<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0038_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>λ</mi><mn>1</mn><mi>D</mi></msubsup><mrow><mo stretchy="false">(</mo><mrow><mi>B</mi><mrow><mo stretchy="false">(</mo><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow></math><tex-math>$\lambda _1^D \left({B\left({p,r} \right)} \right)$</tex-math></alternatives></inline-formula> 
of Laplacian operator for the manifold with Ricci curvature <italic>Rc</italic> ≥ <italic>−K</italic>, by using Li-Yau’s gradient estimate for the heat equation.</p></abstract>

In this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue 
λ1D(B(p,r))$\lambda _1^D \left({B\left({p,r} \right)} \right)$ 
of Laplacian operator for the manifold with Ricci curvature Rc ≥ −K, by using Li-Yau’s gradient estimate for the heat equation.

A new proof of the bound for the first Dirichlet eigenvalue of the Laplacian operator

School of Mathematical Sciences, Ocean University of China, Lane 238, Songling Road, Laoshan District, Qingdao City, Shandong Province, 266100, People’s Republic of China.

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue 
λ1D(B(p,r))$\lambda _1^D \left({B\left({p,r}...

A new proof of the bound for the first Dirichlet eigenvalue of the Laplacian operator

Published Online: Oct 20, 2015

Page range: 129 - 140

Received: Aug 01, 2012

Accepted: Nov 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0038

Keywords
Laplacian operator, Li-Yau gradient estimate, the first Dirichlet eigenvalue

© 2014 Chang-Jun Li et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

A new proof of the bound for the first Dirichlet eigenvalue of the Laplacian operator

Published Online: Oct 20, 2015

Page range: 129 - 140

Received: Aug 01, 2012

Accepted: Nov 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0038

KeywordsLaplacian operator, Li-Yau gradient estimate, the first Dirichlet eigenvalue

© 2014 Chang-Jun Li et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Laplacian operator, Li-Yau gradient estimate, the first Dirichlet eigenvalue