Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

Regularity at infinity of Hadamard manifolds with respect to some elliptic operators and applications to asymptotic Dirichlet problems


Authors: Jaime Ripoll and Miriam Telichevesky
Journal: Trans. Amer. Math. Soc. 367 (2015), 1523-1541
MSC (2010): Primary 58J05; Secondary 35J92, 35J93
DOI: https://doi.org/10.1090/S0002-9947-2014-06001-7
Published electronically: November 12, 2014
MathSciNet review: 3286491
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be Hadamard manifold with sectional curvature $ K_{M}\leq -k^{2}$, $ k>0$. Denote by $ \partial _{\infty }M$ the asymptotic boundary of $ M$. We say that $ M$ satisfies the strict convexity condition (SC condition) if, given $ x\in \partial _{\infty }M$ and a relatively open subset $ W\subset \partial _{\infty }M$ containing $ x$, there exists a $ C^{2}$ open subset $ \Omega \subset M$ such that $ x\in \operatorname *{Int}\left ( \partial _{\infty }\Omega \right ) \subset W$ and $ M\setminus \Omega $ is convex. We prove that the SC condition implies that $ M$ is regular at infinity relative to the operator

$\displaystyle \mathcal {Q}\left [ u\right ] :=\mathrm {{\,div\,}}\left ( \frac {a(\vert\nabla u\vert)}{\vert\nabla u\vert}\nabla u\right ) , $

subject to some conditions. It follows that under the SC condition, the Dirichlet problem for the minimal hypersurface and the $ p$-Laplacian ($ p>1$) equations are solvable for any prescribed continuous asymptotic boundary data. It is also proved that if $ M$ is rotationally symmetric or if $ \inf _{B_{R+1}}K_{M}\geq -e^{2kR}/R^{2+2\epsilon },\,\,R\geq R^{\ast },$ for some $ R^{\ast }$ and $ \epsilon >0,$ where $ B_{R+1}$ is the geodesic ball with radius $ R+1$ centered at a fixed point of $ M,$ then $ M$ satisfies the SC condition.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 58J05, 35J92, 35J93

Retrieve articles in all journals with MSC (2010): 58J05, 35J92, 35J93


Additional Information

Jaime Ripoll
Affiliation: Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 9500, CEP 91540-000, Porto Alegre, Rio Grande do Sul, Brasil
Email: jaime.ripoll@ufrgs.br

Miriam Telichevesky
Affiliation: Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 9500, CEP 91540-000, Porto Alegre, Rio Grande do Sul, Brasil
Email: miriam.telichevesky@ufrgs.br

DOI: https://doi.org/10.1090/S0002-9947-2014-06001-7
Received by editor(s): August 24, 2012
Published electronically: November 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society