Assume that $u\in W^{1,2}(B^n,R)$ is a weak solution to the elliptic pde of type $$\sum_{i,j=1}^n\partial_j \left(a_{ij}(x) \partial_i u(x)\right)=f\in L^p(B^n),$$ where $n/2<p<n$, and $A=(a_{ij}(x))$ is a smooth elliptic symmetric matrix defined in the unit ball $B^n$. Assume also that $u_{\partial B^n}=0$. I need a reference to the following claim $$\Du\_{L^{p^*}}\le C(\f\_{L^p}+\Du\_{L^2}),$$ where $p^*=np/(np)$.

$\begingroup$ If $p^* \leq 2$ the inequality is obvious (omitting $\f\_p$ on the RHS. In the other case, since the coefficients are smooth, you can use the estimate $\u\_{2,p} \leq C\f\_p$ and then Sobolev embedding $\Du\_{p^*} \leq C\u\_{2,p}$ to obatain the inequality without $Du$ on the right hand side (when $p^* >2$ the $W^{2,p}$ solution is in $H^1$). $\endgroup$– Giorgio MetafuneJan 10 at 21:17

$\begingroup$ @Giorgio Metafune, how do you know $\u\_{2,p}\le C\f\_p$? Maybe u is not in W^{2,p}?? $\endgroup$– DejvJan 10 at 21:42

$\begingroup$ Solve in $W^{2,p}$ and you get a solution in $H^1$ which, by uniquesess in $H^1$, coincides with your original solution. This is what I have in mind. $\endgroup$– Giorgio MetafuneJan 10 at 23:22

1$\begingroup$ Giorgio Metafune. It seems you are right. Thank you very much. $\endgroup$– DejvJan 11 at 8:27
1 Answer
you could also give a look to the paper G.Di Fazio L^p estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Un. Mat. Ital. A (7) 10 (1996), no. 2, 409–420 where similar estimates are shown.