Consider the divergence form uniformly elliptic operator $nabla cdot a(x) nabla$

where the coefficient $a$ are smooth and bounded and $D$ is a bounded

and smooth domain of $mathbb R^d$

$$

begin{cases}

nabla cdot a(x) nabla f (x)=g text{ in } D \

f(x) = 0 text{ in } partial D,

end{cases}

$$

where $g$ for some $g$.

Consider now $x_0in D$ and $delta < d(x,partial D)$ and the function $f_{x_0}$ which solves

$$

begin{cases}

nabla cdot a(x_0) nabla f^delta_{x_0} (x)=g text{ in } B(x_0,delta)\

f^delta_{x_0}(x) = f(x) text{ in } partial B(x_0,delta).

end{cases}

$$

I was wondering whether it is possible to bound quantities such as

$$

M(x_0,delta,r,p):=r^{-d}|f-f^delta_{x_0}|_{L^p(B(x_0,r))}

$$

for $r < delta$ and for some $p in (1,infty)$. In particular, I was wondering about the case asymptotic behaviour for $r to 0^+$. That is, can I show that

$$

M(p, gamma):= sup_{x_0 in D} sup_{delta < d(x,partial D)wedge c_a} sup_{r le delta} frac{M(x_0,delta,r,p)}{r^gamma},

$$

is finite for some $gamma>0$ and some constant $c_a >0$? If so, does that bound depends on the smoothness of $g$?

The idea being that if $delta$ is sufficiently small, $a(x)approx a(x_0)$ in the ball $B(x_0,delta)$ and therefore the two equations should behave similarly. I am not sure if this is indeed enough or if I would need to ask $delta$ to vanish as well.

I would appreciate any references or even what are the keywords to find such type of estimates.