Calder\'on-Zygmund estimates for higher order elliptic equations in Orlicz-Sobolev spaces

TL;DR

This paper establishes Calderón-Zygmund estimates for higher order elliptic equations within Orlicz-Sobolev spaces, linking the integrability of the data to the solution's derivatives in a generalized function space setting.

Contribution

It extends Calderón-Zygmund theory to fourth order quasilinear elliptic equations in Orlicz-Sobolev spaces, providing new regularity results for solutions.

Findings

If G(f) is in L^q, then G(Δu) is also in L^q for q ≥ 1.

The results generalize classical estimates to a broader Orlicz space framework.

Abstract

In this paper we obtain Calder\'on-Zygmund estimates for the laplacian of the following fourth order quasilinear elliptic problem $$ \Delta(g(\Delta u)\Delta u) = \Delta(g(\Delta f)\Delta f). $$ where the primitive of $g(t)t$, $G(t)$, is an $N-$function. We prove that if $G(f)\in L^q$, then $G(\Delta u)\in L^q$ for $q\ge 1$.