INTRODUCTION
5
of Green's functions for arbitrary elliptic problems of the form (1), (3) and get
a representation of the solutions by means of these functions. Finally, elliptic
boundary value problems with a complex parameter are considered in Chapter 3.
Such problems arise, in particular, from boundary value problems in a cone if one
applies the Mellin transformation \x\ r A to the principal parts of the operators
L, B, and C. Spectral properties of the so obtained operator pencils are used later
in the study of boundary value problems in domains with conical points.
For the sake of simplicity, Chapters 1-3 deal only with boundary value problems
for differential equations of order 2m, where the order of the derivatives in the
operator B is less than 2m. In Chapter 4 we generalize the results to arbitrary
elliptic boundary value problems for systems of differential equations. Furthermore,
a special section in Chapter 4 is dedicated to boundary value problems in the
variational form.
The second part of the book (Chapters 5-8) is concerned with elliptic boundary
value problems in cylinders, cones and bounded domains with conical points. Chap-
ter 5 deals with boundary value problems in an infinite cylinder C = {(x,i) : x £
ft, oo t +oo}, where Q is a smooth bounded domain in E
n
. First the coeffi-
cients of the differential operators are assumed to be independent from the variable
t. We obtain necessary and sufficient conditions for the unique solvability of such
boundary value problems in weighted Sobolev spaces, where also Sobolev spaces
of small positive and nonpositive orders are involved. Furthermore, we obtain the
asymptotics of the solutions at infinity and derive formulas for the coefficients in the
asymptotics. These formulas contain special solutions of the homogeneous adjoint
problem. Here it turns out to be an advantage that we have considered boundary
value problems of the form (1), (3) with unknowns both in the domain and on the
boundary. Thus, we do not have to restrict ourselves to boundary value problems
for which the classical Green formula is valid. In the case of ^-dependent coefficients
satisfying the so-called stabilization condition at infinity we obtain the Fredholm
property of the operator to the boundary value problem, regularity assertions, and
a priori estimates for the solutions.
The results of Chapter 5 are applied in Chapter 6 to obtain analogous results
for elliptic boundary value problems in infinite cones and bounded domains with
angular or conical points. The main results concerning problems in bounded do-
mains are the Fredholm property of the operator (4) in weighted Sobolev spaces of
arbitrary integer order, regularity assertions and a priori estimates for the solutions,
asymptotic decomposition of the solutions near the conical points, and formulas for
the coefficients in the asymptotics. Moreover, we study the Green functions of the
boundary value problem. Again for the sake of simplicity, we restrict ourselves in
Chapters 5 and 6 to boundary value problems for a 2m order differential equa-
tion with boundary operators Bk of order less than 2m. Chapter 7 is dedicated
to the generalization of the results to elliptic boundary value problems for sys-
tems of differential equations without such restriction on the boundary conditions.
Furthermore, elliptic boundary value problems in variational form are considered
here.
The class of weighted Sobolev spaces used in Chapters 6 and 7 does not con-
tain the usual Sobolev spaces without weight. Therefore, in Chapter 8 we consider
the boundary value problem also in another class of weighted Sobolev spaces with
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