On minimally thin and rarefied sets in \(R^ p\), p\(\geq 2\). (English) Zbl 0594.31014

Let \(p\geq 2\), \(D=\{x\in R^ p:\) \(x_ 1>0\}\) (where \(x=(x_ 1,...,x_ p))\). If u is subharmonic in D, \(y\in \partial D\), put \(u(y)=\lim \sup u(x),\) \(x\to y\), \(x\in D\). If \(u\leq 0\) on \(\partial D\) and if sup u(x)/x\({}_ 1<\infty\), then it is known that \(u(x)/x_ 1\to \alpha\), \(x\to \infty\), \(x\in D-E\), where the exceptional set E is minimally thin at infinity. If \(p\geq 3\), it is also known that \((u(x)-\alpha x_ 1)/| x| \to 0,\) \(x\to \infty\), \(x\in D-F\), where the exceptional set F is rarefied at infinity. In the present paper precise descriptions of the geometric properties of the exceptional sets E and F are given in terms of a covering by balls.
Reviewer: M.Dont


31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions