On singularity structure of minimax solution to Dirichlet problem for eikonal type equation with discontinuous curvature of boundary of boundary set

Abstract

The birth of nonsmooth singularities in the minimax (generalized) solution of the Dirichlet problem for the eikonal equation is due to the existence of pseudo-vertices, the singular points of the boundary of the boundary set. Finding the pseudo-vertices is the first step in the procedure for constructing a singular set for solving a boundary value problem. To find these points, one has to construct local solutions to an equation of the golden ratio type, which establishes a connection between the eikonal operator and the geometry of the boundary set. The problem of identifying local solutions to the equation is related to the problem of finding fixed points of the mappings formed by local reparametrization of the boundary of the boundary set. In this work we obtain necessary conditions for the existence of pseudo-vertices when the smoothness of the curvature of a parametrically given boundary of the boundary set is broken. The conditions are written in various equivalent forms. In particular, we obtain a representation in the form of a convex combination of one-sided derivatives of the curvature. We provide the formulae for the coefficients of a convex combination determined by markers, which scalar characteristics of the pseudo-vertices. We find an algebraic equation, the roots of which are the markers. We adduce an example of the numerical-analytical construction of a minimax solution to the Dirichlet problem and this example demonstrates the effectiveness of the developed methods for solving nonsmooth boundary value problems.