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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