible. In particular, transformation optics has led to the demonstration of invisibility cloaking for microwaves, surface plasmons and infrared light. Here we report the achievement of another "impossible task". We implement, for microwaves, a device that would normally require a dielectric singularity, an infinity in the refractive index. We transmute a singularity in virtual space into a mere topological defect in a real metamaterial. In particular, we demonstrate an omnidirectional retroreflector, a device for faithfully reflecting images and for creating high visibility, from all directions. Our method is robust, potentially broadband and similar techniques could be applied for visible light

Deterministic Quantum Key Distribution Using Gaussian-Modulated Squeezed States

A continuous variable ping-pong scheme, which is utilized to generate deterministically private key, is proposed. The proposed scheme is implemented physically by using Gaussian-modulated squeezed states. The deterministic way, i.e., no basis reconciliation between two parties, leads a two-times efficiency comparing to the standard quantum key distribution schemes. Especially, the separate control mode does not need in the proposed scheme so that it is simpler and more available than previous ping-pong schemes. The attacker may be detected easily through the fidelity of the transmitted signal, and may not be successful in the beam splitter attack strategy.Comment: 7 pages, 4figure