On Duality in Spaces of Solutions to Elliptic Systems

Abstract

Suppose that D is a bounded domain in \(\mathbb{R}^n \) (n≥2) with connected real-analytic boundary, A is an elliptic system with real-analytic coefficients in a neighborhood of the closure \(\overline D \) of D, and sol(A,D) is the space of solutions to the system Au=0 in D furnished with the standard Frechet–Schwartz topology. Then the dual of sol(A,D) represents the space sol(A,\(\overline D \)) of solutions to the system Au=0 in a neighborhood of \(\overline D \) furnished with the standard inductive limit topology over some decreasing net of neighborhoods of \(\overline D \). The corresponding pairing is generated by the inner product in the Lebesgue space L2(D).