Global representation of harmonic and biharmonic functions (Q1362755)

Let \(K\) be a compact set in \(\mathbb{R}^n\) and \(\Omega\) be an open set containing \(K\). It is well known that, if \(u\) is harmonic on \(\Omega\setminus K\), then \(u\) has a unique decomposition of the form \(u=v+w\), where \(v\) is harmonic on \(\Omega\) and \(w\) is a harmonic function on \(\mathbb{R}^n\setminus K\) such that \(w(x)\to 0\) \((n\geq 3)\), or \(w(x)-c\log|x|\to 0\) for some constant \(c\) \((n=2)\), as \(x\to\infty\) [see, for example, Chapter 9 of Harmonic function theory by \textit{S. Axler, P. Bourdon} and \textit{W. Ramey} (Springer, 1992; Zbl 0765.31001)]. The authors give another proof of this result and discuss analogues for biharmonic functions and bisubharmonic distributions.