Refine
Document Type
- Article (3)
- Doctoral Thesis (1)
Language
- English (4)
Is part of the Bibliography
- yes (4)
Keywords
- Boutet de Monvel's calculus (2)
- Boutet de Monvels Kalkül (1)
- Fredholm property (1)
- Kanten-Randwertprobleme (1)
- Kegel space (1)
- Mannigfaltigkeiten mit Kante (1)
- Mannigfaltigkeiten mit Singularitäten (1)
- Mellin quantization (1)
- Mellin symbols with values in the edge calculus (1)
- Parametrices of elliptic operators (1)
- Pseudo-differential operators (1)
- Randwertprobleme (1)
- Singular cones (1)
- algebra (1)
- boundary value problems (1)
- distribution with asymptotics (1)
- edge boundary value problems (1)
- ellipticity (1)
- manifolds with edge (1)
- manifolds with edge and boundary (1)
- manifolds with singularities (1)
- pseudo-differential equation (1)
- pseudo-differentielle Gleichungen (1)
Institute
- Institut für Mathematik (4) (remove)
We study elements of the calculus of boundary value problems in a variant of Boutet de Monvel’s algebra (Acta Math 126:11–51, 1971) on a manifold N with edge and boundary. If the boundary is empty then the approach corresponds to Schulze (Symposium on partial differential equations (Holzhau, 1988), BSB Teubner, Leipzig, 1989) and other papers from the subsequent development. For non-trivial boundary we study Mellin-edge quantizations and compositions within the structure in terms a new Mellin-edge quantization, compared with a more traditional technique. Similar structures in the closed case have been studied in Gil et al.
In the thesis there are constructed new quantizations for pseudo-differential boundary value problems (BVPs) on manifolds with edge. The shape of operators comes from Boutet de Monvel’s calculus which exists on smooth manifolds with boundary. The singular case, here with edge and boundary, is much more complicated. The present approach simplifies the operator-valued symbolic structures by using suitable Mellin quantizations on infinite stretched model cones of wedges with boundary. The Mellin symbols themselves are, modulo smoothing ones, with asymptotics, holomorphic in the complex Mellin covariable. One of the main results is the construction of parametrices of elliptic elements in the corresponding operator algebra, including elliptic edge conditions.
We establish a calculus of boundary value problems (BVPs) on a manifold N with boundary and edge, based on Boutet de Monvel’s theory of BVPs in the case of a smooth boundary and on the edge calculus, where in the present case the model cone has a base which is a compact manifold with boundary. The corresponding calculus with boundary and edge is a unification of both structures and controls different operator-valued symbolic structures, in order to obtain ellipticity and parametrices.
We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.