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Let $p \in (1.\infty)$ and $a_{ij},b_j,c \in C^\infty(\mathbb{R}^N)$ be the bounded coefficients of the elliptic second order differential operator $$[Au](x) = - \mathrm{div}(A(x) \nabla u) + \left\langle b(x), \nabla u \right\rangle + c(x)u(x).$$

Let $f \in L^p(\mathbb{R}^N) \cap C^\infty(\mathbb{R}^N)$ and $u \in L^p(\mathbb{R}^N)$ is a distributional solution of $Au = f$. Clearly it holds $u \in C^\infty(\mathbb{R}^N)$. I have read multiple times that also $u \in W^{2,p}(\mathbb{R}^N)$ and $Au \in L^p(\mathbb{R}^N)$ but have not found any reference yet. Do you know about a good reference?

Partitioning the space one can derive this result from interior estimates. for weak solutions $u \in W^{2,p}$. (Not $W_0^{2,p}$). However, i can only find references for such interior estimates in the case $p = 2$.

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  • $\begingroup$ Have you checked Gilbarg-Trudinger? $\endgroup$ – MaoWao Mar 12 at 15:42
  • $\begingroup$ Yes, was not successful. $\endgroup$ – Lük Mar 18 at 16:43

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