On Weighted Dirac Operators and Their Fundamental Solutions for Anisotropic Media

Abstract

The heat transfer problem in isotropic media has been studied extensively in Clifford analysis, but very little in the anisotropic case for this setting. As a first step in this way, we introduce in this work Dirac operators with weights belonging to the Clifford algebra $${\mathcal {A}}_n$$ , which factor the second order elliptic differential operator $$ {\tilde{\Delta }}_n= div (B \,\nabla ), $$ where $$B \in \mathbb {R}^{n \times n}$$ is a symmetric and positive definite matrix. For these weighted Dirac operators we construct fundamental solutions and get a Borel–Pompeiu and Cauchy integral formula.