Mathematical topics
1 total work
Vol 4
In this monograph the theory of pseudo-differential boundary value problems under the aspect of a calculus for conical singularities is studied. Here the inner normal to the boundary is regarded as the model cone of a wedge with the bondary as edge. The transmission property in Boutet de Monvel's sense as well as the theory of Visik and Eskin are particular cases. The results of Visik and Eskin are considerably extended. The operators belong to an algebra that contains the parametrices of the elliptic elements. The ellipticity refers to the interior and the boundary symbols of highest orders. Boundary value problems are treated as particular edge problems in terms of a calculus of pseudo-differential operators with operator-valued symbols acting in the edge Sobolev spaces of the author. The theory contains the essential ideas for treating the more general case of pseudo-differential operators on manifolds with edges. This also belongs to the motivations of the approach. Similarly to Boutet de Monvel's theory, the corresponding algebra consists of matrices with the additional trace and potential operators.
The concrete operator-valued symbols are presented both with discrete and continuous asymptotics. This corresponds to the nature of the elliptic regularity that is also proved for both variants of asymptotics. Essential ingredients are Mellin operator conventions and weighted Sobolev spaces with asymptotics. The theory applies to a wide range of classical transmission and mixed problems and gives a description of parametrics on a complete symbolic level.
The concrete operator-valued symbols are presented both with discrete and continuous asymptotics. This corresponds to the nature of the elliptic regularity that is also proved for both variants of asymptotics. Essential ingredients are Mellin operator conventions and weighted Sobolev spaces with asymptotics. The theory applies to a wide range of classical transmission and mixed problems and gives a description of parametrics on a complete symbolic level.