Marcinkiewicz Exponent and Boundary Value Problems in Fractal Domains of $\mathbb{R}^{n+1}$

TL;DR

This paper investigates boundary value problems for monogenic functions in fractal domains, introducing a new solvability condition based on the Marcinkiewicz exponent that surpasses previous Minkowski dimension-based criteria.

Contribution

It introduces a novel solvability condition for boundary problems in fractal hypersurfaces using the Marcinkiewicz exponent within Clifford analysis.

Findings

New solvability condition improves previous Minkowski dimension criteria

The Teodorescu transform's properties are key to the analysis

Enhanced understanding of boundary problems in fractal domains

Abstract

This paper aims to study the jump problem for monogenic functions in fractal hypersurfaces of Euclidean spaces. The notion of the Marcinkiewicz exponent has been taken into consideration. A new solvability condition is obtained, basing the work on specific properties of the Teodorescu transform in Clifford analysis. It is shown that this condition improves those involving the Minkowski dimension.