Mollifiers in Clifford Analysis

TL;DR

This paper develops mollifiers within Clifford analysis to approximate solutions of boundary value problems for Dirac-like operators on domains with smooth boundaries, enabling boundary extension of smooth functions.

Contribution

It introduces mollifiers in Clifford analysis and constructs smooth approximations to solutions of non-homogeneous boundary value problems on domains with C^2 boundaries.

Findings

Constructed a sequence of smooth functions approximating gamma-regular functions

Extended smooth functions up to the boundary of C^2 domains

Provided a framework for mollifiers in Clifford analysis

Abstract

We introduce mollifiers in Clifford analysis setting and construct a sequence of $\C^{\infinity}$-functions that approximate a $\gamma$-regular function and a solution to a non homogeneous BVP of an in homogeneous Dirac like operator in certain Sobolev spaces over bounded domains whose boundary is not that wild. One can extend the smooth functions upto the boundary if the domain has a $\C^1$-boundary and this is the case in the paper as we consider a domain whose boundary is a $\C^2$-hyper surface.