Representation and uniqueness theorems for polyharmonic functions
The main results are a representation formula for functions of class \(C^{2m}\) on the open ball \(B(r)\) of centre 0 and radius \(r\) in \(\mathbb{R}^ n\) and a uniqueness theorem for polyharmonic functions vanishing on certain sets. The representation formula, which we now describe, generalizes the Poisson-Jensen formula for functions of class \(C^ 2(B(r))\). Suppose that \(v\in C^{2m} (B(r))\) and that \(v\) is continuous on \(\overline{B(r)}\). Under an integrability hypothesis on the iterated Laplacians \(\Delta^ j v\) \((1\leq j\leq m)\), it is shown that \(v\) has the form \[ v(x)= \sum_{j=0}^{m-1} U_ j(x) (r^ 2- \| x\|^ 2)^ j+ R_ m(x). \] The functions \(U_ j\) are harmonic on \(B(r)\); in fact \(U_ 0\) is the Poisson integral of \(v\) and \[ U_ j(x)= \int_{B(r)} K_{j,n} (x,\xi; r)\Delta^ j v(\xi) d\xi \quad \text{for} \quad 1\leq j\leq m-1, \] where the kernels \(K_{j,n}\) are given explicitly. The function \(R_ m\) is given by \[ R_ m(x)= \int_{B(r)} G_{m,n} (x,\xi;r) \Delta^ m v(\xi) d\xi, \] where \(G_ m\) is expressed explicitly and is shown to be the Green's function for \(\Delta^ m\) in \(B(r)\) for the boundary conditions \((\partial/ \partial\nu)^ j v=0\) \((0\leq j\leq m-1)\). If \(\Delta^ m v\equiv 0\) on \(B(r)\), then \(R_ m\equiv 0\), so that the above representation formula has Almansi's classical representation of polyharmonic functions in terms of harmonic functions as a corollary. The uniqueness result is as follows. Suppose that \(D\) is a domain, that \(B_ j\) \((1\leq j\leq m-1)\) are distinct balls and that \(B_ m\) is a bounded open set such that \(\overline{B}_ j \subset D\) \((1\leq j\leq m)\). Let \(S_ j= \partial B_ j\) and assume that \(S_ m\setminus (\bigcup_{j=1}^{m-1} S_ j)\) is dense in \(S_ m\). If \(u\) is \(m\)- harmonic on \(D\) and \(u=0\) on \(\bigcup_{j=1}^ m S_ j\), then \(u\equiv 0\). It is shown, at least in the plane, that the conclusion fails if \(m\geq 2\) and only \(m-2\) of the \(B_ j\) are balls. It remains an interesting open question whether the balls \(B_ 1,\dots, B_{m-1}\) can be replaced by, for example, ellipsoids.
representation formula for functions of class \(C^{2m}\)
uniqueness theorem for polyharmonic functions
scientific article; zbMATH DE number 528690
scientific article