proving integrability of a function

Question

For a fixed $a\in \mathbb{R^n}$, we set
$$\Gamma(a,x)=\frac{1}{2\pi}log|a-x|\ \ \ \ \text{for $n=2$}$$ $$\Gamma(a,x)=\frac{1}{\omega_n(2-n)}|a-x|^{2-n}\ \ \ \ \text{for $n\ge 3$}$$ where $\omega_n$ denotes the surface are of the unit sphere in $\mathbb{R^n}$. Suppose $\Omega$ is a bounded open set in $\mathbb{R^n}$. Let $u\in C^1(\bar{\Omega})\cap C^2(\Omega)$.
Could any one give me some hints to prove that $f(x)=\Gamma(a,x)\Delta u(x)$ is Lebesgue integrable in $\Omega$. Thank you !

Answer 1

The only issue to check is the origin singularity at $a \in \Omega$. So take the ball centred at $a$ with radius $\epsilon$, call it $B$. Use Holder's inequality to notice $$ \int_B |f(x)| dx \leq C(B,u) \int_B \Gamma(a,x) dx$$ where $C$ is a constant depending on $\Omega$ and $u$. Swapping to spherical coordinates around $a$ we see $dx = r^{n-1} dr d\Theta$, where $d \Theta$ is the spherical contribution. Thus $$ \int_B \Gamma(a,x) dx = \int_{S^{n-1}} \int_0^\epsilon \frac{r}{\omega_n (2-n)} dr d \Theta < \infty $$ for $n \geq 3$. You can now check the $n=2$ explicitly if you like.