Regularity from $p$-harmonic potentials to $\mathrm{\infty }$-harmonic potentials in convex rings

The exploration of shape metamorphism, surface reconstruction, and image interpolation raises fundamental inquiries concerning the $C^1$ and higher-order regularity of $\mathrm{\infty }$-harmonic potentials – a specialized category of $\mathrm{\infty }$-harmonic functions. Additionally, it prompts questions regarding their corresponding approximations using $p$-harmonic potentials. It is worth noting that establishing $C^1$ and higher-order regularity for $\mathrm{\infty }$-harmonic functions remains a central concern within the realm of $\mathrm{\infty }$-Laplace equations and $L^{\mathrm{\infty }}$-variational problems. In this study, we investigate the regularity properties from $p$-harmonic potentials to $\mathrm{\infty }$-harmonic potentials within arbitrary convex ring domains $\mathrm{\Omega }={\mathrm{\Omega }}_0\mathrm{\backslash }{\bar{\mathrm{\Omega }}}_1$ in ${\mathbb{R}}^n$. Here ${\mathrm{\Omega }}_0$ is a bounded convex domain in ${\mathbb{R}}^n$ and ${\bar{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_0$ is a compact convex set. We prove the interior $C^1$ and some Sobolev regularity for $\mathrm{\infty }$-harmonic potentials.