# The set of $p$-harmonic functions in $B_{1}$ is total in $C^{k}(\bar{B}_{1})$

## TL;DR

This paper proves that the set of p-harmonic functions within the unit ball is dense in the space of k-times continuously differentiable functions on its closure, extending understanding of fractional p-Laplacian operators.

## Contribution

It establishes the density of p-harmonic functions in $C^{k}(ar{B}_{1})$, providing new insights into the approximation properties of fractional p-Laplacian solutions.

## Key findings

- p-harmonic functions form a dense subset in $C^{k}(ar{B}_{1})$
- The result applies to fractional p-Laplacian operators with $0<s<1<p<
olinebreak\infty$
- Advances the understanding of approximation capabilities of p-harmonic functions

## Abstract

Let $(-\Delta_{p})^{s}$, with $0<s<1<p<\infty$, be the fractional $p$-Laplacian operator. We prove that the span of $p$-harmonic functions in $B_{1}$ is dense in $C^{k}(\bar{B}_{1})$.