A periodic Faddeev-type solution operator

Abstract

We construct periodic solution operators for the equation Δu + 2iζ · ∇u = f in a bounded domain with the help of Fourier series. We prove that the L2-norms of these operators converge to zero if the parameter |Im ζ| goes to infinity. Then we apply these operators to show that functions u ∈ C20(Rd) satisfying an inequality |Δu(χ)| ≤ M |u(χ)\ in Rd must vanish everywhere. We extend this result to other second order elliptic differential operators with constant coefficients replacing the Laplacian. Finally, we use the solution operators to derive that the span of products of solutions to differential equations is dense in L1. © 1996 Academic Press, Inc.