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In this paper, we would like to give an answer to \textbf{Problem 1} belowissued firstly in [J. Mao, Eigenvalue estimation and some results on finitetopological type, Ph.D. thesis, IST-UTL, 2013]. In fact, by imposing someconditions on the mean curvature of the initial hypersurface and thecoefficient function of the forcing term of a forced mean curvature flowconsidered here, we can obtain that the first eigenvalues of the Laplace andthe $p$-Laplace operators are monotonic under this flow. Surprisingly, duringthis process, we get an interesting byproduct, that is, without any complicatedconstraint, we can give lower bounds for the first nonzero closed eigenvalue ofthe Laplacian provided additionally the second fundamental form of the initialhypersurface satisfies a pinching condition.

Monotonicity of the first eigenvalue of the Laplace and the p-Laplace operators under a forced mean curvature flow