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

Abstract

Let D : = ∑ i = 1 n ∂ ∂ x i e i be the Dirac operator in R n and let P ( X ) = a m X m + ⋯ + a 1 X 1 + a 0 be a polynomial with complex coefficients. Differential equations of the form P ( 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 μ and 1 where μ ∈ ] 0 , 1 [ . We further give formulas for the Szegö 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 Szegö kernel for solutions to polynomial Dirac equations of degree n ⩾ 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.