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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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