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January, 2004 Reduction of orders in boundary value problems without transmission property
J. Math. Soc. Japan 56(1): 65-85 (January, 2004). DOI: 10.2969/jmsj/1191418696


Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems.

We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.


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Gohar HARUTJUNJAN. Bert-Wolfgang SCHULZE. "Reduction of orders in boundary value problems without transmission property." J. Math. Soc. Japan 56 (1) 65 - 85, January, 2004.


Published: January, 2004
First available in Project Euclid: 3 October 2007

zbMATH: 1050.58019
MathSciNet: MR2023454
Digital Object Identifier: 10.2969/jmsj/1191418696

Primary: 35J40 , 35K25 , 35S15 , 58J05

Keywords: boundary value problems , elliptic operators , order reduction , Volterra symbols

Rights: Copyright © 2004 Mathematical Society of Japan


Vol.56 • No. 1 • January, 2004
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