Transmission of harmonic functions through quasicircles on compact Riemann surfaces

Eric Schippers; Wolfgang Staubach

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arXiv · 1810.02147

Transmission of harmonic functions through quasicircles on compact Riemann surfaces

Eric Schippers·Wolfgang Staubach

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Abstract

Let $R$ be a compact surface and let $\Gamma$ be a Jordan curve which separates $R$ into two connected components $\Sigma_1$ and $\Sigma_2$. A harmonic function $h_1$ on $\Sigma_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $\Gamma$. We show that if $\Gamma$ is a quasicircle, then there is a unique harmonic function $h_2$ of bounded Dirichlet norm on $\Sigma_2$ whose boundary values agree with those of $h_1$. Furthermore, the resulting map from the Dirichlet space of $\Sigma_1$ into $\Sigma_2$ is bounded with respect to the Dirichlet semi-norm.

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BibTeX RIS

Eric Schippers, Wolfgang Staubach. 2018-10-04. Transmission of harmonic functions through quasicircles on compact Riemann surfaces. https://arxiv.org/abs/1810.02147

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