New solutions for critical Neumann problems in ℝ2

We consider the elliptic equation {-\Delta u+u=0} in a bounded, smooth domain Ω in {\mathbb{R}^{2}} subject to the nonlinear Neumann boundary condition {\frac{\partial u}{\partial\nu}=\lambda ue^{u^{2}}} , where ν denotes the outer normal vector of {\partial\Omega} . Here {\lambda>0} is a small parameter. For any λ small we construct positive solutions concentrating, as {\lambda\to 0} , around points of the boundary of Ω.

An Obstacle Avoidance Algorithm for Path Planning Based on Inspection MBD Model

Aiming at measuring path planning, based on inspection MBD (Model Based Definition) models of the measured parts, size and position information of the parts features are generated. Global coordinate equations of the features are established. In measuring process, probe heads should move along the outer normal vector, and the path should be the shortest. Under these constraint condictons, the obstacle avoidance path are created between the probe head starting points and the inspected features origin points. Finally, an example is given to verify the algorithm, which offers a base for the whole path planning considering the change of the probe heads and their direction.

Existence of outgoing solutions for perturbations of and applications to the scattering matrix

In this paper we shall prove an existence theorem and give applications of an outgoing solution of the following problem:where L(x, x) is a second order elliptic differential operator with a potential term q(x), is an exterior domain of ℝn (where n 2) with the C2-class boundary , k is an element of the complex plane or of a logarithmic Riemann surface, and B is either a Dirichlet boundary condition or of the form Bu = vj(x) ajk(x) ku + (x)u with the unit outer normal vector v(x) = (vl,, vn) at x.

Scattering Theory And Spectral Representations For General Wave Equations With Short Range Perturbations

In this paper we shall develop the scattering theory introduced by Lax and Phillips [5] for the following general wave equation; where Ω is an exterior domain Rn(n ≥ 3) with the smooth boundary δΩ and B is either a Dirichlet boundary condition or of the form Bu = Vi(x)aij(x)δju+σ(x)u with the unit outer normal vector v(x) = (v1 , … , vn) at x ∈ δ Ω. The precise assumptions on α(x), aij(x),q(x), σ(x) are denoted below. If Ω is an inhomogeneous medium with the density ρ (x), the propagation of waves is described by (1.1) with a(x) = a(x)2 ρ(x), aij(x) = ρ-l(x) δij and q(x) = 0 with the velocity a(x).