4
INTRODUCTION
real-valued functions satisfying the condition |6i| + I62I 7^ 0 on dft. Then an adjoint
problem is
Av = / in f£,
v -bi vi=gu - ^-(62^1) = 92 onffl.
Clearly, if 61 7^ 0 on cft7, then the unknown v\ on dft can be eliminated. However,
if 61 = 0 on a nonempty subset of dQ, then the adjoint problem can not be written
in the form (1), (2).
Note that the operators
(4)
A=(LB
I
of the boundary value problem (1), (3) form a subalgebra of Boutet de Monvel's
algebra for pseudodifferential boundary value problems which consists of matrices
of the form
L + G K
^ 1 B C
where L is a pseudodifferential operator, G is a singular Green operator, B is a trace
operator acting from Q to dfl, C is a pseudodifferential operator on the boundary
dQ, and K is a Poisson operator acting from dVt to O.
The inclusion of adjoint boundary value problems has several advantages. For
example, it is not necessary to construct a regularizer in order to prove the Fredholm
property of the operator (4). It suffices to prove a priori estimates and regularity
assertions for the solutions of the boundary value problem (1), (3). The cokernel
of the operator (4) can be described by the solutions of the homogeneous formally
adjoint problem. Furthermore, with the help of the adjoint problem we are able
to construct an extension of the operator (4) to Sobolev spaces of an arbitrary
order. For elliptic boundary value problems of the form (1), (2) this extension
was constructed in papers of Ya. A. Roitberg [201] - [205] and Ya. A. Roitberg,
Z. G. ShefteP [207], [208].
The structure of the book. The book consists of three parts. In the first part
(Chapters 1-4) we consider the boundary value problem (1), (3) in a domain with
smooth boundary. We give a detailed proof for the equivalence of the ellipticity,
the Fredholm property of the operator (4) and the validity of a priori estimates for
the solutions in corresponding Sobolev spaces. In these assertions the operator (4)
is considered in Sobolev spaces of both positive and negative order.
The main step in the proof of the Fredholm property is the derivation of neces-
sary and sufficient conditions for the unique solvability of boundary value problems
with constant coefficients in the half-space xn 0 in Chapter 2. Here we use
Sobolev spaces of functions which are periodic in the variables x\,... ,x
n
_i. The
theorem on the unique solvability of boundary value problems with constant co-
efficients in the half-space implies, in particular, regularity assertions and a priori
estimates for the solutions. These are extended to elliptic problems with variable
coefficients in the half-space.
Chapter 3 deals with elliptic boundary value problems in smooth bounded
domains. Based on the results of Chapter 2, we obtain the Fredholm property
and regularity assertions for the solution. Furthermore, we prove the existence
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