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INTRODUCTION

Parallel with the theory of general elliptic boundary value problems in smooth

domains, such problems in domains with singularities on the boundary were in-

vestigated. The question on the behaviour of solutions of elliptic boundary value

problems near boundary singularities is of great importance for many applications,

e.g., in aerodynamics, hydrodynamics, fracture machanics. The treatment of ellip-

tic boundary value problems in domains with singularities required a new theory.

On one hand, the methods which were developed for domains with smooth bound-

aries cannot be directly applied to domains with singularities. On the other hand,

many results of the theory for smooth domains are not true if the boundary of the

domain contains singularities.

The pioneering work in the development of a general theory for elliptic bound-

ary value problems in domains with angular and conical points was done by G. I. Es-

kin [65, 66], Ya. B. Lopatinskh [128], and V. A. Kondrat'ev [98, 99]. The first

two authors investigated boundary value problems in plane domains with angular

points applying the Mellin transformation to an integral equation on the bound-

ary. V. A. Kondrat'ev considered elliptic boundary value problems in domains of

arbitrary dimension with conical points. He applied the Mellin transformation to a

model problem connected with the boundary value problem and proved the Fred-

holm property of the operator of the boundary value problem in weighted and usual

L2 Sobolev spaces of positive integer order. Furthermore, he described the asymp-

totics of the solutions near conical points. Analogous asymptotic representations

for solutions of elliptic boundary value problems in an infinite cylinder were found

in 1963 by S. Agmon and L. Nirenberg [9].

The gap between V. A. Kondrat'ev's theory and applications was narrowed in

the works of V. G. Maz'ya and B. A. Plamenevskh [142, 143, 144, 147, 149].

These two authors extended the results of V. A. Kondrat'ev to other function spaces

(Lp Sobolev spaces, Holder classes, spaces with inhomogeneous norms), calculated

the coefficients in the asymptotics and described the singularities of the Green

functions. In the monograph of M. Dauge [53] L2 Sobolev spaces of fractional

order were admitted. Several papers of V. G. Maz'ya, B. A. Plamenevskh [150,

151, 152], V. A. Kozlov, V. G. Maz'ya [111, 112, 113], V. A. Kozlov (e.g.,

[105, 106, 107, 108]), V. A. Kozlov, J. Rofimann [115, 116]), and M. Costabel,

M. Dauge [51] contain a detailed analysis of the singularities of the solutions to

elliptic boundary value problems near conical points. Other results in this field are

estimates of the Lp-means (V. A. Kozlov, V. G. Maz'ya [109, 110]), the Miranda-

Agmon maximum principle (V. G. Maz'ya, B. A. Plamenevskh [150], V. G. Maz'ya,

J. Rofimann [155]), and the construction of stable asymptotics (V. G. Maz'ya,

J. Rofimann [156], M. Costabel, M. Dauge [52]).

We also mention the books of P. Grisvard [79, 80], A. Kufner, A.-M. Sandig

[121], V. G. Maz'ya, S. A. Nazarov, B. A. Plamenevskh [138], S. A. Nazarov,

B. A. Plamenevskh [182] and S. Nicaise [186], where different aspects of the theory

of elliptic boundary value problems in domains with angular and conical points are

considered.

A theory of elliptic boundary value problems in domains with cusps or in quasi-

cylindrical domains was established in papers of V. I. Feigin [69], L. A. Bagirov and

V. I. Feigin [23], V. G. Maz'ya and B. A. Plamenevskh [139, 147], A. B. Movchan

and S. A. Nazarov [170, 171, 173], J.-L. Steux [239, 240], and M. Dauge [56].

Elliptic equations on manifolds with cusps were studied further by B. W. Schulze,

B. Sternin, and V. Shatalov [227, 228].