LAUSR

We consider a bulk charge potential of the form $$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$u(x)=∫Ωg(y)F(x−y)dy,x=(x1,x2,x3)∈R3, 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 $$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$∂2u(x)∂xi∂xj=h∫Σg(y)∂2F(x−y)∂xi∂xjdy−ni(x)nj(x)g(x)+γij(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.