Asymptotic behavior and degeneracy of biharmonic functions on Riemannian manifolds
Lung Ock Chung · Kodai Mathematical Journal · 1976
One of the most fascinating results in harmonic classification theory is the is the identity O% D =O% C , where H stands for the class of harmonic functions h, dh=0, with Δ^dδ+δd the Laplace-Beltrami operator, and HD, HC are the subclasses of functions which are Dirichlet finite, or bounded Dirichlet finite, respectively.For any class F of functions, O Fi O F denote the classes of Riemannian manifolds on which FdR or F(tR respectively, and 0$, 0$ are the corresponding subclasses of manifolds of dimension N^2.A striking phenomenon in biharmonic classification theory is that, in contrast with the harmonic case, the inclusion O H 2 D (Zθ H 2c is strict, with H 2 the class of nonharmonic biharmonic functions.This has been, however, known only in the 2-dimensional case, in which it was established by undoubtedly the most intricate counterexample in all classification theory (Nakai-Sario [6]).The technique of complex analysis used therein is not available for an arbitrarily high dimension.Combining certain recent results in the biharmonic classification of the Poincare Λf-ball for the subclasses H 2 D, H 2 B of H 2 functions which are Dirichlet finite or bounded, respectively (Hada-Sario-Wang [2], [3]), one can draw the conclusion that O%2 D (Zθ%2 C is strict for Λ/>5.However, for N-3, 4, the reasoning fails and the question remains unsettled.The first purpose of the present paper is to give a complete and unified solution to this problem by proving the strict inclusion for any dimension Λ^2.We shall, in fact, show more generally that On the other hand, from recent results on the Poincare ΛΓ-ball (Hada-Sario-Wang [2], [3]), we infer that 0%2 D