On the asymptotics of a solution to a second-order elliptic equation with a small parameter and a piecewise smooth boundary function

We study the asymptotic behavior of a solution to the first boundary value problem for a second-order elliptic equation in the case when the small parameter is a factor at only one of the highest derivatives and the limit equation is an ordinary differential equation. We consider the case when the boundary function is piecewise smooth. Moreover, the point where the smoothness is violated is a point of jump discontinuity and coincides with the point where a characteristic of the limit equation touches the boundary from inside. Although the limit equation has the same order as the original equation, the problem under consideration is bisingular. We investigate the asymptotic behavior of the solution to this problem using the method of matched asymptotic expansions.