Asymptotic Analysis of Singularities for Pseudodifferential Equations in Canonical Non-Smooth Domains

Pseudodifferential operator theory was apparently developed no later than half a century ago (Vishik and Eskin,

1965

; Eskin,

1981

; Mikhlin,

1962

; Mikhlin and Prössdorf,

1986

; Malgrange,

1962

; Calderon and Zygmund,

1957

; Hörmander,

1986–1988

; Treves,

1982

; Taylor,

1981

; Rempel and Schulze,

1982

; Sternin,

1966

), but it is not so young because its basic achievements were invented in 1960s or 1970s. The main point of this theory is a symbolic calculus for pseudodifferential operators and boundary value problems for pseudodifferential equations on manifolds with a smooth boundary. This is not so good for manifolds with a non-smooth boundary although there are many studies on this problem (see, for example, the papers of B.-W. Schulze and his colleagues). The first paper in this direction was Kondratiev’s paper of 1967, in which the general boundary value problem for a partial differential operator in a cone was studied. But in this paper, and almost all of he following studies, the conical singularity is geometrically treated as direct product of a manifold with a smooth boundary and a half-line. The author suggests another point of view (for elliptic equations in this time) related to the concept of wave factorization of elliptic symbol (Vasil’ev,

2000

; Vasilyev,

1998

). From this point of view the conical singularity is something indivisible, and it cannot be reduced to other cases which were studied earlier. “Curiously enough,” V.A. Kondratiev said to me (in answer to my question) “there is nothing interesting any more in the theory of boundary value problems in domains with non-smooth boundary” (December 2004). This urged the author to collect his own outlines into one but still unfinished fragment. The author hopes that in the future it will be possible to obtain a “non-smooth” version of Sternin’s results (Sternin,

1966

) by the wave factorization method.