Regularity from p-harmonic potentials to ∞-harmonic potentials in convex rings

The exploration of shape metamorphism, surface reconstruction, and image interpolation raises fundamental inquiries concerning the (Formula presented.) and higher order regularity of (Formula presented.) -harmonic potentials — a specialized category of (Formula presented.) -harmonic functions. Additionally, it prompts questions regarding their corresponding approximations using (Formula presented.) -harmonic potentials. It is worth noting that establishing (Formula presented.) and higher order regularity for (Formula presented.) -harmonic functions remains a central concern within the realm of (Formula presented.) -Laplace equations and (Formula presented.) -variational problems. In this paper, for (Formula presented.) -harmonic potentials (Formula presented.) in arbitrary convex rings (Formula presented.) of (Formula presented.), we establish the (Formula presented.) -regularity of (Formula presented.) and its (Formula presented.) -approximation by (Formula presented.) -harmonic potentials (Formula presented.) in (Formula presented.). This answers an open problem by Lindgren and Lindqvist[Discrete Contin. Dyn. Syst. 39 (2019), no. 8, 4731–4746; Adv. Math. 378 (2021), Paper No. 107526, 24 pp]. We also obtain the (Formula presented.) -regularity of (Formula presented.) and its weak (Formula presented.) -approximation by (Formula presented.), where (Formula presented.). The distributional second-order derivative (Formula presented.) is further proved to be a Radon measure and be approximated by (Formula presented.) weakly in a measure-theoretic sense. Moreover, in the special case that (Formula presented.) for some (Formula presented.), we show (Formula presented.). Finally, in planar convex rings, we prove that (Formula presented.) -harmonic potentials are twice differentiable almost everywhere, providing optimal results in this context. The second-order derivatives contribute to the absolutely continuous part of (Formula presented.), enabling (Formula presented.).