The asymptotic Dirichlet problems on manifolds with unbounded negative curvature
2018 ◽
Vol 167
(01)
◽
pp. 133-157
Keyword(s):
AbstractElton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable under the curvature condition −Ce(2−η)r(x) ≤ KM(x) ≤ −1 with η > 0. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack inequalities under the curvature condition −Ce(2/3−η)r(x) ≤ KM(x) ≤ −1 with η > 0. This implies that there is a natural homeomorphism between the Martin boundary and the geometric boundary of M. As far as we know, this is the first result of this kind under unbounded curvature conditions. Our proof is a modification of an argument due to M. T. Anderson and R. Schoen.
Continuous solutions and approximating scheme for fractional Dirichlet problems on Lipschitz domains
2018 ◽
Vol 149
(2)
◽
pp. 533-560
2021 ◽
Vol 66
(1)
◽
pp. 95-103
2019 ◽
Vol 53
(3)
◽
pp. 987-1003
◽
2014 ◽
Vol 66
(2)
◽
pp. 429-452
◽
1996 ◽
Vol 22
(3)
◽
pp. 151-173
◽
2013 ◽
Vol 2013
(679)
◽
pp. 223-247
◽
1988 ◽
Vol 31
(3)
◽
pp. 345-351
◽
2020 ◽
pp. 2050006
◽