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Differential Equations

, Volume 54, Issue 9, pp 1236–1255 | Cite as

On the Surface Integral Approximation of the Second Derivatives of the Potential of a Bulk Charge Located in a Layer of Small Thickness

  • A. V. Setukha
Integral Equations
  • 3 Downloads

Abstract

We consider a bulk charge potential of the form
$$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$
where Ω is a layer of small thickness h > 0 located around the midsurface Σ, which can be either closed or open, and F(xy) is a function with a singularity of the form 1/|xy|. 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
$$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$
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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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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