Reproducing kernel functions of solutions to polynomial Dirac equations in the annulus of the unit ball in R-n and applications to boundary value problems

Journal of Mathematical Analysis and Applications vol:358 issue:2 pages:281-293

Abstract:

Let ${\bf D}:= \sum_{i=1}^n \frac{\partial }{\partial x_i} e_i$ be the Dirac operator in $\R^n$ and let $P(X) = a_m X^m + \ldots + a_1 X_1 + a_0$ be a polynomial with complex coefficients. Differential equations of the form $P({\bf D})f = 0$ are called polynomial Dirac equations. In this paper we consider Hilbert spaces of Clifford algebra valued functions that satisfy such a polynomial Dirac equation in annuli of the unit ball in $\R^n$. We determine an explicit formula for the Bergman kernel for solutions of complex polynomial Dirac equations of any degree $m$ in the annulus of radii $\mu$ and $1$ where $\mu \in ]0,1[$. We further give formulas for the Szeg\"o kernel for solutions to polynomial Dirac equations of degree $m < 3$ in the annulus. This includes the Helmholtz and the Klein-Gordon equation as special cases. We further show the non-existence of the Szeg\"o kernel for solutions to polynomial Dirac equations of degree $n \ge 3$ in the annulus. As an application we give an explicit representation formula for the solutions of the Helmholtz and the Klein-Gordon equation in the annulus in terms of integral operators that involve the explicit formulas of the Bergman kernel that we computed.