Anisotropic singularity of solutions to elliptic equations in a measure framework

In this article we study the weak solutions of elliptic equation $$\displaylines{ -\Delta u=2\frac{\partial \delta_0}{\partial \nu }\quad \text{in }\Omega,\cr u=0\quad \text{on }\partial\Omega, }$$ where $\Omega$ is an open bounded $C^2$ domain of $\mathbb{R}^N$ with $N\ge 2$ containing the origin, $\nu$ is a unit vector and $\frac{\partial\delta_0}{\partial \nu}$ is defined in the distribution sense, i.e. $$ \langle\frac{\partial \delta_0}{\partial \nu},\zeta\rangle =\frac{\partial\zeta(0)}{\partial \nu} , \quad \forall \zeta\in C^1_0(\Omega). $$ We prove that this problem admits a unique weak solution u in the sense that $$ \int_\Omega u(-\Delta)\xi dx=2\frac{\partial \xi(0)}{\partial \nu},\quad \forall \xi\in C^2_0(\Omega). $$ Moreover, u has an anisotropic singularity and can be approximated, as $t\to 0^+$, by the solutions of $$\displaylines{ -\Delta u=\frac{\delta_{t\nu}-\delta_{-t\nu}}{t}\quad \text{in }\Omega,\cr u=0\quad \text{on }\partial\Omega. }$$