Integration in Variational Inequalities on Spatial Grids

We prove an analog of the classical Jacobi theorem concerning the positive definiteness of the second variation for a functional defined on functions of branching argument belonging to a spatial grid (a geometric graph). The singularities of the corresponding analog of the Jacobi equation (and of the Euler equation) are generated by the procedure of integration by parts, which leads to differentiation with respect to measures glued (joined) together.