A duality theorem for solutions of elliptic equations

Pierre Blanchet

doi:10.1155/s0161171290000114

Abstract

LetLbe a second order linear partial differential operator of elliptic type on a domainΩofℝmwith coefficients inC∞(Ω). We consider the linear space of all solutions of the equationLu=0onΩwith the topology of uniform convergence on compact subsets and describe the topological dual of this space. It turns out that this dual may be identified with the space of solutions of an adjoint equation near the boundary modulo the solutions of this adjoint equation on the entire domain.

Highlights

AND PRELIMINARY NOTIONS.Let H(R) denote the linear space of harmonic functions on a noncompact Riemann surface R with the topology of uniform convergence on compact subsets and let us consider the linear space H(R)’ of all continuous linear functionals on H(R)
Nakai and Sario [1] showed that this dual can be identified with a quotient space of harmonic functions as follows: H(R)’ H((R).)/H(R), where (R), represents the Alexandroff ideal boundary point of R and where H((R)) is the space of germs of functions "harmonic at (R)"
We endow L(f/) with the topology of uniform convergence on compact subsets of f/ and we denote by L(f/)’ the linear space of all continuous linear functionals on L(t)

Summary

Introduction

AND PRELIMINARY NOTIONS.Let H(R) denote the linear space of harmonic functions on a noncompact Riemann surface R with the topology of uniform convergence on compact subsets and let us consider the linear space H(R)’ of all continuous linear functionals on H(R). We endow L(f/) with the topology of uniform convergence on compact subsets of f/ and we denote by L(f/)’ the linear space of all continuous linear functionals on L(t) Let L be a second order linear partial differential operator of elliptic type on a domain 12 of Im with coefficients in C(R)(t2).

Results

Conclusion