Degeneration of the Second Order Conditions in Pseudo Vertexes of Boundary Set Construction for Eikonal Equation

Conditions of non smooth singularities appearance for generalized solutions of the first order PDE are researched. Boundary Dirichlet problem for eikonal type equation is considered. The subject of study is a pseudo vertex of boundary set, which is necessary for analytical and numerical constructing of branches of the singular set, the solution of boundary problem loses smoothness on this set. Conditions providing correctness of limit transition in a definition of a pseudo vertex are obtained. Necessary conditions for pseudo vertex existence in case of smooth boarder are studied. It is proved that necessary conditions of the second order have a degenerated type. The opportunities of eikonal equations solutions logistics are suggested. These solutions construction are useful for minimization of outlay of stocks, technical support centers and gas station nets projection engineering.