# fa.functional analysis – Reference request: Is if possible to estimate the local behaviour of the solution of \$nabla cdot a(x) nabla f=g\$ via constant coefficients?

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.

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