Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MONOTONICITY OF THE FIRST EIGENVALUE OF THE LAPLACE AND THE p-LAPLACE OPERATORS UNDER A FORCED MEAN CURVATURE FLOW

  • Mao, Jing (Faculty of Mathematics and Statistics Key Laboratory of Applied Mathematics of Hubei Province Hubei University)
  • Received : 2017.11.12
  • Accepted : 2018.02.21
  • Published : 2018.11.01
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Abstract

In this paper, we would like to give an answer to Problem 1 below issued firstly in [17]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the p-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicate constraint, we can give lower bounds for the first nonzero closed eigenvalue of the Laplacian provided additionally the second fundamental form of the initial hypersurface satisfies a pinching condition.

Keywords

Acknowledgement

Supported by : NSF of China, Fok Ying-Tong Education Foundation

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