A theorem on the potential of a double layer
It is well known that the double layer potential \(D\mu(P)=\int_ S\mu(q)\cdot(\partial/\partial n_ q)(1/| P-q|)dS_ q\) is discontinuous at the boundary \(S\) crossing: \(\lim_{\varepsilon\to 0}\mu(p\pm\varepsilon n_ p)=D\mu(p)\pm 2\pi\mu(p)\), \((\varepsilon>0)\). The author establishes a similar discontinuous behaviour for the surface gradients in the case of closed surfaces \(S\): \[ \overline\nabla(D\mu)\pm(p)=\oint_ S[\mu(q)-\mu(p)]\overline\nabla_ pK(q,p) dS_ q\pm 2\pi\overline\nabla\mu(p). \] The following notations are used: \(P\) --- a point in \(R_ +\) or \(R_ -\); \(R_ +,R_ -\) --- the interior and exterior domains, respectively, defined by \(S\); \(S\) --- a Lyapunov surface in \(R^ 3\); \(p,q\) --- points on \(S\); \(\mu(q)\in C'(S)\) --- the density of the double layer potential; \(n_ q\) --- the inner unit normal at \(q\); \(K(p,q)=(\partial/\partial n_ q)\) \((1/| P-q|)\) --- the kernel of the integral in the double layer potential definition. The author compares the presented jump relation with other ones, established by other authors, and shows its advantages.
jump relations in the double layer potential
jump relations for the gradient of double layer potential
scientific article; zbMATH DE number 57732
scientific article