On solvability in the small of higher order elliptic equations in Orlicz-Sobolev spaces

TL;DR

This paper investigates the local solvability of higher-order elliptic equations with nonsmooth coefficients within Orlicz-Sobolev spaces, extending classical $L_p$ results by considering more general function spaces with specific coefficient restrictions.

Contribution

It establishes local solvability of higher-order elliptic equations in Orlicz-Sobolev spaces under new conditions, generalizing classical $L_p$ space results.

Findings

Proves local solvability in Orlicz-Sobolev spaces.

Identifies conditions on coefficients and Boyd indices.

Extends classical $L_p$ solvability results.

Abstract

In this article, we consider a higher-order elliptic equation with nonsmooth coefficients with respect to Orlicz spaces on the domain $\Omega\subset\mathbb{R}^{n}$. The separable subspace of this space is distinguished in which infinitely differentiable and compactly supported functions are dense; Sobolev spaces generated by these subspaces are determined. We demonstrate the local solvability of the equation in Orlicz-Sobolev spaces under specific restrictions on the coefficients of the equation and the Boyd indices of the Orlicz space. This result strengthens the previously known classical $L_{p}$ analog.