An higher integrability result for the second derivatives of the solutions to a class of elliptic PDE’s

Abstract

In this paper we establish an higher integrability result for second derivatives of the local solution of elliptic equation

$$\begin{aligned} \text {div} (A(x, Du)) =0 \,\,\,\,\,\, \hbox {in } \Omega \end{aligned}$$

(0.1)

where $\Omega \subseteq \mathbb {R}^n$, $n\ge 2$ and $A(x,\xi )$ has linear growth with respect to $\xi$ variable. Concerning the dependence on the x-variable, we shall assume that, for the map $x \rightarrow A(x,\xi )$, there exists a non negative function k(x), such that

$$\begin{aligned} |D_x A(x, \xi )| \leqslant k(x)\,(1+ |\xi |) \end{aligned}$$

(0.2)

for every $\xi \in \mathbb {R}^n$ and a.e. $x \in \Omega$. It is well known that there exists a relationship between this condition and the regularity of the solutions of the equation. Our pourpose is to establish an higher integrability result for second derivatives of the local solution, by assuming k(x) in a suitable Zygmund class.