Elliptic Mixed, Transmission and Singular Crack Problems

Boundary value problems in weighted edge spaces

“…The technicalities for g = 0 may be found, for instance, in [9], see also [7]. An inspection of the details shows that the case of arbitrary g works in a similar manner.…”

“…We now state the Fredholm indices of (32) for the case α = π which is a result from [2] for i = 3; for the cases i = 1, 2 the values can be obtained by similar methods, see also [7,Section 5.3.4]. …”

Recent developments on pseudo-differential operators (II)

“…This problem has been studied by many mathematicians, cf. Vishik, Eskin [60] or Eskin [18], see also [24], [36], and [47]. In the present paper we develop an approach that is based on a Mellin quantisation, already indicated in Dines, Liu, and Schulze [15].…”

Recent developments on Pseudo-Differential Operators (I)

“…Singularities of higher order appear in connection with mixed and transmission or crack problems, cf. the monographs [11], [8], or the paper [2]. It turns out that the details require new structures, not only new classes of weighted Sobolev spaces and edge spaces, as established in [17], but also new techniques of proving continuity of operators in those spaces, see the paper [23] of Seiler.…”

“…In addition, as we shall see below, the class of such 2 × 2 block matrix operators also produces so-called Green Concerning details we refer to [1] or [8]. For any case G and T are asked to be of type d ∈ Z + , see the explanations below.…”