Universal Bounds for the Low Eigenvalues of Neumann Laplacians in N Dimensions
Mark S. Ashbaugh, Rafael D. Benguria · SIAM Journal on Mathematical Analysis · 1993
The authors consider bounds on the Neumann eigenvalues of the Laplacian on domains in $I\mathbb{R}^n $ in the light of their recent results on Dirichlet eigenvalues, in particular, their proof of the Payne-Pólya–Weinberger conjecture via spherical rearrangement. They prove the bound ${1 / {\mu _1 }} + {1 / {\mu _2 }} \geq {A / {2\pi }}$ for the first two nonzero Neumann eigenvalues for an arbitrary bounded domain $\Omega $ in two dimensions and also the stronger (and optimal) bound $\mu _2 \leq \pi (j'_{1,1} )^{{2 / A}} $ for domains having a 4-fold rotational symmetry. (Here $(j'_{1,1} ) \approx 1.84118$ denotes the first positive zero of the derivative of the Bessel function $J_1 (x)$ and A is the area of the domain $\Omega $.) The authors also obtain analogues of these results for domains in $I\mathbb{R}^n $. Previous results in this vein are due to Szegö, who proved ${{\mu _1 \leq \pi (j'_{1,1} )^2 } / A}$ and ${1 / {\mu _1 }} + {1 / {\mu _2 }} \geqslant {{2A} / {\pi (j'_{1,1} )^2 }}$ for simply connected domains in $I\mathbb{R}^2 $, and to Weinberger, who proved the general result$\mu _1 \leq ({{C_n } / {|\Omega |}})^{{2 / n}} p_{{n / {2,1}}}^2 $ for arbitrary domains in $I\mathbb{R}^n $ (here ${{C_n = \pi ^{{n / 2}} } {{C_n = \pi ^{{n / 2}} } {\Gamma ({n / {2 + 1}})}}} $$=$ volume of the unit ball in $I\mathbb{R}^n $, and $p_{ u ,k} $ denotes the kth positive zero of the derivative of $x^{1 - u } J_ u (x)$, where $J_ u (x)$ represents the standard Bessel function of the first kind of order v).