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Estimation of Green’s function on piecewise Dini-smooth bounded Jordan domains. (English) Zbl 1283.81071
Summary: We establish inequalities for Green functions on general bounded piecewise Dini-smooth Jordan domains in \(\mathbb R^2\). This enables us to prove a new version of the 3G Theorem which generalizes its previous version given in [M. Selmi, Potential Anal. 13, No. 1, 81–102 (2000; Zbl 0959.31004)]. Using these results, we give a comparison theorem for the Green kernel of \(\Delta \) and the Green kernel of \(\Delta -\mu \), where \(\mu \) is a nonnegative and exact Radon measure.

MSC:
30C35 General theory of conformal mappings
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31A35 Connections of harmonic functions with differential equations in two dimensions
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