Difference between revisions of "Neumann eigenvalue"
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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|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 <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 $\mu _ { 1 } = 0 < \ldots < \mu _ { N }$, 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 | 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 | + | 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]]]: | 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). | A proof of Pólya's conjecture for both Dirichlet and Neumann eigenvalues would imply Friedlander's result (a5). | ||
====References==== | ====References==== | ||
− | <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 |
Neumann eigenvalue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_eigenvalue&oldid=49869