$p$-Energy forms on fractals: recent progress

TL;DR

This survey addresses the problem of developing self-similar $p$-energy forms for $p\in(1,\infty)\setminus\{2\}$ on self-similar fractals, extending the classical Dirichlet form theory ($p=2$). The main approach surveys constructions that yield a self-similar $p$-energy form $(\mathcal{E}_{p},\mathcal{F}_{p})$ on p.-c.f. sets and a broad class of infinitely ramified fractals via inductive limits and energy measures, together with a generalized contraction property and differentiability of $\mathcal{E}_{p}$. The key contributions include existence results on large classes, the definition and properties of the associated $p$-energy measures $\mu^{p}_{\langle u\rangle}$ with a chain rule, and the development of strong comparison principles and uniqueness/rigidity results in symmetric settings, illustrated in the 2D Sierpinski gasket. The article emphasizes how these results illuminate $p$-harmonic functions and the nonlinear potential theory on fractals and connect to geometric questions such as the Ahlfors regular conformal dimension.

Abstract

In this article, we survey recent progress on self-similar $p$-energy forms on self-similar fractals, where $p\in(1,\infty)$. While for $p=2$ the notion of such forms coincides with that of self-similar Dirichlet forms and there have been plenty of studies on them since the late 1980s, studies on the case of $p\in(1,\infty)\setminus\{2\}$ was initiated much later in 2004 by Herman, Peirone and Strichartz [Potential Anal. 20 (2004), 125--148] and Strichartz and Wong [Nonlinearity 17 (2004), 595--616] and no essential progress on this case had been made since then until a few years ago. The recent progress by Kigami, Shimizu, Cao--Gu--Qiu and Murugan--Shimizu has established the existence of such $p$-energy forms on general post-critically finite (p.-c.f.) self-similar sets and on large classes of low-dimensional infinitely ramified self-similar sets, and the authors have proved further detailed properties of these forms and associated $p$-harmonic functions, mainly for p.-c.f. self-similar sets. This article is devoted to a review of these results, focusing on the most recent developments by the authors and illustrating them in the simplest non-trivial setting of the two-dimensional standard Sierpiński gasket.