Introduction
The goal of this book is a systematic and self-contained exposition of a theory of
linear elliptic boundary value problems in domains with isolated singularities on
the boundary.
Roots of the theory. Elliptic boundary value problems play an important
role in mathematical physics. Starting with the 18th century an innumerable num-
ber of works was dedicated to special boundary value problems. The theory of
general elliptic boundary value problems in smooth domains was developed in the
second half of the 20th century by I. G. Petrovskh [195], M. I. Vishik [246],
Ya. B. Lopatinskh [127], L. Hormander [86, 87], S. Agmon [3, 6], S. Agmon,
A. Doughs, L. Nirenberg [7, 8], F. E. Browder [37, 38], M. Schechter [213, 217],
J. Peetre [192], A. I. Koshelev [102], Yu. M. Berezanskh [26, 27], V. A. Solonnikov
[236], Ya. A. Roitberg [201] - [205], Ya. A. RoTtberg, Z. G. Sheftel' [207, 208], J.
Necas [184], J.-L. Lions, E. Magenes [126], B.-W. Schulze, G. Wildenhain [229],
I. V. Gel'man, V. G. Maz'ya [74], and others.
Fundamental results in this theory are:
a priori estimates for the solutions in different function spaces
the Fredholm property of the operator corresponding to the boundary value
problem
regularity assertions for the solutions
The construction of parametrices (approximately inverse operators) to the operators
of elliptic boundary value problems in domains with smooth boundaries resulted in
the development of the theory of pseudodifferential boundary value problems. This
theory has its origin in papers of A. P. Calderon, A. Zygmund [42], M. I. Vishik,
G. I. Eskin [247, 248, 249], G. I. Eskin [67, 68], and L. Boutet de Monvel [34, 35].
One of the most important results in the theory of pseudodifferential boundary
value problems is the calculation of the index for elliptic boundary value prob-
lems in topological terms. A formula for the index of boundary value problems
in two-dimensional domains was found by A. I. Vol'pert [253]. M. F. Atiyah and
I. M. Singer [18] calculated the index of elliptic operators on compact manifolds
without boundary, while an index formula for elliptic boundary value problems
was derived by M. F. Atiyah and R. Bott [17]. We refer further to the papers of
M. S. Agranovich [10], A. P. Calderon [41], and L. Boutet de Monvel [35]). A
more recent treatment of the theory of pseudodifferential boundary value problems
is given, e.g., in the monographs of S. Rempel, B.-W. Schulze [200] and G. Grubb
[81].
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