Optimal global regularity for minimal graphs over convex domains in hyperbolic space

2021 ◽  
Author(s):  
You Li ◽  
Yannan Liu
Keyword(s):  
Hyperbolic Space ◽  
Global Regularity ◽  
Convex Domains ◽  
Minimal Graphs

scholarly journals Gradient estimates for the constant mean curvature equation in hyperbolic space

2019 ◽  
Vol 150 (6) ◽  
pp. 3216-3230
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gradient Estimates ◽  
Maximum Principles ◽  
Convex Domains ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

AbstractWe establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of Φ-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature H satisfies H < 1 under suitable boundary conditions.

scholarly journals Refined asymptotics for minimal graphs in the hyperbolic space

2017 ◽  
Vol 62 (2) ◽  
pp. 381-390 ◽  
Author(s):  
Weiming Shen ◽  
Yue Wang
Keyword(s):  
Hyperbolic Space ◽  
Minimal Graphs

scholarly journals Existence and uniqueness of minimal graphs in hyperbolic space

2000 ◽  
Vol 4 (3) ◽  
pp. 669-694 ◽  
Author(s):  
Ricardo Sa Earp ◽  
Eric Toubiana
Keyword(s):  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Minimal Graphs

scholarly journals On the regularity of the total variation minimizers

2019 ◽  
Vol 23 (01) ◽  
pp. 1950082
Author(s):  
Alessio Porretta
Keyword(s):  
Image Processing ◽  
Total Variation ◽  
Source Term ◽  
Global Regularity ◽  
Regular Domain ◽  
Unique Minimizer ◽  
Convex Domains ◽  
Lipschitz Norm ◽  
Dimension Formula ◽  
Regularity Results

We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by Rudin, Osher and Fatemi. In particular, we show that if the source term [Formula: see text] is locally (respectively, globally) Lipschitz, then the solution has the same regularity with local (respectively, global) Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by Caselles, Chambolle and Novaga for dimension [Formula: see text] and (in case of the global regularity) for convex domains.

scholarly journals Global regularity estimates for Neumann problems of elliptic operators with coefficients having a BMO anti-symmetric part in NTA domains

2012 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sibei Yang ◽  
Dachun Yang ◽  
Wenxian Ma
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Global Regularity ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Symmetric Part ◽  
Convex Domains ◽  
Regularity Estimates ◽  
Neumann Boundary Value Problems ◽  
Second Order Elliptic Equations

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ n\ge2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^n $\end{document}</tex-math></inline-formula> be a bounded NTA domain. In this article, the authors study (weighted) global regularity estimates for Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. Precisely, for any given <inline-formula><tex-math id="M4">\begin{document}$ p\in(2,\infty) $\end{document}</tex-math></inline-formula>, via a weak reverse Hölder inequality with the exponent <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula>, the authors give a sufficient condition for the global <inline-formula><tex-math id="M6">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimate and the global weighted <inline-formula><tex-math id="M7">\begin{document}$ W^{1,q} $\end{document}</tex-math></inline-formula> estimate, with <inline-formula><tex-math id="M8">\begin{document}$ q\in[2,p] $\end{document}</tex-math></inline-formula> and some Muckenhoupt weights, of solutions to Neumann boundary value problems in <inline-formula><tex-math id="M9">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. As applications, the authors further obtain global regularity estimates for solutions to Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both a small <inline-formula><tex-math id="M10">\begin{document}$ \mathrm{BMO} $\end{document}</tex-math></inline-formula> symmetric part and a small <inline-formula><tex-math id="M11">\begin{document}$ \mathrm{BMO} $\end{document}</tex-math></inline-formula> anti-symmetric part, respectively, in bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, <inline-formula><tex-math id="M12">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> domains, or (semi-)convex domains, in weighted Lebesgue spaces. The results given in this article improve the known results by weakening the assumption on the coefficient matrix.</p>

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