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On a condition of thinness at infinity. (English) Zbl 0691.31002
Let E be a domain in \({\mathbb{R}}^ m\) (m\(\geq 3)\). E is said to be thin at a point \(x_ 0\) if there exists a superharmonic function u, defined in \(B(x_ 0,r)=\{x\in {\mathbb{R}}^ m| \quad | x-x_ 0| <r\}\) for some r, such that \[ \liminf_{y\to x_ 0,y\in B(x_ 0,r)\cap E,y\neq x_ 0}u(y)\quad >\quad u(x_ 0). \] This is the definition given by Brelot to characterize regularity of a point for the Dirichlet problem. Using the Riesz decomposition theorem it is possible to prove that the “lim inf” can be taken to be infinite. This fact implies that the solid angle subtended by \(\partial B(x_ 0,r)\cap E\) at the point \(x_ 0\) is very small. In fact, it tends to zero with r. More precisely, if \(\theta (r)=\sigma [\partial B(x_ 0,r)\cap E]/\sigma [\partial B(x_ 0,r)],\) where \(\sigma\) is the Lebesgue measure on \(\partial B(x_ 0,r)\), then \(\theta\) (r)\(\to 0\), as \(r\to 0\). Therefore the following natural question arises: how big can \(\theta\) (r) be when E is thin at \(x_ 0?\) In this paper, we give an answer to this question which is best possible.
Reviewer: G.A.Cámera

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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