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We consider quasicomplexes of Boutet de Monvel operators in Sobolev spaces on a smooth compact manifold with boundary. To each quasicomplex we associate two complexes of symbols. One complex is defined on the cotangent bundle of the manifold and the other on that of the boundary. The quasicomplex is elliptic if these symbol complexes are exact away from the zero sections. We prove that elliptic quasicomplexes are Fredholm. As a consequence of this result we deduce that a compatibility complex for an overdetermined elliptic boundary problem operator is also Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes of Boutet de Monvel operators.
The inhomogeneous ∂-equations is an inexhaustible source of locally unsolvable equations, subelliptic estimates and other phenomena in partial differential equations. Loosely speaking, for the anaysis on complex manifolds with boundary nonelliptic problems are typical rather than elliptic ones. Using explicit integral representations we assign a Fredholm complex to the Dolbeault complex over an arbitrary bounded domain in C up(n).
The paper deals with the calculation of the fractional part of the η-invariant for elliptic self-adjoint operators in topological terms. The method used to obtain the corresponding formula is based on the index theorem for elliptic operators in subspaces obtained in [1], [2]. It also utilizes K-theory with coefficients Zsub(n). In particular, it is shown that the group K(T*M,Zsub(n)) is realized by elliptic operators (symbols) acting in appropriate subspaces.
An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.
An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.
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.
On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof.
We investigate general Shapiro-Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. This is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension, and we then deal with boundary value problems for cusp differential operators. We introduce an adapted Boutet de Monvel’s calculus of pseudodifferential boundary value problems, and construct parametrices for elliptic cusp operators within this calculus. Fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows.
Um beim Berufseinstieg erfolgreich als Informatiker wirken zu können, reicht es oft nicht aus nur separierte Kenntnisse über technische und theoretische Grundlagen, Programmiersprachen, Werkzeuge und Selbst- und Zeitmanagement zu besitzen. Vielmehr sollten Absolventen diese Kenntnisse praktisch miteinander verzahnt einsetzen können. An der Universität wird Studierenden leider selten die Möglichkeit geboten, diese verschiedenen Bereiche der Informatik miteinander integriert auszuüben. Dafür entwickeln wir seit über zwei Dekaden ein Lehr- und Lernkonzept zur Unterstützung praktischer Softwareentwicklungsveranstaltungen und setzen dieses um. Dadurch bieten wir angehenden SoftwareentwicklerInnen und ProjektmanagerInnen eine Umgebung, in der sie neues, praktisch relevantes Wissen erwerben können, sich selbst praktisch erproben und ihr Wissen konkret einsetzen können. Hier legen wir einen Schwerpunkt auf das Arbeiten im Team. Das hier vorgestellte Konzept kann auf ähnliche Lehrveranstaltungen übertragen und aufgrund seiner Modularisierung verändert und erweitert werden.
We consider edge-degenerate families of pseudodifferential boundary value problems on a semi-infinite cylinder and study the behavior of their push-forwards as the cylinder is blown up to a cone near infinity. We show that the transformed symbols belong to a particularly convenient symbol class. This result has applications in the Fredholm theory of boundary value problems on manifolds with edges.
The ellipticity of boundary value problems on a smooth manifold with boundary relies on a two-component principal symbolic structure (σψ; σ∂), consisting of interior and boundary symbols. In the case of a smooth edge on manifolds with boundary we have a third symbolic component, namely the edge symbol σ∧, referring to extra conditions on the edge, analogously as boundary conditions. Apart from such conditions in integral form' there may exist singular trace conditions, investigated in [6] on closed' manifolds with edge. Here we concentrate on the phenomena in combination with boundary conditions and edge problem.
The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy σ = (σψ, σ∧), where the second component takes value in operators on the infinite model cone of the local wedges. In general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the elliptcity of the principal edge symbol σ∧ which includes the (in general not explicitly known) number of additional conditions on the edge of trace and potential type. We focus here on these queations and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems, and we establish relations of elliptic operators for different weights, via the spectral flow of the underlying conormal symbols.
Elliptic equations on configurations W = W1 ∪ ... ∪ Wn with edge Y and components Wj 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 W\Y. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics.
Contents: 1 Introduction. Denfitions and Discussions 2 Solvability of the Cauchy Type Functional Equations 2.1 The Case of a P-configuration 2.2 The Case of a Z-configuration 2.3 Multiplicative Cauchy type functional equations 3 Problems in Analysis Reducing to Cauchy Type Functional Equations 3.1 Some problems in Integral Geometry and Cauchy Functional Equations 3.2 First Boundary Problem for Hyperbolic Differential Equations and Cauchy Type Functional Equations 4 Functional Equations Determining Polynomials
Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write Ssub(A)(D) for the space of solutions to thesystem Au = 0 in a domain D ⊂ X. Using reproducing kernels related to various Hilbert structures on subspaces of Ssub(A)(D) we show explicit identifications of the dual spaces. To prove the "regularity" of reproducing kernels up to the boundary of D we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the ∂-Neumann problem. The duality itself takes place only for those domains D which possess certain convexity properties with respect to A.
We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge - de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.
Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this paper we give a geometric interpretation of higherindex DAEs and indicate problems arising in connection with such DAEs by means of several examples.
Contents: Chapter 7: The Index Problemon Manifolds with Singularities Preface 7.1. The Simplest Index Formulas 7.1.1. General properties of the index 7.1.2. The index of invariant operators on the cylinder 7.1.3. Relative index formulas 7.1.4. The index of general operators on the cylinder 7.1.5. The index of operators of the form 1 + G with a Green operator G 7.1.6. The index of operators of the form 1 + G on manifolds with edges 7.1.7. The index on bundles with smooth base and fiber having conical points 7.2. The Index Problem for Manifolds with Isolated Singularities 7.2.1. Statement of the index splitting problem 7.2.2. The obstruction to the index splitting 7.2.3. Computation of the obstruction in topological terms 7.2.4. Examples. Operators with symmetries 7.3. The Index Problem for Manifolds with Edges 7.3.1. The index excision property 7.3.2. The obstruction to the index splitting 7.4. Bibliographical Remarks