In this paper we show that for Ω \Omega , a starlike Lipschitz domain, the dual of the space of harmonic functions in L p ( Ω ) {L^p}(\Omega ) need not be the harmonic functions in L q ( Ω ) {L^q}(\Omega ) , where 1 / p + 1 / q = 1 1/p + 1/q = 1 . We show that, as a consequence, the harmonic Bergman projection for Ω \Omega need not extend to a bounded operator on L p ( Ω ) {L^p}(\Omega ) for all 1 > p > ∞ 1 > p > \infty . The duality result is a partial answer to a question of Nakai and Sario [9] posed initially in the Proceedings of the London Mathematical Society in 1978. We treat the duality question as a biharmonic problem, and our result follows from the failure of uniqueness for the biharmonic Dirichlet problem in domains with sharp intruding corners.