On the fundamental solutions of a class of elliptic quartic operators in dimension 3

Abstract

The (uniquely determined) even and homogeneous fundamental solutions of the linear elliptic partial differential operators with constant coefficients of the form ∑ j=1 3∑ k=1 3 c jk ∂ j 2 ∂ k 2 are represented by elliptic integrals of the first kind. Thereby we generalize Fredholm's example ∂ 1 4+ ∂ 2 4+ ∂ 3 4, which, up to now, was the only irreducible homogeneous quartic operator in 3 variables the fundamental solution of which was known to be expressible by tabulated functions.