On the asymptotc behaviour of the spectral function and discrete spectrum of the Laplace operator on a fundamental domain of Lobachevsky's plane

1982 ◽  
Vol 52 (1) ◽  
pp. 195-219 ◽  
Author(s):  
B. M. Levitan
Keyword(s):  
Spectral Function ◽  
Discrete Spectrum ◽  
Laplace Operator ◽  
Fundamental Domain ◽  
The Laplace Operator

scholarly journals An upper bound for λ1 for Γ(q) and Γ0(q)

1990 ◽  
Vol 33 (2) ◽  
pp. 241-250
Author(s):  
C. J. Mozzochi
Keyword(s):  
Discrete Spectrum ◽  
Laplace Operator ◽  
Trace Formula ◽  
Positive Constant ◽  
Upper Bound ◽  
Selberg Trace Formula ◽  
Absolute Positive Constant ◽  
Image Position ◽  
The Laplace Operator

Under the assumption of the Selberg conjecture I establish by means of the Selberg trace formula the following:Theorem. Let Γ denote Γ(q) or Γ0(q), q square-free. Let Δq denote the Laplace operator on L2(Γ\H), and let Σq denote its discrete spectrum. Then there exists an absolute positive constant A such that for q≧A

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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