A higher Dimensional Marcinkiewicz Exponent and the Riemann Boundary Value Problems for Polymonogenic Functions on Fractals Domains

TL;DR

This paper extends the Marcinkiewicz exponent to higher dimensions and applies it to solve Riemann boundary value problems for polymonogenic functions on fractal domains in Euclidean space, establishing conditions for existence and uniqueness.

Contribution

It introduces a higher-dimensional Marcinkiewicz exponent and applies it to boundary value problems on fractal domains, providing new existence and uniqueness results.

Findings

Established sufficient conditions for solution existence and uniqueness.

Described a class of hypersurfaces with refined results.

Extended Marcinkiewicz exponent to higher dimensions.

Abstract

We use a high-dimensional version of the Marcinkiewicz exponent, a metric characteristic for non-rectifiable plane curves, to present a direct application to the solution of some kind of Riemann boundary value problems on fractal domains of Euclidean space $\mathbb{R}^{n+1}, n\geq2$ for Clifford algebra-valued polymonogenic functions with boundary data in classes of higher order Lipschitz functions. Sufficient conditions to guarantee the existence and uniqueness of solution to the problems are proved. To illustrate the delicate nature of this theory we described a class of hypersurfaces where the results are more refined than those that exist in literature.