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Modern Aspects of the Theory of Partial Differential Equations pp 41–64Cite as

Edge-degenerate Operators at Conical Exits to Infinity

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Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 216))

Abstract

We develop elements of a calculus of pseudo-differential operators on an infinite cylinder \(B^\asymp:=\mathbb{R}\times B \ni (t, \cdot)\) where the cross section B is a compact manifold with smooth edge Y. The space \(B^\asymp\) is regarded as a manifold with edge \(Y^\asymp\) with conical exits to infinity \(t \longrightarrow \pm \infty.\) The amplitude functions are families of operators in the edge algebra on B depending on parameters \((t, \top, \zeta), \zeta \ne 0.\) We impose a special degenerate behaviour for \(|t|\longrightarrow \infty,\) motivated by the structure of principal edge symbols of the next higher corner calculus, consisting of operators on an infinite singular cone with base B and axial variable t. In this framework we study ellipticity and parametrices.

Mathematics Subject Classification (2000). Primary 35S35; Secondary 35J70.

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Authors and Affiliations

  1. Am Neuen Palais 10, D-14469, Potsdam, Germany

    B.-Wolfgang Schulze

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  1. B.-Wolfgang Schulze
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Correspondence to B.-Wolfgang Schulze .

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Editors and Affiliations

  1. Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, United Kingdom

    Michael Ruzhansky

  2. , Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

    Jens Wirth

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Schulze, BW. (2011). Edge-degenerate Operators at Conical Exits to Infinity. In: Ruzhansky, M., Wirth, J. (eds) Modern Aspects of the Theory of Partial Differential Equations. Operator Theory: Advances and Applications(), vol 216. Springer, Basel. https://doi.org/10.1007/978-3-0348-0069-3_3

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