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Abstract
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We study the convexity of the
first eigenfunction of the drifting Laplacian operator with zero Dirichlet boundary
value provided a suitable assumption to the drifting term is added. We firstly
generalize some results of N. Korevaar and S.-T. Yau to gain a Hessian estimate of
the first eigenfunction. As an application, we use this Hessian estimate to get a lower
bound of the difference of the first and second eigenvalues of the drifting Laplacian.
At the end we also find a lower bound when the Hessian estimate does not
hold.
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Keywords
eigenvalue, drifting Laplacian, Hessian
estimate, fundamental gap
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Mathematical Subject Classification
Primary: 35P15
Secondary: 35P15
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Milestones
Received: 2 June 2008
Revised: 4 June 2008
Accepted: 25 November 2008
Published: 4 March 2009
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