View source for Rayleigh-Faber-Krahn inequality

An inequality concerning the lowest eigenvalue of the [[Laplace operator|Laplace operator]], with Dirichlet boundary condition, on a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r1300401.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r1300402.png" />).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r1300403.png" /> be the Dirichlet eigenvalues of the Laplacian in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r1300404.png" />, i.e.,

<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/r/r130/r130040/r1300405.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<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/r/r130/r130040/r1300406.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

(Cf. also [[Dirichlet boundary conditions|Dirichlet boundary conditions]]; [[Dirichlet eigenvalue|Dirichlet eigenvalue]].) Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r1300407.png" /> is the Laplace operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r1300408.png" /> is an open bounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r1300409.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004010.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004011.png" />, the Dirichlet eigenvalues are proportional to the square of the eigenfrequencies of an elastic, homogeneous, vibrating membrane with fixed boundary (cf. also [[Natural frequencies|Natural frequencies]]).

The Rayleigh–Faber–Krahn inequality for the membrane (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004012.png" />) states 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/r/r130/r130040/r13004013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004014.png" /> is the first zero of the Bessel function of order zero (cf. also [[Bessel functions|Bessel functions]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004015.png" /> is the area of the membrane. Equality is attained in (a3) if and only if the membrane is circular. In other words, among all membranes of given area, the circle has the lowest fundamental frequency. This inequality was conjectured by Lord Rayleigh [[#References|[a14]]], based on exact calculations for simple domains and a variational argument for near circular domains. In 1918, R. Courant [[#References|[a5]]] proved the weaker result that among all membranes of the same perimeter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004016.png" />, the circular one yields the least lowest eigenvalue, i.e.,

<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/r/r130/r130040/r13004017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

with equality if and only if the membrane is circular. Rayleigh's conjecture was proven independently by G. Faber [[#References|[a6]]] and E. Krahn [[#References|[a7]]]. The corresponding [[Isoperimetric inequality|isoperimetric inequality]] in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004018.png" />,

<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/r/r130/r130040/r13004019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

was proven by Krahn [[#References|[a8]]]. In (a5), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004020.png" /> is the first positive zero of the Bessel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004022.png" /> is the volume of the domain and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004023.png" /> is the volume of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004024.png" />-dimensional unit ball. Equality is attained in (a5) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004025.png" /> is a ball.

The proof of the Rayleigh–Faber–Krahn inequality rests upon two facts: a variational characterization for the lowest Dirichlet eigenvalue and the properties of symmetric decreasing rearrangements of functions. The variational characterization of the lowest eigenvalue is given by

<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/r/r130/r130040/r13004026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

Concerning decreasing rearrangements, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004027.png" /> be a measurable subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004028.png" />, then the symmetrized domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004029.png" /> is a ball with the same measure as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004031.png" /> is a real-valued measurable function defined on a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004032.png" />, its spherical decreasing rearrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004033.png" /> is a function defined on the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004034.png" /> centred at the origin and having the same measure as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004035.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004036.png" /> depends only on the distance from the origin, is decreasing away from the origin and is equi-measurable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004037.png" />. See [[#References|[a13]]], [[#References|[a18]]], [[#References|[a4]]] for properties of rearrangements of functions. Since the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004038.png" /> and its spherical decreasing rearrangement are equi-measurable, their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004039.png" />-norms are the same. What Faber and Krahn actually proved is that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004040.png" />-norm of the gradient of a function decreases under rearrangements (see [[#References|[a18]]] for details, and also [[#References|[a9]]] for a different approach to this fact). The fact that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004041.png" />-norm of the gradient of a function decreases under rearrangements, combined with the variational characterization (a6), immediately gives the Rayleigh–Faber–Krahn inequality.

==Isoperimetric inequalities for the lowest eigenvalue.==
There are several isoperimetric inequalities for the lowest eigenvalue of boundary value problems, similar to the Rayleigh–Faber–Krahn inequality. The lowest non-trivial [[Neumann eigenvalue|Neumann eigenvalue]] also satisfies an isoperimetric inequality. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004042.png" /> be the Neumann eigenvalues of the Laplace operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004043.png" />, i.e.,

<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/r/r130/r130040/r13004044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

<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/r/r130/r130040/r13004045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004046.png" />, G. Szegö [[#References|[a17]]] proved

<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/r/r130/r130040/r13004047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004048.png" />, with equality if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004049.png" /> is a circle. The corresponding result for dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004050.png" />,

<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/r/r130/r130040/r13004051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>

was proven by H.F. Weinberger [[#References|[a19]]], with equality if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004052.png" /> is a ball. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004053.png" /> is the volume of the unit ball in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004054.png" />. In (a9) and (a10), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004055.png" /> denotes the first positive zero of the derivative of the Bessel function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004056.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004057.png" />, Weinberger [[#References|[a19]]] also proved

<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/r/r130/r130040/r13004058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>

with equality if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004059.png" /> is a circle.

There is also an analogue of the Rayleigh–Faber–Krahn inequality for domains in spaces of constant curvature [[#References|[a15]]]. The optimal Rayleigh–Faber–Krahn inequalities for domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004060.png" /> was proven by E. Sperner [[#References|[a16]]].

