Calderón-Zygmund estimates for higher order systems with p(x) growth

Abstract

For weak solutions $u \in W^{m,1}(Ω;\R^N)$ of higher order systems of the type \int_Ω< A(x,D^m u),D^m ϕ> dx = \int_Ω< |F|^{p(x)-2}F,D^m ϕ> dx, for all $ϕ\in C^{\infty}_c(Ω;\R^N), m > 1$ with variable growth exponent $p:Ω\to (1,\infty)$ we prove that if $|F|^{p(\cdot)} \in L^q_{loc}(Ω)$ with $1 < q < \frac{n}{n-2} + δ$, then $|D^m u|^{p(\cdot)} \in L^q_{loc}(Ω)$. We should note that we prove this implication both in the non-degenerate and in the degenerate case.