anisotropic singularity of solutions to elliptic equations in a measure framework
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ID: 241865
2015
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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
$$\displaylines{
-\Delta u=\frac{\delta_{t\nu}-\delta_{-t\nu}}{t}\quad \text{in }\Omega,\cr
u=0\quad \text{on }\partial\Omega.
}$$
| Reference Key |
wang2015electronicanisotropic
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|
|---|---|
| Authors | ;Wanwan Wang;Huyuan Chen;Jian Wang |
| Journal | icsoft 2006 - 1st international conference on software and data technologies, proceedings |
| Year | 2015 |
| DOI |
DOI not found
|
| URL | |
| Keywords | Keywords not found |
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