L^p regularity of homogeneous elliptic differential operators with constant coefficients on R^N

TL;DR

This paper establishes necessary and sufficient conditions for the regularity of solutions to homogeneous elliptic differential operators with constant coefficients on bR^N, linking growth at infinity to membership in L^p spaces.

Contribution

It provides a comprehensive growth condition characterization for the L^p regularity of derivatives of solutions to elliptic operators, extending previous results to systems and exterior domains.

Findings

Characterizes when derivatives of solutions are in L^p based on growth conditions.

Extends results to elliptic systems with different orders.

Provides necessary and sufficient growth conditions for regularity.

Abstract

Let $A$ be a homogeneous elliptic differential operator of order $m$ on $% \Bbb{R}^{N}$ with constant complex coefficients. A partial version of the main result is as follows: Suppose that $u\in L_{loc}^{1}$ and that $Au\in L^{p}$ for some $1<p<\infty .$ Then, all the partial derivatives of order $m$ of $u$ are in $L^{p}$ if and only if $|u|$ grows slower than $|x|^{m}$ at infinity, provided that growth is measured in an $L^{1}$-averaged sense over balls with increasing radii. The necessity provides an alternative answer to the pointwise growth question investigated with mixed success in the literature. Only a few special cases of the sufficiency are already known, mostly when $A=\Delta .$ The full result gives a similar necessary and sufficient growth condition for the derivatives of $u$ of any order $k\geq 0$ to be in $L^{p}$ when $Au$ satisfies a suitable (necessary) condition. This is generalized to exterior domains under mandatory restrictions on $N$ and $p$ and to Douglis-Nirenberg elliptic systems whose entries are homogeneous operators with constant coefficients and possibly different orders.