Year: 2023
Type: preprint
Abstract: The exploration of shape metamorphism, surface reconstruction, and image interpolation raises fundamental inquiries concerning the $C^1$ and higher-order regularity of $\infty$-harmonic potentials -- a specialized category of $\infty$-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 $\infty$-harmonic functions remains a central concern within the realm of $\infty$-Laplace equations and $L^\infty$-variational problems. In this study, we investigate the regularity properties from $p$-harmonic potentials to $\infty$-harmonic potentials within arbitrary convex ring domains $Ω=Ω_0\backslash \overline Ω_1$ in $\mathbb R^n$. Here $Ω_0$ is a bounded convex domain in $\mathbb R^n$ and $\overlineΩ_1\subset Ω_0$ is a compact convex set. We prove the interior $C^1$ and some Sobolev regularity for $\infty$-harmonic potentials.
Source: arXiv (Cornell University)
Authors: Peng, Fa, Zhang, Yi Ru-Ya, Zhou, Yuan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4387635235