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Sharp estimates for the Green function in Lipschitz domains. (English) Zbl 0971.31005

The main result of this paper is a precise estimate for the Green function of a bounded Lipschitz domain in \(\mathbb{R}^{d}\) for \(d\geq 3\). The author provides a careful analysis of the dependence of the constants appearing in the estimates. Applications are given to provide short proofs of precise estimates for the Martin kernel of Lipschitz domains, for the Green function of \(C^{1,1}\) domains, and a short proof of the 3G theorem. Moreover, the author shows how these results are connected to the boundary Harnack principle.

MSC:

31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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