问题及解答

证明:

\[
\int_0^r(\frac{1}{\rho^{n-1}}\int_{B_{\rho}}\Delta u dx)d\rho\geqslant\frac{1}{\omega_n r^{n-2}}\int_{B_r}\Delta u
\]

这里 $\Delta u$ 是 $B_r$ 上的非负函数 (即 $\Delta u\geqslant 0$ in $B_r$).

令 $\varphi(\rho)=\int_{B_{\rho}}\Delta u dx$,

\[
\begin{split}
&\int_0^r\frac{1}{\rho^{n-1}}\varphi(\rho)d\rho\geqslant\frac{1}{\omega_n r^{n-2}}\varphi(r)\\
\Longleftrightarrow &\omega_n r^{n-2}\int_0^r \frac{1}{\rho^{n-1}}\varphi(\rho)d\rho\geqslant\varphi(r).
\end{split}
\]