A biharmonic normal operator

Abstract

When a biharmonic singularity s(x) is given on a boundary neighborhood A of a Riemannian manifold R, there arises a rather natural question about the biharmonic extendability of this singularity to p(x) which is biharmonic on all of R. For harmonic singularities s(x)^H(A), the question was answered by L. Sario (1952), who showed that although s(x) may not be harmonically extendable, nevertheless, in terms of the regular singularity L(f), s+L(f) is so extendable. Here, L: C(dA)-+H(A) is a bounded linear operator resembling the Dirichlet operator and is called normal. Analogously, in terms of H\A), the set of biharmonic singularities on A, a biharmonic normal operator L: C(dA)xC(dA)-+H 2 (A) is to resemble a Dirichlet operator and as an operator, is to be bounded. The purpose of the present effort is then to establish that, given a biharmonic normal operator L, each biharmonic singularity s(x) has a biharmonic extension modulo a regular biharmonic function L(f, g).Examples may be obtained by applying the extension process to particular choices of L and s(x) in particular, when s(x) has a fundamental biharmonic singularity at a, then a biharmonic Green's function with singularity at a is obtained.In his basic paper [4], L. Sario introduced a normal operator whose purpose was the construction of harmonic functions with certain prescribed behavior near the ideal boundary of a Riemann surface W. A full account of the applications of this operator, as well as an account of its own intrinsic interest, are given in, among others, the monograph of Rodin and Sario [1]. The issue of primary concern is, how does one accomplish showing that given a harmonic singularity s(z) defined on a boundary neighborhood W'dW, there is a harmonic p(z) defined on a Riemann surface W for which p{z)-s{z) is a regular singularity L(f). Here, in terms of a, the compact border of the bordered W, L is a continuous linear mapping similar to a Dirichlet operator. Explicitly, L is a linear mapping from C(α) to the set of regular singularities H B (W')\ that is, the set of bounded harmonic functions with 0 flux. In this sense, p(z) is said