Moreover, there are many works concerning other boundary singularities, such
as edges, polyhedral vertices, and domains of class
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-
(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
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
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