In [[#References|[a14]]], Lord Rayleigh also conjectured an isoperimetric inequality for the lowest eigenvalue, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004061.png" />, of the clamped plate. The eigenvalue problem for the clamped plate is given by

<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/r/r130/r130040/r13004062.png" /></td> </tr></table>

with

<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/r/r130/r130040/r13004063.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004064.png" /> is a bounded open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004065.png" />. Rayleigh's conjecture for the clamped plate reads

<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/r/r130/r130040/r13004066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004067.png" /> is a ball of the same area as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004068.png" />. Rayleigh's conjecture was proven by N. Nadirashvili [[#References|[a12]]]. Equality is attained in (a12) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004069.png" /> is a circle. For dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004070.png" />, the corresponding isoperimetric inequality was proven by M.S. Ashbaugh and R.D. Benguria [[#References|[a2]]]. To prove the analogous result for dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004071.png" /> and higher is still an open problem (as of 2000, see [[#References|[a3]]] however).

Back in the membrane problem, if one goes beyond the lowest eigenvalue, there are several isoperimetric inequalities as well as a number of open problems. The simplest combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004072.png" /> satisfies the following inequality [[#References|[a1]]]:

<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/r/r130/r130040/r13004073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>

in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004074.png" /> dimensions, where equality is obtained if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004075.png" /> is a ball. Stability results for both the Rayleigh–Faber–Krahn inequality (a3), (a4) and inequality (a13) have been obtained by A.D. Melas [[#References|[a11]]] (in simple words,  "stability"  means that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004076.png" /> is convex and the appropriate left-hand side on either (a3), (a4) or (a13) is not too different from its corresponding isoperimetric value, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004077.png" /> is approximately a ball).

====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">  M.S. Ashbaugh,   R.D. Benguria,   "On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions"  ''Duke Math. J.'' , '''78'''  (1995)  pp. 1–17</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.S. Ashbaugh,   R.S. Laugesen,   "Fundamental tones and buckling loads of clamped plates"  ''Ann. Scuola Norm. Sup. Pisa Cl. Sci. (Ser. IV)'' , '''XXIII'''  (1996)  pp. 383–402</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Bandle,   "Isoperimetric inequalities and applications" , ''Adv. Publ. Program'' , Pitman  (1980)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Courant,   "Beweis des Satzes, dass von allen homogenen Membranen gegebenen Umfanges und gegebener Spannung die Kreisförmige den tiefsten Grundton besitzen"  ''Math. Z.'' , '''1'''  (1918)  pp. 321–328</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Faber,   "Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt"  ''Sitzungsber. Bayer. Akad. Wiss. München, Math.-Phys. Kl.''  (1923)  pp. 169–172</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Krahn,   "Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises"  ''Math. Ann.'' , '''94'''  (1925)  pp. 97–100</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Krahn,   "Über Minimaleigenschaft der Kugel in drei und mehr Dimensionen"  ''Acta Comm. Univ. Tartu (Dorpat)'' , '''A9'''  (1926)  pp. 1–44  (English transl.: Ü. Lumiste and J. Peetre (eds.), Edgar Krahn, 1894-1961, A Centenary Volume, IOS Press, 1994, Chap. 6, pp. 139-174)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.H. Lieb,   "Existence and uniqueness of the minimizing solution of Chocquard's nonlinear equation"  ''Stud. Appl. Math.'' , '''57'''  (1977)  pp. 93–105</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  Ü. Lumiste,   J. Peetre,   "Edgar Krahn, 1894-1961, A Centenary Volume" , IOS Press  (1994)  pp. Chap. 6</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A.D. Melas,   "The stability of some eigenvalue estimates"  ''J. Diff. Geom.'' , '''36'''  (1992)  pp. 19–33</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  N.S. Nadirashvili,   "Rayleigh's conjecture on the principal frequency of the clamped plate"  ''Arch. Rational Mech. Anal.'' , '''129'''  (1995)  pp. 1–10</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  G. Pólya,   G. Szegö,   "Isoperimetric inequalities in mathematical physics" , ''Ann. of Math. Stud.'' , '''27''' , Princeton Univ. Press  (1951)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  J.W.S. Rayleigh,   "The theory of sound" , London  (1894/96)  pp. 339–340  (Edition: Second)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  E. Schmidt,   "Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sfärischen Raum jeder Dimensionzahl"  ''Math. Z.'' , '''49'''  (1943)  pp. 1–109</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  E. Sperner,   "Zur Symmetrisierung von Funktionen auf Sphären"  ''Math. Z.'' , '''134'''  (1973)  pp. 317–327</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  G. Szegö,   "Inequalities for certain eigenvalues of a membrane of given area"  ''J. Rat. Mech. Anal.'' , '''3'''  (1954)  pp. 343–356</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  G. Talenti,   "Elliptic Equations and Rearrangements"  ''Ann. Scuola Norm. Sup. Pisa'' , '''3''' :  4  (1976)  pp. 697–718</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  H.F. Weinberger,   "An isoperimetric inequality for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130040/r13004078.png" />-dimensional free membrane problem"  ''J. Rat. Mech. Anal.'' , '''5'''  (1956)  pp. 633–636</TD></TR></table>