Construction of a~minimax solution for an eikonal-type equation

P. D. Lebedev ; A. A. Uspenskii ; V. N. Ushakov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 2, pp. 182-191

Résumé

A formula for a minimax (generalized) solution of the Cauchy–Dirichlet problem for an eikonal-type equation is proved in the case of an isotropic medium providing that the edge set is closed; the boundary of the edge set can be nonsmooth. A technique of constructing a minimax solution is proposed that uses methods from the theory of singularities of differentiable mappings. The notion of a bisector, which is a representative of symmetry sets, is introduced. Special points of the set boundary–pseudovertices–are singled out and bisector branches corresponding to them are constructed; the solution suffers a “gradient catastrophe” on these branches. Having constructed the bisector, one can generate the evolution of wave fronts in smoothness domains of the generalized solution. The relation of the problem under consideration to one class of time-optimal dynamic control problems is shown. The efficiency of the developed approach is illustrated by examples of analytical and numerical construction of minimax solutions.