INTRODUCTION

3

Moreover, there are many works concerning other boundary singularities, such

as edges, polyhedral vertices, and domains of class

C0'1,

which are not studied here.

The class of boundary value problems. A principal new feature of this

book in comparison with other monographs and papers on elliptic boundary value

problems in domains with conical points is the consideration of solutions in Sobolev

spaces of both positive and negative order.

In this book we consider boundary value problems for differential equations.

Avoiding the use of pseudodifferential operators ensures a more elementary charac-

ter of the book. Moreover, in most of applications of the theory of elliptic boundary

value problems only differential operators occur.

Pseudo-differential operators on manifolds with conical points were studied by

R. Melrose and G. Mendoza [164], B. A. Plamenevskii [199], and B. W. Schulze

[222]-[225]. A. O. Derviz [59], E. Schrohe and B.-W. Schulze [218, 219] extended

the results to pseudo-differential boundary value problems on manifolds with conical

points. They constructed algebras of pseudo-differential boundary value problems

and parametrices for elliptic elements. Studying the structure of the parametrices,

they obtained regularity assertions and the asymptotics of the solutions near the

conical point. We refer further to the work [163] of R. Melrose which is dedicated

to index theorems of Atiyah-Patodi-Singer type for pseudo-differential operators on

manifolds with conical points.

A boundary value problem in the classical form consists of a differential equa-

tion (or a system of differential equations)

(1) Lu = f

for the unknown function (vector-function) u in a domain Q C W1 and some con-

ditions

(2) Bu = g

which have to be satisfied on the boundary dfl. Here B is a vector (or matrix)

differential operator. The equations (2) are called boundary conditions.

In contrast to other monographs, we consider boundary conditions, where addi-

tionally to the unknown functions in the domain ft also an unknown vector-function

u on the boundary dVt appears, i.e., boundary conditions of the form

(3) Bu + Cu = g on dQ.

Here B is a vector (or a matrix) differential operator on ft and C is a matrix

differential operator on dft. Naturally, boundary value problems of the form (1),

(2) are contained in the class of the problems (1), (3).

The reason for considering the boundary value problems of the form (1), (3),

which appeared first in the works of B. Lawruk [124], is that the adjoint problem

belongs to the same class of problems. This is not true if we restrict ourselves

to classical boundary value problems (1), (2). Let us consider, e.g., the Laplace

equation in a plane domain ft with the boundary condition

. du

1

du ^

where djdv denotes the derivative in the direction of the exterior normal, d/dr

denotes the derivative in the tangential direction to dft, and b\, 62 are smooth