Boundary value problems in oscillating cuspidal wedges
type: Item
instance of: scholarly article
title: Boundary value problems in oscillating cuspidal wedges
zbMATH Open document ID: 1083.35151
author: B.-W. Schulze
author: Vladimir Rabinovich
author: N. N. Tarkhanov
published in: Rocky Mountain Journal of Mathematics
publication date: 2005-03-10T00:00:00Z
review text: This paper deals with pseudodifferential boundary value problems in domains with cuspidal wedges. A simple example of singular spaces studied here is the so-called canonical wedge \[ W= \{(\varphi(z) x,y,z)\in \mathbb{R}^3;\,|x|\leq 1,\,y\in \mathbb{R},\,z\geq 0\}\quad\text{in }\mathbb{R}^3, \] where \(\varphi\in C^\infty(\mathbb{R}^+)\), \(\varphi(z)> 0\) for \(z> 0\), \(\varphi(0)= 0\). The behaviour of \(\varphi(z)\) near \(z= 0\) specifies the singularity of \(W\) along the edge \(\{(0,y,0)\in \mathbb{R}^3\}\). As in general \(\varphi'(z)\) need not have any limit when \(z\to 0\), hence \(W\) may oscillate close to the edge. The results of this paper are based on an interplay between some classes of operator-valued symbols and the ``order reduction'' symbol \(\Lambda_B\), \(B\) being a \(C^\infty\) compact manifold. In the case of cuspidal wedges the authors develop a calculus of pseudodifferential operators where the behaviour of the corresponding symbols is controlled by a special operator valued function \(\lambda(t,\tau)\) depending on the variables \(t,\tau\in\mathbb{R}\). Moreover, the symbols under investigation vary slowly close to singularities. In Theorem 17.1 a criterion for the Fredholm property of differential operators on manifolds with oscillating cuspidal edges is shown. Part IV of the paper is devoted to boundary value problems including Dirichlet and Neumann ones in domains with cuspidal wedges.
Mathematics Subject Classification ID: 35S15
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 2143235
zbMATH Keywords: manifolds with edges and cuspidal wedges
zbMATH Keywords: operator-valued symbols
zbMATH Keywords: Fredholm property
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069808
OpenAlex ID: W2058787188
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DOI: 10.1216/RMJM/1181069808
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description: scientific article; zbMATH DE number 2143235