# On the lack of interior regularity of the $p$-Poisson problem with $p>2$

## TL;DR

This paper demonstrates that solutions to the $p$-Poisson problem with $p>2$ can have limited interior regularity, showing the optimality of existing regularity theorems through explicit constructions.

## Contribution

It constructs specific right-hand sides to show the sharpness of known regularity results for the $p$-Poisson problem, confirming their optimality.

## Key findings

- Solutions have limited Besov regularity depending on the right-hand side.
- The results confirm the optimality of Savaré's shift theorem.
- The findings apply to all integrability parameters and are of local nature.

## Abstract

In this note we are concerned with interior regularity properties of the $p$-Poisson problem $\Delta_p(u)=f$ with $p>2$. For all $0<\lambda\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+\lambda$ such that the (Besov-) smoothness of corresponding solutions $u$ is essentially limited to $1+\lambda / (p-1)$. The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savar\'e [J. Funct. Anal. 152:176-201, 1998], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130:298-329, 2016]. Keywords: Nonlinear and adaptive approximation, Besov space, regularity of solutions, $p$-Poisson problem.