Consider the scalar elliptic equation of divergence form
$$div((1+a)\nabla\pi)=div F\ \ in\ \ R^3,$$
where $a$ is a Schwartz function with $1+a\geq c=const>0$, $F=(F\_1,F\_2,F\_3)$ is a vector-valued Schwartz function.


Now can we give a counterexample to show that for $1<p<\infty, p\neq2$, the solution map $F\mapsto\nabla\pi$ is not bounded on $L^p$, namely, there does not hold
$$\|\nabla\pi\|\_{L^p}\leq C\|F\|\_{L^p}.$$


Let's give a remark. Denote by $Q=\nabla(-\Delta)^{-1}div$, a $C-Z$ singular integral operator on $L^p$, then
$$(1+a)\nabla\pi=[a,Q]\nabla\pi+QF,$$
where $[a,Q]\nabla\pi=aQ\nabla\pi-Q(a\nabla\pi)$ is a commutator. 


By *R. R. Coifman, R. Rochberg, and Guido Weiss*, [**Factorization theorems for Hardy spaces in several variables**](http://www.ams.org/mathscinet-getitem?mr=412721), *Ann. of Math. (2)* **103** (1976), no. 3, 611--635, the commutator operator $[a,Q]$ is bounded on $L^p$ if and only if $a\in BMO$
$$\|[a,Q]\nabla\pi\|\_{L^p}\leq C\|a\|\_{BMO}\|\nabla\pi\|\_{L^p}.$$ Thus if $\|a\|\_{BMO}$ is sufficiently small, then the gradient estimate is valid. But what happens if we remove the smallness condition for $a$?



$L^p$ estimates for elliptic equation of divergence form
You may also give a look to


G.Di Fazio Lp estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Un. Mat. Ital. A (7) 10 (1996), no. 2, 409–420


where leading coefficients are VMO and boundedness is obtained for any $1<p<+\infty$.


