Newtonian capacity and quasi-balayage (Q2719805)

One of the topics of this paper concerns whether harmonic functions on an unbounded domain \(\Omega\) in \(\mathbb{R}^n\) can be approximated arbitrarily closely, in the \(L^1\) norm, by potentials of compactly supported signed measures that are \(O(|x|^{-m})\) as \(|x|\to\infty\) for some (large) \(m\). This is shown to be possible provided \(\text{cap} (\overline{B(0,R})\setminus \Omega)/ \operatorname {cap} \overline{B(0,R)}\) has a positive upper limit as \(R\to\infty\), where \(\text{cap}(\cdot)\) denotes Newtonian (if \(n\geq 3)\) or logarithmic (if \(n=2\)) capacity. (There is also a minor topological restriction on \(\Omega\).) Another part of the paper applies the notion of ``quasi-balayage'' to estimate a \(C^\infty\) function \(\varphi\), of compact support in \(\mathbb{R}^n\), in terms of a bound on its Laplacian: here it is required that the set of critical points of \(\varphi\) is sufficiently large (again in terms of capacity). Such results are relevant to the study of the regularity of free boundaries in obstacle problems and Hele-Shaw flows. The paper complements, and in some respects improves on, recent work of the third author [Multivariate Approximation, W. Haussmann (ed.) et al., Akademie Verlag, Berlin, Math. Res. 101, 203-230 (1997; Zbl 0899.31002)].