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Consider a bounded domain $\Omega \subset \mathbf{R} ^ { n }$ with a piecewise smooth boundary $\partial \Omega$. A number $\mu$ is a Neumann eigenvalue of $\Omega$ if there exists a function $u \in C ^ { 2 } ( \Omega ) \cap C ^ { 0 } ( \overline { \Omega } )$ (a Neumann eigenfunction) satisfying the following Neumann boundary value problem (cf. also [[Neumann boundary conditions|Neumann boundary conditions]]):

\begin{equation} \tag{a1} - \Delta u = \mu u \text { in } \Omega, \end{equation}

\begin{equation} \tag{a2} \frac { \partial u } { \partial n } = 0 \text { in } \partial \Omega, \end{equation}

where $\Delta$ is the [[Laplace operator]] (i.e., $\Delta = \sum _ { i = 1 } ^ { n } \partial ^ { 2 } / \partial x _ { i } ^ { 2 }$). For more general definitions, see [[#References|[a8]]]. Neumann eigenvalues (with $n = 2$) appear naturally when considering the vibrations of a free membrane (cf. also [[Natural frequencies|Natural frequencies]]). In fact, for $n = 2$ the non-zero Neumann eigenvalues are proportional to the square of the eigenfrequencies of the membrane with free boundary. Provided $\Omega$ is bounded and the boundary $\partial \Omega$ is sufficiently regular, the Neumann Laplacian has a discrete spectrum of infinitely many non-negative eigenvalues with no finite accumulation point:

\begin{equation} \tag{a3} 0 = \mu _ { 1 } ( \Omega ) \leq \mu _ { 2 } ( \Omega ) \leq \dots \end{equation}

($\mu _ { k } \rightarrow \infty$ as $ k  \rightarrow \infty$). The Neumann eigenvalues are characterized by the max-min principle [[#References|[a3]]]:

\begin{equation} \tag{a4} \mu _ { k } = \operatorname { sup } \operatorname { inf } \frac { \int _ { \Omega } ( \nabla u ) ^ { 2 } d x } { \int _ { \Omega } u ^ { 2 } d x }, \end{equation}

where the $\operatorname { inf }$ is taken over all $u \in H ^ { 1 } ( \Omega )$ orthogonal to $\varphi _ { 1 } , \dots , \varphi _ { k - 1 } \in H ^ { 1 } ( \Omega )$, and the $\operatorname {sup}$ is taken over all the choices of $\{ \varphi _ { i } \} _ { i = 1 } ^ { k - 1 }$. For simply-connected domains the first eigenfunction $u_1$, corresponding to the eigenvalue $\mu _ { 1 } = 0$ is constant throughout the domain. All the other eigenvalues are positive. While Dirichlet eigenvalues satisfy stringent constraints (e.g., $\lambda _ { 2 } / \lambda _ { 1 }$ cannot exceed $2.539\dots$ for any bounded domain in $\mathbf{R} ^ { 2 }$, [[#References|[a1]]]; see also [[Dirichlet eigenvalue|Dirichlet eigenvalue]]), no such constraints exist for Neumann eigenvalues, other than the fact that they are non-negative. In fact, given any finite sequence $\mu _ { 1 } = 0 < \ldots < \mu _ { N }$, there is an open, bounded, smooth, simply-connected domain of $\mathbf{R} ^ { 2 }$ having this sequence as the first $N$ Neumann eigenvalues of the Laplacian on that domain [[#References|[a2]]]. Though it is obvious from the variational characterization of both Dirichlet and Neumann eigenvalues (see (a4)) that $\mu _ { k } \leq \lambda _ { k }$, L. Friedlander [[#References|[a4]]] proved the stronger result

\begin{equation} \tag{a5} \mu _ { k + 1 } \leq \lambda _ { k } ,\, k = 1, 2,\dots . \end{equation}

How far the first non-trivial Neumann eigenvalue is from zero for a convex domain in $\mathbf{R} ^ { 2 }$ is given through the optimal inequality [[#References|[a7]]]

\begin{equation} \tag{a6} \mu _ { 1 } \geq \frac { \pi ^ { 2 } } { d ^ { 2 } }, \end{equation}

where $d$ is the diameter of the domain. There are many more isoperimetric inequalities for Neumann eigenvalues (see [[Rayleigh–Faber–Krahn inequality|Rayleigh–Faber–Krahn inequality]]).

