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A fine limit property of functions superharmonic outside a manifold. (English) Zbl 0759.31002
Author’s abstract: Let \((x,x')\) denote a typical point of \(\mathbb{R}^ n=\mathbb{R}^ k\times\mathbb{R}^{n-k}\) where \(n\geq 3\) and \(1\leq k\leq n-2\). Let \(E=\{| x'|<f(x)\}\) where \(f:[0,\infty[\to[0,\infty[\) is increasing. A necessary and sufficient condition is given for \(E\) to be thin at the origin. This, in turn, is used to study the behaviour of functions \(u\) which are superharmonic on the complement of a \(C^ 2\) \(k\)-dimensional manifold \(S\). In particular it is shown that if \(u^ -\) does not grow too quickly near \(S\), then \(| x-y|^{n-2}u(x)\) has a finite non-negative fine limit as \(x\to y\) for any \(y\in S\).
MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
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