Abstract
We consider a bulk charge potential of the form
where Ω is a layer of small thickness h > 0 located around the midsurface Σ, which can be either closed or open, and F(x − y) is a function with a singularity of the form 1/|x − y|. We prove that, under certain assumptions on the shape of the surface Σ, the kernel F, and the function g at each point x lying on the midsurface Σ (but not on its boundary), the second derivatives of the function u can be represented as
where the function γij(x) does not exceed in absolute value a certain quantity of the order of h2, the surface integral is understood in the sense of Hadamard finite value, and the ni(x), i = 1, 2, 3, are the coordinates of the normal vector on the surface Σ at a point x.
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Original Russian Text © A.V. Setukha, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 9, pp. 1262–1281.
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Setukha, A.V. On the Surface Integral Approximation of the Second Derivatives of the Potential of a Bulk Charge Located in a Layer of Small Thickness. Diff Equat 54, 1236–1255 (2018). https://doi.org/10.1134/S0012266118090112
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DOI: https://doi.org/10.1134/S0012266118090112