Hodge decomposition and solution formulas for some first-order time-dependent parabolic operators with non-constant coefficients

Abstract

In this paper, we present a Hodge decomposition for the \(L_p\)-space of some parabolic first-order partial differential operators with non-constant coefficients. This is done over different types of domains in Euclidean space \(\mathbb{R }^n\) and on some conformally flat cylinders and the \(n\)-torus associated with different spinor bundles. Initially, we apply a regularization procedure in order to control the non-removable singularities over the hyperplane \(t=0\). Using the setting of Clifford algebras combined with a Witt basis, we introduce some specific integral and projection operators. We present an \(L_p\)-decomposition where one of the components is the kernel of the regularized parabolic Dirac operator with non-constant coefficients. After that, we study the behavior of the solutions and the validity of our results when the regularization parameter tends to zero. To round off, we give some analytic solution formulas for the special context of domains on cylinders and \(n\)-tori.