Jump relations of the quadruple layer potential on a regular surface in three dimensions
Suppose that \(S\) is a sufficiently smooth regular surface in \(\mathbb R^3\). Denote the unit normal vector to \(S\) at \(x\) by \(N(x)\). Denote \(G(x,t)=| x-t| ^{-1}\), \(\partial^2 G(x,t)/\partial N(t)^2 =N(t)\cdot \nabla_t \nabla_t G(x,t) \cdot N(t)= 3\langle N(t),X-T\rangle ^2 | x-t| ^{-5} -| x-t| ^{-3}\). Let \(u(x)=\int\sigma (t)[\partial^2 G(x,t)/\partial N(t)^2] \,dt\) be the quadruple layer potential with density \(\sigma \), where \(\sigma \) is a twice continuously differentiable real function on \(S\). It is proved the jump relations at \(x\in S\): \[ \lim u(x\pm \varepsilon N(x))= \pm 2\pi (k_1(x)+k_2(x))\sigma (x)+ f.p. \int \sigma (t)[\partial^2 G(x,t)/ \partial N(t)^2] \,dt \text{ as }\varepsilon \to 0_+. \] Here \(k_1(x)\), \(k_2(x)\) are the principle curvatures of \(S\) at \(x\) and \(f.p.\int \) denotes the finite part of the integral.
scientific article; zbMATH DE number 5083831