u is nonnegative, continuous function on the unit ball $B_1\subset \mathbb{R}^n$. $\Delta u\ge-1$ in the sense of distributions. Then the function $u+|x|^2/2n$ is subharmonic in $B_1$. Subharmonicity of $u+|x|^2/2n$ implies \begin{align*} \max_{B_{1/2}} (u+|x|^2/2n) &\le C \int_{B_1\backslash B_{1/2}}(u+|x|^2/2n)dx\\ &\le C \int_{\partial B_1}(u+|x|^2/2n)d\sigma \end{align*} where $C$ is a positive constant depend only on $n$.
I have two problems:
- The first inequality implied by the Poisson integral. Let $K_R(x,y)=\frac{R^2-|x|^2}{n\omega_nR|x-y|^n}$, $x\in B_R$, $y\in\partial B_R$. Denote $K_R(x,y)[u]=\int_{\partial B_R }K_R(x,y)u(y)d\sigma_y$. By maximum principle: $u\le K_R(x,y)[u] $ in $\overline B_R$ for any $R<1$. Thus $$u\le K_R(x,y)[u] \le \frac{4^n2/3}{7^nn\omega_n }\int_{\partial B_R}u(y)d\sigma_y$$ for $x\in B_{1/2}$, $R\in [2/3,3/4]$. Integrate the above inequality by $r$ from [2/3,3/4]. Then we can obtain the first inequality. Is this right?
- How to get the second inequality?