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We study pseudodifferential operators on a cylinder IR x B with cross section B that conical singularities. Configurations of that kind are the local model of cornere singularities with base spaces B. Operators A in our calculus are assumed to have symbols α which are meromorphic in the complex covariable with values in the space of all cone operators on B. In case α is dependent of the axial variable t ∈ IR, we show an explicit formula for solutions of the homogeneous equation. Each non-bjectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula.
We show relative index formulas for boundary value problems in cylindrical domains and Sobolev spaces with different weights at too. The amplitude functions are meromorphic in the axial covariable and take values in the space of boundary value problems on the cross section of the cylinder. Copyright (c) 2005 John Wiley & Sons, Ltd
Mixed elliptic problems for differential operators A in a domain Q with smooth boundary Y are studied in the form Au = f in Omega, T+/-u = g+/- on Y+/-, where Y+/- subset of Y are manifolds with a common boundary Z, such that Y- boolean OR Y+ = Y and Y- boolean AND Y+ = z, with boundary conditions T+/- on Y+/- (with smooth coefficients up to Z from the respective side) satisfying the Shapiro-Lopatinskij condition. We consider such problems in standard Sobolev spaces and characterise natural extra conditions on the interface Z with an analogue of Shapiro-Lopatinskij ellipticity for an associated transmission problem on the boundary; then the extended operator is Fredholm. The transmission operators on the boundary with respect to Z belong to a complete pseudo-differential calculus, a modification of the algebra of boundary value problems without the transmission property. We construct parametrices of elliptic elements in that calculus, and we obtain parametrices of the original mixed problems under additional conditions on the interface. We consider the Zaremba problem and other mixed problems for the Laplace operator, determine the number of extra conditions and calculate the index of associated Fredholm operators. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim