scholarly journals Sharp gradient estimates on weighted manifolds with compact boundary

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ha Tuan Dung ◽  
Nguyen Thac Dung ◽  
Jiayong Wu
Keyword(s):  
Dirichlet Boundary ◽  
Gradient Estimates ◽  
Metric Measure Spaces ◽  
Liouville Theorems ◽  
Ancient Solutions ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Sharp Growth ◽  
Weighted Manifolds ◽  
Compact Boundary

<p style='text-indent:20px;'>In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.</p>

scholarly journals Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abimbola Abolarinwa ◽  
Akram Ali ◽  
Ali Alkhadi
Keyword(s):  
Lower Bound ◽  
Vector Fields ◽  
First Eigenvalue ◽  
Metric Measure Spaces ◽  
Cheeger Constant ◽  
Weighted Mean Curvature ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Geometric Constant ◽  
Weighted Manifolds

AbstractWe establish new eigenvalue inequalities in terms of the weighted Cheeger constant for drifting p-Laplacian on smooth metric measure spaces with or without boundary. The weighted Cheeger constant is bounded from below by a geometric constant involving the divergence of suitable vector fields. On the other hand, we establish a weighted form of Escobar–Lichnerowicz–Reilly lower bound estimates on the first nonzero eigenvalue of the drifting bi-Laplacian on weighted manifolds. As an application, we prove buckling eigenvalue lower bound estimates, first, on the weighted geodesic balls and then on submanifolds having bounded weighted mean curvature.

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