Refine
Has Fulltext
- no (10) (remove)
Year of publication
- 2006 (10) (remove)
Document Type
- Article (5)
- Monograph/Edited Volume (5)
Language
- English (10)
Is part of the Bibliography
- yes (10)
Keywords
Institute
Elliptic equations on configurations W = W-1 boolean OR (. . .) boolean OR W-N with edge Y and components W-j of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here edges. Starting from edge-degenerate operators on Wj, j = 1, . . . , N, we construct an algebra with extra 'transmission' conditions on Y that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator- valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on WY. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics
We study pseudo-differential operators on a cylinder R x B where B has conical singularities. Configurations of that kind are the local model of corner singularities with cross section B. Operators in our calculus are assumed to have symbols a which are meromorphic in the complex covariable with values in the algebra of all cone operators on B. We show an explicit formula for solutions of the homogeneous equation if a is independent of the axial variable t is an element of R. Each non-bijectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula
Ellipticity of operators on a manifold with edges can be treated in the framework of a calculus of 2 X 2-block matrix operators with trace and potential operators on the edges. The picture is similar to the pseudodifferential analysis of boundary-value problems. The extra conditions satisfy an analogue of the Shapiro-Lopatinskij condition, provided a topological obstruction for the elliptic edge-degenerate operator in the upper left corner vanishes; this is an analogue of a condition of Atiyah and Bott in boundary-value problems. In general, however, we need global projection data, similarly to global boundary conditions, known for Dirac operators or other geometric operators. The present paper develops a new calculus with global projection data for operators on manifolds with edges. In particular, we show the Fredholm property in a suitable scale of spaces and construct parametrices within the calculus
We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y, i.e., Au = f in int X, T (+/-) u = g(+/-) on int Y+/-, where Y is subdivided into subsets Y+/- with an interface Z and boundary conditions T+/- on Y+/- that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z subset of Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T- Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in Bull. Sci. Math. ( to appear). With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.