# $L^p$ estimates for elliptic equation of divergence form

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, 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 $a$?

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 $1<p<+\infty$.

Paper

N.G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 1963, 17, No. 3, 189–206.

contains example which shows that boundedness is valid in the interval near $p=2$ only. In this example $a$ is the matrix with bounded entries. Similar result for isotropic case may be found in

D. Faraco, Milton’s conjecture on the regularity of solutions to isotropic equations, Ann. Inst. H. Poincar´e, Anal. Non Lin´eaire, 2003, 20, No. 5, 889–909.