On the domains of first order differential operators on the Sierpi\'{n}ski gasket

TL;DR

This paper investigates the structure and properties of first order differential operators on the Sierpiński gasket, revealing self-similarity, domain representations, and examples with discontinuities in a fractal setting.

Contribution

It introduces a framework for first order differential operators on the Sierpiński gasket and provides a pointwise representation and examples with various discontinuities.

Findings

Self-similarity of one-forms on the gasket

Pointwise representation of domain elements via normal derivatives

Examples with different types of discontinuities

Abstract

We study the first order structure of $L^2$-differential one-forms on the Sierpi\'{n}ski gasket. We consider piecewise energy finite functions related to one-forms, normal parts, and show self-similarity properties of one-forms as in the case of energy finite functions on the Sierpi\'{n}ski gasket. We introduce first order differential operators, taking functions into functions, that can be understood as total derivatives with respect to some reference function or form. The main result is a pointwise representation result for certain elements in the domain of said first order differential operator by a ratio of normal derivatives. At last we provide three classes of examples in the domain of said first order differential operator with different types of discontinuities.