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Contributions to Books:

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.

English abstract:
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.

Electronic version of the publication:

Created from the Publication Database of the Vienna University of Technology.