An interface problem in a domain of R3

Boundary-contact problems for domains with edge singularities

“…In contrast to the case of differential transmission problems as in Section 1.2 we have not at once associated families (14) that are holomorphic in z, but we need a so-called Mellin quantization to pass from p to families h of that kind. In order to formulate a corresponding result we need to say what we understand by a holomorphic family of transmission problems.…”

“…Problems of this kind have been investigated by several authors, in different context, partly under specific assumptions on the geometry or the involved dimensions, cf. Lemrabet [14], Escauriaza, Fabes, and Verchota [6], Torres and Welland [21], Chkadua [3,4], Li and Vogelius [13], Li and Nirenberg [12], Nicaise and Sändig [16] (numerical method), Heinrich, Nicaise and Weber [8] (the Fourier-finite-element method and singular functions of non-tensorial type), Kapanadze and Schulze [10] (the latter paper studies the case with conical singularities at the interfaces).…”

Composite finite elements for elliptic interface problems

“…For a characterization of the singularities that may appear if the conditions (R1)-(R3) are not satisfied, we refer the reader to [2,6,10,11] among many others. A comprehensive regularity analysis for the three-dimensional case is more technical and beyond the scope of this paper; we refer to [12] for necessary conditions under which H 2 (Ω 1 ∪ Ω 2 )-regularity is achieved. If the geometric setting allows H 2 (Ω 1 ∪ Ω 2 )-regularity, then the proof of (2.11) could be treated in a similar way as in Theorem 4.1.…”

Component mode synthesis and eigenvalues of second order operators : discretization and algorithm

“…By hypothesis, we already know that UijB H l + a (f2 i ), a >0. On the other hand, it is possible to predict the minimal regularity of the static modes ü r t in the case of model problems (see Lemrabet [1977Lemrabet [ , 1978). One can prove that the function u r i is smooth only in the interior of the interface F. In any case, if there exists a real number a > 0 such that all the functions w k , u t j and ü r t belong to the space H 1 + a ({2 ), where H s (f2 ) dénotes the usual Besov space defined by the Kmethod (cf.…”