Vol. 48, No. 1, 1973

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Existence of Dirichlet finite biharmonic functions on the Poincaré 3-ball

Leo Sario and Cecilia Wang

Vol. 48 (1973), No. 1, 267–274
Abstract

In an earlier study we discussed the existence of quasiharmonic functions, i.e., solutions of Δu = 1. We showed, in particular, that there exist Dirichlet finite quasiharmonic functions on the Poincaré 3-ball

Bα : {|x| < 1,ds = (1 − |x|2)α|dαj|}

if and only if α (35,1). We now ask: Is the existence of these functions entailed by that of Dirichlet biharmonic functions? This is known to be the case for dimension 2. We shall show that, perhaps somewhat unexpectedly, it is no longer true for dimension 3.

Mathematical Subject Classification 2000
Primary: 53C20
Secondary: 31B30
Milestones
Received: 19 July 1972
Published: 1 September 1973
Authors
Leo Sario
Cecilia Wang