Electron. J. Diff. Equ., Vol. 2015 (2015), No. 230, pp. 1-12.

Anisotropic singularity of solutions to elliptic equations in a measure framework

Wanwan Wang, Huyuan Chen, Jian Wang

Abstract:
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
$$\displaylines{ -\Delta u=\frac{\delta_{t\nu}-\delta_{-t\nu}}{t}\quad \text{in }\Omega,\cr u=0\quad \text{on }\partial\Omega. }$$

Submitted May 11, 2015. Published September 10, 2015.
Math Subject Classifications: 35R06, 35B40, 35Q60.
Key Words: Anisotropy singularity; weak solution; uniqueness.

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Wanwan Wang
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: wwwang2014@yeah.net
Huyuan Chen
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: chenhuyuan@yeah.net
Jian Wang
Institute of Technology
East China Jiaotong University
Nanchang, Jiangxi 330022, China
email: jianwang2007@126.com

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