$p$-Energy forms on fractals: recent progress

TL;DR

This paper reviews recent advances in the theory of self-similar p-energy forms on fractals, highlighting new existence results and properties for p-harmonic functions, especially on p.c.f. fractals like the Sierpiński gasket.

Contribution

It summarizes recent progress on existence and properties of p-energy forms on fractals, extending the theory beyond the classical p=2 case to general p in (1,∞).

Findings

Existence of p-energy forms on p.c.f. fractals established.

Detailed properties of p-harmonic functions analyzed.

Applications illustrated on the Sierpiński gasket.

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 Sierpi\'{n}ski gasket.