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Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions

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MM was partially supported by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-TE-2016-2233 and JDR by CONICET grant PIP GI No 112201-50100036CO (Argentina) and by UBACyT grant 20020160100155BA (Argentina)

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  • We deal with monotonicity with respect to $ p $ of the first positive eigenvalue of the $ p $-Laplace operator on $ \Omega $ subject to the homogeneous Neumann boundary condition. For any fixed integer $ D>1 $ we show that there exists $ M\in[2 e^{-1}, 2] $ such that for any open, bounded, convex domain $ \Omega\subset{{\mathbb R}}^D $ with smooth boundary for which the diameter of $ \Omega $ is less than or equal to $ M $, the first positive eigenvalue of the $ p $-Laplace operator on $ \Omega $ subject to the homogeneous Neumann boundary condition is an increasing function of $ p $ on $ (1, \infty) $. Moreover, for each real number $ s>M $ there exists a sequence of open, bounded, convex domains $ \{\Omega_n\}_n\subset{{\mathbb R}}^D $ with smooth boundaries for which the sequence of the diameters of $ \Omega_n $ converges to $ s $, as $ n\rightarrow\infty $, and for each $ n $ large enough the first positive eigenvalue of the $ p $-Laplace operator on $ \Omega_n $ subject to the homogeneous Neumann boundary condition is not a monotone function of $ p $ on $ (1, \infty) $.

    Mathematics Subject Classification: Primary: 35P30, 47J10, 49R05; Secondary: 49J40, 58C40.


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