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Angular limits of double layer potentials. (English) Zbl 0838.31006
The double layer potential on a Borel set \(A \subset \mathbb{R}^n\) and angular limits of the double layer potential are investigated under very general assumptions. Given \(\eta \in \partial A\) it is supposed that \(G = \mathbb{R}^n \backslash A\) has locally finite perimeter in \(\mathbb{R}^n \backslash \{\eta\}\). Let \(q : \partial A \to \langle 0, + \infty \rangle\) be a lower-semicontinuous function which is bounded and strictly positive on \(\partial A \backslash \{\eta\}\). Some necessary and sufficient geometric conditions are given for the existence of angular limits in \(\eta\) of the double layer potential with any continuous density \(f : \partial A \to \mathbb{R}\) such that \[ f(\xi) - f(\eta) = o \bigl( q(\xi) \bigr) \quad \text{as} \quad \xi \to \eta,\quad \xi \in \partial A. \]
Reviewer: M.Dont (Praha)

MSC:
31B25 Boundary behavior of harmonic functions in higher dimensions
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References:
[1] Ju. D. Burago, V.G. Maz’ya: Some problems of potential theory and function theory for domains with nonregular boundaries. Zapiski Naučnych Seminarov LOMI 3 (1967).
[2] M. Chlebík: Tricomi potencials. Thesis, Mathematical Institute of the Czechoslovak Academy of Sciences, Praha, 1988.
[3] M. Dont: Non-tangential limits of the double layer potentials. Časopis pro pěstování matematiky 97 (1972), 231-258. · Zbl 0237.31012
[4] H. Federer: The Gauss-Green theorem. Trans. Amer. Math. Soc. 58 (1945), 44-76. · Zbl 0060.14102
[5] H. Federer: Geometrie Measure Theory, Springer-Verlag. 1969. · Zbl 0176.00801
[6] J. Král: On the logarithmic potential. Comment. Math. Univ. Carolinae 3 (1962), no. 1, 3-10.
[7] J. Král: The Fredholm method in potential theory. Trans. Amer. Math. Soc. 125 (1966), 511-547. · Zbl 0149.07906
[8] J. Král: Limits of double layer potentials. Accad. Naz. dei Lincei, Rendiconti Cl. Sci. fis. mat. e. natur., ser. 8, vol. 48 (1970), 39-42. · Zbl 0195.11605
[9] J. Král: Flows of heat and the Fourier problem. Czechoslovak Math. J. 20(95) (1970), 556-598. · Zbl 0213.38203
[10] J. Král: On boundary behaviour of double layer potentials. Trudy Seminara S.L. Soboleva, Novosibirsk, 1976, pp. 19-34.
[11] J. Král: Integral Operators in Potential Theory. Lecture Notes in Math., vol. 823, Springer-Verlag, 1980.
[12] J. Král, J. Lukeš: Integrals of the Cauchy type. Czechoslovak Math. J. 22 (1972), 663-682.
[13] J. Lukeš: A note on integral of the Cauchy type. Comment. Math. Univ. Carolinae 9 (1968), 563-570.
[14] J.D. Machavariani: On lower and upper nontangential limits of the logarithmic potential of a double distribution. Amer. Math. Soc. Transl. 131 (1986), no. 2, 109-120. · Zbl 0596.31001
[15] V.G. Maz’ya: Boundary Integral Equations, in Analysis IV, Encyclopaedia of Mathematical Sciences. vol. 27, Springer-Verlag, 1991.
[16] S. Saks: Theory of the Integral. Dover Publications, New York, 1964. · Zbl 1196.28001
[17] J. Veselý: On the limits of the potential of the double distribution. Comment. Math. Univ. Carolinae 10 (1969), no. 2, 189-194. · Zbl 0176.41301
[18] J. Veselý: Angular limits of double layer potentials (in Czech with an English summary). Časopis pro pěstování matematiky 95 (1970), 379-401.
[19] W.P. Ziemer: Weakly Differentiable Functions. Springer-Verlag, 1989. · Zbl 0692.46022
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