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The invertibility of the double layer potential operator in the space of continuous functions defined over a polyhedron. The panel method. Erratum. (English) Zbl 0921.31004
From the introduction: In his paper [ibid. 45, No. 1-4, 135-177 (1992; Zbl 0749.31003)] the author proved that the spectral radius $$r_W$$ of the double layer operator \begin{aligned} Wx(Q) &:= [1/2- d_\Omega (Q)] x(Q)+ \frac{1}{4\pi} \int_S \frac{n_P\cdot(Q-P)} {| P-Q|^3} x(P)d_PS,\\ d_\Omega(Q) &:= \lim_{\varepsilon \to\infty} \frac{|\{P\in\Omega:| P-Q|<\varepsilon\}|} {|\{P\in \mathbb{R}^3:| P-Q|< \varepsilon\}|} \end{aligned} defined over the boundary $$S$$ of an infinite polyhedral cone $$\Omega$$ is less than $$1/2$$. Here $$n_P$$ denotes the unit vector of the interior normal to $$\Omega$$ at $$P$$ and $$| Z|$$ is the Lebesgue measure of $$Z$$ for any $$Z\subseteq \mathbb{R}^3$$. The operator $$W$$ is considered in the space $$C(S)$$ of all continuous functions over $$S$$ vanishing at infinity.
Unfortunately, the article contains two serious errors. First, in the proof of Lemma 1.5 it was asserted that the spectral radius $$r_{\Phi(B)}$$ considered in the image algebra $$\Phi({\mathcal A})$$ is equal to the radius $$r_{\Phi(B)}$$ considered in $$X_{\xi\in \mathbb{R}}{\mathcal L}(C(\Gamma))$$. However, $$\Phi({\mathcal A})$$ is not a closed subalgebra of $$X_{\xi\in \mathbb{R}}{\mathcal L}(C(\Gamma))$$ and we actually do not know any proof of this fact. A second error occurs in the proof of Lemma 3.1. Namely, the second equation on page 158 is divided by $$(e^{2\pi/| \xi|})^{i\xi}-1=0$$ to obtain the third equation. The author is grateful to R. Spencer (University of Minnesota) who has observed this last error and has sent me an improvement.

##### MSC:
 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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##### References:
 [1] kral J., Integral operators in potential theory (1980) [2] Mazya V. G., Boundary integral equations 27 (1991) [3] DOI: 10.1080/00036819208840093 · Zbl 0749.31003
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