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We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator-valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator A(s) in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of A(s) as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices P-s of A(s) interpreted as Mellin-edge representations of P. Copyright (c) 2005 John Wiley & Sons, Ltd
Necessary and sufficient conditions for the representation of the index of elliptic operators on manifolds with edges in the form of the sum of homotopy invariants of symbols on the smooth stratum and on the edge are found. An index formula is obtained for elliptic operators on manifolds with edges under symmetry conditions with respect to the edge covariables
We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ ψ(A), σ ^(A), σ ^(A)), where σ ψ is the interior symbol and σ ^(A)(y, η), (y, η) 2 T*Y \ 0, the (operator-valued) edge symbol of ‘first generation’, cf. [15]. The novelty here is the edge symbol σ^ of ‘second generation’, parametrised by (z, Ϛ) 2 T*Z \ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.
In this paper we establish the regularity, exponential stability of global (weak) solutions and existence of uniform compact attractors of semiprocesses, which are generated by the global solutions, of a two-parameter family of operators for the nonlinear 1-d non-autonomous viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.