Realization theorems of geometric singularities for Hamilton-Jacobi equations
The geometric solution of Hamilton-Jacobi equations, constructed by the method of characteristics, is initially smooth, but post a critical time is multi-solved. The viscosity solution is continuous and coincides with the smooth geometric solution until the first critical time. The problem is to construct the viscosity solutions around each singularity of the geometric solution. So it is necessary to classify the singularities of geometric solutions. It has been shown by the authors [J. Differ. Equations 118, No. 1, 166-193 (1995; Zbl 0837.35091)] that the geometric solution is locally an one-parameter Legendrian unfolding and they presented the classifications of singularities of generic Legendrian unfoldings. The main results are the realization theorems for one-parameter Legendrian unfoldings as geometric solutions for a given Hamilton-Jacobi equation (with nonconvex Hamiltonian). The authors can also specify the point where the generic bifunctions of geometric singularities occur.
classifications of singularities of generic Legendrian unfoldings
generic bifunctions of geometric singularities
scientific article; zbMATH DE number 1123631