NDLI: The geometric Neumann problem for the Liouville equation

Content Provider	SpringerLink
Author	Mira, Pablo Jiménez, Asun Gálvez, José A.
Copyright Year	2011
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 $$\left\{\begin{array}{lll}{\Delta v+2 Ke^v=0\quad {\rm in}\,\Omega\subseteq \mathbb{R}^2_+=\{(s, t)\in \mathbb{R}^2: t >0 \},}\\ {\frac{\partial v}{\partial t}=c_1e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s >0 \},}\\ {\frac{\partial v}{\partial t}=c_2e^{v/2}\quad\quad\quad{\rm on}\,\partial\Omega\cap\{s <0 \},}\end{array}\right.$$ 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.
Ending Page	599
Page Count	23
Starting Page	577
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3-4
Volume Number	44
Language	English
Publisher	Springer-Verlag
Publisher Date	2011-08-25
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Boundary value problems for second-order elliptic equations Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Theoretical, Mathematical and Computational Physics Analysis Second-order elliptic equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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