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

Integral Equations

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## Abstract

We consider a bulk charge potential of the form where Ω is a layer of small thickness where the function

$$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$

*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$$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$

*γ*_{ij}(*x*) does not exceed in absolute value a certain quantity of the order of*h*^{2}, the surface integral is understood in the sense of Hadamard finite value, and the*n*_{i}(*x*),*i*= 1, 2, 3, are the coordinates of the normal vector on the surface Σ at a point*x*.## Preview

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