Regularity of solutions to the polyharmonic equation 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.