FIXED POINTS IN THE CONSTRUCTION OF A MINIMAX SOLUTION FOR A CLASS OF BOUNDARY VALUE PROBLEMS FOR HAMILTON–JACOBI EQUATIONS

This paper deals with analytical and numerical methods for constructing a minimax (generalized) solution to the Dirichlet problem for the Hamilton–Jacobi equation. The case of a closed planar nonconvex boundary set is considered, where the boundary points have a smoothness defect in the coordinate functions with respect to third-order derivatives. These points belong to the pseudo-vertices of the boundary set. Pseudo-vertices generate branches of a singular set, which are one-dimensional manifolds where the smoothness of the minimax solution breaks down. To construct a branch of a singular set, it is necessary to find markers, i.e., numerical characteristics of the corresponding pseudo-vertex. The markers (left and right ones) establish a link between the characteristics of the Hamilton–Jacobi equation and the geometry of the boundary set. For the markers, a relation with the structure of the equation at a fixed point is obtained. An iterative procedure for calculating a solution based on Newton's method is proposed. The convergence of the procedure to the pseudo-vertex marker is proved. An example of constructing a minimax solution is given, demonstrating the effectiveness of the developed approaches for solving nonsmooth boundary value problems.