Regularity of solutions to the polyharmonic equation in general domains

TL;DR

This paper proves optimal regularity results for solutions to the polyharmonic equation in arbitrary domains, establishing boundedness of derivatives up to a certain order and providing sharp estimates for the Green function.

Contribution

It demonstrates the boundedness of specific derivatives of polyharmonic solutions in general domains without geometric restrictions, and offers sharp Green function estimates.

Findings

Boundedness of [m - n/2 + 1/2] derivatives for solutions

Sharp estimates on the polyharmonic Green function

Results are optimal and cannot be improved in general domains

Abstract

The present paper establishes boundedness of $[m-\frac n2+\frac 12]$ derivatives for the solutions to the polyharmonic equation of order $2m$ in arbitrary bounded open sets of $\RR^n$, $2\leq n\leq 2m+1$, without any restrictions on the geometry of the underlying domain. It is shown that this result is sharp and cannot be improved in general domains. Moreover, it is accompanied by sharp estimates on the polyharmonic Green function.