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

Abstract

For weak solutions $u\in W^{m,1}(\Omega ;{\mathbb{R}}^N)$ 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 $\varphi \in C_c^{\infty }(\Omega ;{\mathbb{R}}^N),m>1$ with variable growth exponent $p:\Omega \to (1,\infty )$ we prove that if $|F{|}^{p(\cdot )}\in L_{loc}^q(\Omega )$ with $1<q<\frac{n}{n-2}+\delta$, then $|D^{m}u{|}^{p(\cdot )}\in L_{loc}^q(\Omega )$. We should note that we prove this implication both in the non-degenerate and in the degenerate case.

29 pages