Concentration solutions to the singularly prescribed Gaussian and geodesic curvatures problem [PDF]
Journal of Differential Equations, 2021 We consider the following Liouville-type equation with exponential Neumann boundary condition: $$ - \tilde u = \varepsilon^2 K(x) e^{2\tilde u}, \quad x\in D, \qquad \frac{\partial \tilde u}{\partial n} + 1 = \varepsilon (x) e^{\tilde u}, \quad x\in\partial D, $$ where $D\subset \mathbb R^2$ is the unit disc, $\varepsilon^2 K(x)$ and $\varepsilon (
Wang, Liping, Zhao, Chunyi
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A problem of prescribing Gaussian curvature on $S^2$ [PDF]
Proceedings of the American Mathematical Society, 2001 Summary: A class of functions \(K(x)=K(x_1,x_2,x_3)\) and the corresponding solutions of \[ \Delta u + K(x)e^{2u}=1, \quad x\in S^2, \] are obtained as a special case of the solutions of \[ \Delta^mu+K(x)e^{au}=f(x),\qquad x=(x_1,x_2,\dots,x_n)\in \mathbb{R}^n, \] where \(\Delta^m\) is defined as \(\Delta(\Delta^{m-1})\).
Goyal, Sulbha, Goyal, Vinod
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A Note on the Problem of Prescribing Gaussian Curvature on Surfaces [PDF]
Transactions of the American Mathematical Society, 1995 The paper treats the problem of prescribing the Gaussian curvature on a surface of genus at least two. Let \((M,g)\) be a closed connected 2- dimensional Riemannian manifold with Gaussian curvature \(k\). If \(g' = e^{2n} \cdot g\) is a conformally equivalent metric, then its curvature is given by \(k' = e^{-2n}\) \((k - \Delta n)\).
Ding, Wei-Yue, Liu, Jia-Quan
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A flow approach to the prescribed Gaussian curvature problem in $\mathbb{H}^{n+1}$
, 2022 In this paper, we study the following prescribed Gaussian curvature problem $$K=\frac{\tilde{f}(θ)}{ϕ(ρ)^{α-2}\sqrt{ϕ(ρ)^2+|\bar{\nabla}ρ|^2}},$$ a generalization of the Alexandrov problem ($α=n+1$) in hyperbolic space, where $\tilde{f}$ is a smooth positive function on $\mathbb{S}^{n}$, $ρ$ is the radial function of the hypersurface, $ϕ(ρ)=\sinhρ$ and
Li, Haizhong, Zhang, Ruijia
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Existence and non existence results for the singular Nirenberg problem [PDF]
, 2016 In this paper we study the problem, posed by Troyanov (Trans AMS 324: 793–821, 1991), of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities.
DE MARCHIS, Francesca +1 more
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Minimal resonances in annular non-Euclidean strips [PDF]
, 2010 Differential growth processes play a prominent role in shaping leaves and biological tissues. Using both analytical and numerical calculations, we consider the shapes of closed, elastic strips which have been subjected to an inhomogeneous pattern of ...
Bryan Gin-ge Chen +6 more
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Isolated singularities of the prescribed mean curvature equation in Minkowski $3$-space [PDF]
, 2017 We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3 ...
Gálvez, José A. +2 more
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A gradient flow for the prescribed Gaussian curvature problem on a closed Riemann surface with conical singularity [PDF]
Differential Geometry and its Applications, 2019 15 ...
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Prescribing Gaussian and Geodesic Curvature on Disks
Advanced Nonlinear Studies, 2018 In this paper, we consider the problem of prescribing the Gaussian and geodesic curvature on a disk and its boundary, respectively, via a conformal change of the metric.
Cruz-Blázquez Sergio, Ruiz David
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Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the Torus [PDF]
, 2014 In the spirit of the paper "Large conformal metrics of prescribed Gauss curvature on surfaces of higher genus" by Borer-Galimberti-Struwe, where we dealt with the case of a closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the ...
Galimberti, Luca
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