COMPLETENESS THEOREMS IN THE UNIFORM NORM CONNECTED TO ELLIPTIC EQUATIONS OF HIGHER ORDER WITH CONSTANT COEFFICIENTS

Abstract

We consider the problem of the completeness of the system [Formula: see text] in [C0(Σ)]m, where {ωk} is a basis of polynomial solutions of the elliptic equation ∑|α|≤2maαDαu = 0, aαare real constants, Σ is the boundary of a bounded domain in ℝnand ∂νdenotes the normal derivative. If E satisfies a Gårding inequality and [Formula: see text] is connected, we show that such a completeness property holds if and only if all the irreducible factors of the characteristic polynomial of the differential operator vanish at the origin. The proof hinges on some jump formulas obtained for general potentials generated by measures.