J. Melenk:

"Mapping properties of combined field Helmholtz boundary integral operators";

in: "ASC Report 01/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

For the Helmholtz equation (with wavenumber $k$) and analytic curves or surfaces $\Gamma$

we analyze the mapping properties of the single layer, double layer as well

combined potential boundary integral operators.

A $k$-explicit regularity theory for the single layer and double layer potentials is developed,

in which these operators are decomposed into three parts: the first part is the single or

double layer potential for the Laplace equation, the second part is an operator with

finite shift properties, and the third part is an operator that maps into a space

of piecewise analytic functions. For

all parts, the $k$-dependence is made explicit.

We also develop a $k$-explicit regularity theory for the inverse of the combined potential

operator $A = \pm 1/2 + K - {\bf i} \eta V$ and its adjoint, where $V$ and $K$ are the single layer

and double layer operators for the Helmholtz kernel and $\eta \in \BbbR$ is a coupling parameter

with $|\eta| \sim |k|$. The decomposition of the inverses $A^{-1}$ and $(A^\prime)^{-1}$ takes

the form of a sum of two operators $A_1$, $A_2$ where $A_1:H^s(\Gamma)\rightarrow H^s(\Gamma)$

with bounds independent of $k$ and a smoothing operator $A_2$ that maps into a space of analytic

functions on $\Gamma$. The $k$-dependence of the mapping properties of $A_2$ is made explicit.

http://www.asc.tuwien.ac.at/preprint/2010/asc01x2010.pdf

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