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 L ²(D).