# Global elliptic regularity theory on $\mathbb{R}^n$ or interior estimates for elliptic pde when $p \neq 2$

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 $$p = 2$$.

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