This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

We consider the linearized elasticity system in a multidomain of ${\bf R}^3$ . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius $r^{\varepsilon}$ . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and $r^{\varepsilon}$ tend to zero simultaneously, with $r^{\varepsilon}\gg\varepsilon^2$ , we identify the limit problem. This limit problem involves six junction conditions.

In this work a new concept of designing two degree of freedom (2-DOF) planar parallel manipulators (PPMs) is presented. With this design the manipulator's workspace can be increased by increasing the number of cells in the manipulator. A general dynamic model is formulated for the manipulator with any number of cells. The model is adapted for SCARA and ADEPT configurations, and a new approach for balancing these type of manipulators is proposed.