zbMATH — the first resource for mathematics

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)

31B25 Boundary behavior of harmonic functions in higher dimensions
Full Text: EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.