Boundary value problem for Fuchsian partial differential equations and reflection of singularities

Abstract

Introduction. We study linear partial differential equations of Fuchsian type with respect to a hypersurface in R. The notion of Fuchsian equations was introduced by Baouendi–Goulaouic [1] and is almost equivalent to the notion of equations with regular singularities in the weak sense defined by Kashiwara– Oshima [2]. We note here that these notions have been generalized to systems and to submanifolds of arbitrary codimension by Oshima [10] and Laurent–Monteiro Fernandes [3]. First, we present a method for defining boundary values of hyperfunction solutions to a Fuchsian partial differential equation with respect to a hypersurface. For single partial differential equations with regular singularities (in the weak sense), there have been two methods to define boundary values of hyperfunction solutions: one is a very general but rather abstract method of Kashiwara–Oshima [2], and the other is a very concrete but rather restrictive (the characteristic exponents must be constant) method by Oshima [9]. Our method is close to the latter, but not restrictive. Our main purpose is to study the reflection of singularities of hyperfunction solutions to a Fuchsian linear partial differential equation under some boundary condition. One of the most typical examples is the equation