For large values of $k$, H. Weyl proved [[#References|[a9]]]

\begin{equation} \tag{a7} \mu _ { k + 1 } \approx \frac { 4 \pi ^ { 2 } k ^ { 2 / n } } { ( C _ { n } | \Omega | ) ^ { 2 / n } }, \end{equation}

where $| \Omega |$ and $C _ { n } = \pi ^ { n / 2 } / \Gamma ( n / 2 + 1 )$ are, respectively, the volumes of $\Omega$ and of the unit ball in ${\bf R} ^ { n }$.

For any plane-covering domain (i.e., a domain that can be used to tile the plane without gaps, nor overlaps, allowing rotations, translations and reflections of itself), G. Pólya [[#References|[a6]]] proved that

\begin{equation} \tag{a8} \mu _ { k + 1 } \leq \frac { 4 \pi k } { A } , k = 0,1 , \ldots , \end{equation}

and conjectured the same bound for any bounded domain in $\mathbf{R} ^ { 2 }$. This is equivalent to saying that the Weyl asymptotics of $\mu _ { k }$ is an upper bound for $\mu _ { k }$. The analogous conjecture in dimension $n$ is

\begin{equation} \tag{a9} \mu _ { k + 1 } \leq \frac { 4 \pi ^ { 2 } k ^ { 2 / n } } { ( C _ { n } | \Omega | ) ^ { 2 / n } } ,\, k = 0,1\dots. \end{equation}

The most significant result towards the proof of Pólya's conjecture for Neumann eigenvalues is the result by P. Kröger [[#References|[a5]]]:

\begin{equation*} \sum _ { i = 1 } ^ { k } \mu _ { i } \leq \frac { n } { n + 2 } \frac { 4 \pi ^ { 2 } k ^ { 2 / n } } { ( C _ { n } | \Omega | ) ^ { 2 / n } } , k = 1,2, \dots . \end{equation*}

A proof of Pólya's conjecture for both Dirichlet and Neumann eigenvalues would imply Friedlander's result (a5).

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  M.S. Ashbaugh,   R.D. Benguria,   "A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions"  ''Ann. of Math.'' , '''135'''  (1992)  pp. 601–628</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  Y. Colin de Vérdiere,   "Construction de laplaciens dont une partie finie du spectre est donnée"  ''Ann. Sci. École Norm. Sup.'' , '''20''' :  4  (1987)  pp. 599–615</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  R. Courant,   D. Hilbert,   "Methoden der mathematischen Physik" , '''I''' , Springer  (1931)  (English transl.: Methods of Mathematical Physics, vol. I., Interscience, 1953)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  L. Friedlander,   "Some inequalities between Dirichlet and Neumann eigenvalues"  ''Arch. Rational Mech. Anal.'' , '''116'''  (1991)  pp. 153–160</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P. Kröger,   "Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean Space"  ''J. Funct. Anal.'' , '''106'''  (1992)  pp. 353–357</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  G. Polya,   "On the eigenvalues of vibrating membranes"  ''Proc. London Math. Soc.'' , '''11''' :  3  (1961)  pp. 419–433</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  L.E. Payne,   H.F. Weinberger,   "An optimal Poincaré inequality for convex domains"  ''Arch. Rational Mech. Anal.'' , '''5'''  (1960)  pp. 286–292</td></tr>
<tr><td valign="top">[a8]</td> <td valign="top">  M. Reed,   B. Simon, "Methods of modern mathematical physics IV: Analysis of operators" , Acad. Press  (1978)</td></tr>
<tr><td valign="top">[a9]</td> <td valign="top">  H. Weyl, "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen"  ''Math. Ann.'' , '''71'''  (1911)  pp. 441–479</td></tr>
</table>