here we discuss homogeneous equations with no lower-order terms.
$Lu \equiv -D_{j}(a_{ij}(x)D_{j}u)$in $B_{1}(0)\subset \mathbb{R}^n$
where $a_{ij} \in L^{\infty}(B_{1})$ satisfies
$\lambda |\xi|^{2} \leq a_{ij}(x)\xi_{i} \xi_{j}\leq \Lambda |\xi|^{2}$ for all $x \in B_{1}(0)$ e $\xi \in \mathbb{R}^{n}$
for some positive constants $\lambda$ and $\Lambda$.
Remark : $B_{1}$ is a unit ball centered in $0$.
Done this. I would like to help in the following theorem
Theorem - Suppose $Lu = 0$ weakly in $B_{1}$. Then there holds
$\sup\limits_{B_{1/2}}|u(x)| + \sup\limits_{x,y \in B_{1/2}} \frac{ |u(x)-u(y)|}{|x-y|^{\alpha}} \leq c(n,\frac{\Lambda}{\lambda})||u||_{L^{2}(B_{1})} $
with $\alpha=\alpha(n,\frac{\Lambda}{\lambda}) \in (0,1)$
My work: In particular, $u$ is a weakly subsolution then by local boundedness theorem (Elliptic partial differential equation- QING HAN / FANGHUA LIN -chapter 4 -pg 67) is valid
$\sup\limits_{B_{1/2}}|u(x)|\leq c(n,\frac{\Lambda}{\lambda})||u||_{L^{2}(B_{1})}$
Now, I believe that, I should to use the oscillation theorem (about oscillation theorem pg.80) but i don't know
Thanks