Abstract
Let Ω denote the upper half-plane \({\mathbb{R}_+^2}\) or the upper half-disk \({D_{\varepsilon}^+\subset \mathbb{R}_+^2}\) of center 0 and radius \({\varepsilon}\). In this paper we classify the solutions \({v\in\;C^2(\overline{\Omega}\setminus\{0\})}\) to the Neumann problem
where \({K, c_1, c_2 \in \mathbb{R}}\), with the finite energy condition \({\int_{\Omega} e^v < \infty}\)As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.
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Gálvez, J.A., Jiménez, A. & Mira, P. The geometric Neumann problem for the Liouville equation. Calc. Var. 44, 577–599 (2012). https://doi.org/10.1007/s00526-011-0445-4
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DOI: https://doi.org/10.1007/s00526-011-0445-4