The Zaremba problem with singular interfaces as a corner boundary value problem
- 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.
Author details: | Gohar Harutyunyan, Bert-Wolfgang SchulzeGND |
---|---|
DOI: | https://doi.org/10.1007/s11118-006-9020-6 |
ISSN: | 0926-2601 |
Title of parent work (English): | Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis |
Publisher: | Springer |
Place of publishing: | Dordrecht |
Publication type: | Article |
Language: | English |
Date of first publication: | 2006/10/12 |
Publication year: | 2006 |
Release date: | 2020/05/03 |
Tag: | Zaremba problem; corner Sobolev spaces with double weights; pseudo-differential boundary value problems |
Volume: | 25 |
Number of pages: | 43 |
First page: | 327 |
Last Page: | 369 |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |