NDLI: Transmission Problems and Boundary Operator Algebras

Content Provider	Springer Nature Link
Author	Axelsson, Andreas
Copyright Year	2004
Abstract	We examine the operator algebra $$\mathcal{A}$$ behind the boundary integral equation method for solving transmission problems. A new type of boundary integral operator, the rotation operator, is introduced, which is more appropriate than operators of double layer type for solving transmission problems for first order elliptic partial differential equations. We give a general invertibility criteria for operators in $$\mathcal{A}$$ by defining a Clifford algebra valued Gelfand transform on $$\mathcal{A}$$ . The general theory is applied to transmission problems with strongly Lipschitz interfaces for the two classical elliptic operators $$\bar \partial $$ and Δ. We here use Rellich techniques in a new way to estimate the full complex spectrum of the boundary integral operators. For $$\bar \partial $$ we use the associated rotation operator to solve the Hilbert boundary value problem and a Riemann type transmission problem. For the Helmholtz equation, we demonstrate how Rellich estimates give an angular spectral estimate on the rotation operator, which with the general spectral mapping properties in $$\mathcal{A}$$ translates to a hyperbolic spectral estimate for the double layer potential operator.
Ending Page	164
Page Count	18
Starting Page	147
File Format	PDF
ISSN	0378620X
e-ISSN	14208989
Journal	Integral Equations and Operator Theory
Issue Number	2
Volume Number	50
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	2004-01-01
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) Transmission problem double layer potential Hilbert problem Analysis Calderón projection Clifford algebra Integral equations with kernels of Cauchy type Gelfand transform
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Analysis

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