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Consider a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300601.png" /> with a piecewise smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300602.png" />. A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300603.png" /> is a Neumann eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300604.png" /> if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300605.png" /> (a Neumann eigenfunction) satisfying the following Neumann boundary value problem (cf. also [[Neumann boundary conditions|Neumann boundary conditions]]):
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300607.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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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]]):
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300608.png" /> is the [[Laplace operator|Laplace operator]] (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n1300609.png" />). For more general definitions, see [[#References|[a8]]]. Neumann eigenvalues (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006010.png" />) appear naturally when considering the vibrations of a free membrane (cf. also [[Natural frequencies|Natural frequencies]]). In fact, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006011.png" /> the non-zero Neumann eigenvalues are proportional to the square of the eigenfrequencies of the membrane with free boundary. Provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006012.png" /> is bounded and the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006013.png" /> is sufficiently regular, the Neumann Laplacian has a discrete spectrum of infinitely many non-negative eigenvalues with no finite accumulation point:
+
\begin{equation} \tag{a1} - \Delta u = \mu u \text { in } \Omega, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a2} \frac { \partial u } { \partial n } = 0 \text { in } \partial \Omega, \end{equation}
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006015.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006016.png" />). The Neumann eigenvalues are characterized by the max-min principle [[#References|[a3]]]:
+
where $\Delta$ is the [[Laplace operator|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a3} 0 = \mu _ { 1 } ( \Omega ) \leq \mu _ { 2 } ( \Omega ) \leq \dots \end{equation}
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006018.png" /> is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006019.png" /> orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006020.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006021.png" /> is taken over all the choices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006022.png" />. For simply-connected domains the first eigenfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006023.png" />, corresponding to the eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006024.png" /> is constant throughout the domain. All the other eigenvalues are positive. While Dirichlet eigenvalues satisfy stringent constraints (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006025.png" /> cannot exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006026.png" /> for any bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006027.png" />, [[#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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006028.png" />, there is an open, bounded, smooth, simply-connected domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006029.png" /> having this sequence as the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006030.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006031.png" />, L. Friedlander [[#References|[a4]]] proved the stronger result
+
($\mu _ { k } \rightarrow \infty$ as $ k  \rightarrow \infty$). The Neumann eigenvalues are characterized by the max-min principle [[#References|[a3]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\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}
  
How far the first non-trivial Neumann eigenvalue is from zero for a convex domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006033.png" /> is given through the optimal inequality [[#References|[a7]]]
+
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006018.png"/> 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 $\mu _ { 1 } = 0 &lt; \ldots &lt; \mu _ { N }$, 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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
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\begin{equation} \tag{a5} \mu _ { k + 1 } \leq \lambda _ { k } ,\, k = 1, 2,\dots . \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006035.png" /> is the diameter of the domain. There are many more isoperimetric inequalities for Neumann eigenvalues (see [[Rayleigh–Faber–Krahn inequality|Rayleigh–Faber–Krahn inequality]]).
+
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]]]
  
For large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006036.png" />, H. Weyl proved [[#References|[a9]]]
+
\begin{equation} \tag{a6} \mu _ { 1 } \geq \frac { \pi ^ { 2 } } { d ^ { 2 } }, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
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]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006039.png" /> are, respectively, the volumes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006040.png" /> and of the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006041.png" />.
+
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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006043.png" />. This is equivalent to saying that the Weyl asymptotics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006044.png" /> is an upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006045.png" />. The analogous conjecture in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006046.png" /> is
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
+
\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]]]:
 
The most significant result towards the proof of Pólya's conjecture for Neumann eigenvalues is the result by P. Kröger [[#References|[a5]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130060/n13006048.png" /></td> </tr></table>
+
\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).
 
A proof of Pólya's conjecture for both Dirichlet and Neumann eigenvalues would imply Friedlander's result (a5).
  
 
====References====
 
====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>
+
<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>

Revision as of 15:19, 1 July 2020

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):

\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 [a8]. Neumann eigenvalues (with $n = 2$) appear naturally when considering the vibrations of a free membrane (cf. also 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 [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 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 $\mathbf{R} ^ { 2 }$, [a1]; see also 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 [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 [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 [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).

For large values of $k$, H. Weyl proved [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 [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 [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

[a1] 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
[a2] 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
[a3] R. Courant, D. Hilbert, "Methoden der mathematischen Physik" , I , Springer (1931) (English transl.: Methods of Mathematical Physics, vol. I., Interscience, 1953)
[a4] L. Friedlander, "Some inequalities between Dirichlet and Neumann eigenvalues" Arch. Rational Mech. Anal. , 116 (1991) pp. 153–160
[a5] P. Kröger, "Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean Space" J. Funct. Anal. , 106 (1992) pp. 353–357
[a6] G. Polya, "On the eigenvalues of vibrating membranes" Proc. London Math. Soc. , 11 : 3 (1961) pp. 419–433
[a7] L.E. Payne, H.F. Weinberger, "An optimal Poincaré inequality for convex domains" Arch. Rational Mech. Anal. , 5 (1960) pp. 286–292
[a8] M. Reed, B. Simon, "Methods of modern mathematical physics IV: Analysis of operators" , Acad. Press (1978)
[a9] H. Weyl, "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen" Math. Ann. , 71 (1911) pp. 441–479
How to Cite This Entry:
Neumann eigenvalue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_eigenvalue&oldid=12806
This article was adapted from an original article by Rafael D. Benguria (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article