A shortwave source near a smooth, convex hypersurface and the spectral function of the Laplace operator on a Riemannian manifold

1983 ◽  
Vol 22 (1) ◽  
pp. 1082-1086
Author(s):  
Ya. V. Kurylev
Keyword(s):  
Riemannian Manifold ◽  
Spectral Function ◽  
Laplace Operator ◽  
Convex Hypersurface ◽  
Smooth Convex ◽  
The Laplace Operator

scholarly journals Some Inequalities for the Omori-Yau Maximum Principle

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Kyusik Hong
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Laplace Operator ◽  
Order Term ◽  
Complete Riemannian Manifold ◽  
Sufficient Condition ◽  
Zeroth Order ◽  
Exhaustion Function ◽  
Semielliptic Operator ◽  
The Laplace Operator

We generalize A. Borbély’s condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semielliptic operatorLwith bounded coefficients and no zeroth order term. Also, we consider a new sufficient condition for the existence of a tamed exhaustion function. From these results, we may remark that the existence of a tamed exhaustion function is more general than the hypotheses in the version of the Omori-Yau maximum principle that was given by A. Ratto, M. Rigoli, and A. G. Setti.

Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

1949 ◽  
Vol 1 (3) ◽  
pp. 242-256 ◽  
Author(s):  
S. Minakshisundaram ◽  
Å. Pleijel
Keyword(s):  
Differential Equation ◽  
Riemannian Manifold ◽  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Line Element ◽  
Metric Tensor ◽  
Local Coordinates ◽  
Image Position ◽  
The Laplace Operator ◽  
Beltrami Laplace Operator

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (xi), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … xN) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

scholarly journals A Note on Killing Calculus on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 307
Author(s):  
Sharief Deshmukh ◽  
Amira Ishan ◽  
Suha B. Al-Shaikh ◽  
Cihan Özgür
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Laplace Operator ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Adjoint Operator ◽  
Dimensional Sphere ◽  
Killing Vector Field ◽  
Smooth Functions ◽  
The Laplace Operator

In this article, it has been observed that a unit Killing vector field ξ on an n-dimensional Riemannian manifold (M,g), influences its algebra of smooth functions C∞(M). For instance, if h is an eigenfunction of the Laplace operator Δ with eigenvalue λ, then ξ(h) is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian Hh(ξ,ξ) of a smooth function h∈C∞(M) defines a self adjoint operator ⊡ξ and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold (M,g). We study several properties of functions associated to the unit Killing vector field ξ. Finally, we find characterizations of the odd dimensional sphere using properties of the operator ⊡ξ and the nontrivial solution of Fischer–Marsden differential equation, respectively.

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