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Institute
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.
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
Differential and pseudo-differential operators on a manifold with (regular) geometric singularities can be studied within a calculus, inspired by the concept of classical pseudo-differential operators on a C1 manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy σ = (σj)0≤j≤k, with k being the order of the singularity and σk operator-valued for k ≥ 1. The symbols determine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by various quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give examples and also comment on new challenges and interesting problems of the recent development.
We establish a new calculus of pseudodifferential operators on a manifold with smooth edges and study ellipticity with extra trace and potential conditions (as well as Green operators) at the edge. In contrast to the known scenario with conditions of that kind in integral form we admit in this paper ‘singular’ trace, potential and Green operators, which are related to the corresponding operators of positive type in Boutet de Monvel’s calculus for boundary value problems.
On a manifold with edge we construct a specific class of (edgedegenerate) elliptic differential operators. The ellipticity refers to the principal symbolic structure σ = (σψ, σ^) of the edge calculus consisting of the interior and edge symbol, denoted by σψ and σ^, respectively. For our choice of weights the ellipticity will not require additional edge conditions of trace or potential type, and the operators will induce isomorphisms between the respective edge spaces.
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
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 elliptic boundary value problems in a wedge with additional edge conditions of trace and potential type. We compute the (difference of the) number of such conditions in terms of the Fredholm index of the principal edge symbol. The task will be reduced to the case of special opening angles, together with a homotopy argument.