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Article Contents

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

• * Corresponding author

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)

• 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.

 Citation:

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