ON THE FIRST EIGENVALUE OF THE LAPLACIAN FOR POLYGONS EMANUEL INDREI

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ON THE FIRST EIGENVALUE OF THE LAPLACIAN FOR

POLYGONS

EMANUEL INDREI

Abstract. In 1947, P´olya proved that if n= 3,4 the regular polygon

Pnminimizes the principal frequency of an n-gon with given area α >

0 and suggested that the same holds when n≥5. In 1951,P´olya &

Szeg¨o discussed the possibility of counterexamples in the book “Isoperi-

metric Inequalities In Mathematical Physics.” This paper constructs ex-

plicit (2n−4)–dimensional polygonal manifolds M(n, α) and proves the

existence of a computable N≥5 such that for all n≥N, the admissible n-

gons are given via M(n, α) and there exists an explicit set An(α)⊂ M(n, α)

such that Pnhas the smallest principal frequency among n-gons in An(α).

Inter-alia when n≥3, a formula is proved for the principal frequency of a

convex P∈ M(n, α) in terms of an equilateral n-gon with the same area;

and, the set of equilateral polygons is proved to be an (n−3)–dimensional

submanifold of the (2n−4)–dimensional manifold M(n, α) near Pn. If

n= 3, the formula completely addresses a 2006 conjecture of Antunes

and Freitas and another problem mentioned in “Isoperimetric Inequalities

In Mathematical Physics.” Moreover, a solution to the sharp polygonal

Faber-Krahn stability problem for triangles is given and with an explicit

constant. The techniques involve a partial symmetrization, tensor calculus,

the spectral theory of circulant matrices, and W2,p/BMO estimates. Last,

an application is given in the context of electron bubbles.

1. Introduction

The principal frequency Λ of a domain Ω is the frequency of the gravest

proper tone of a uniform and uniformly stretched elastic membrane in equilib-

rium and ﬁxed along the boundary ∂Ω. In 1877, Lord Rayleigh observed that

of all membranes with a given area, the circle generates the minimum principal

frequency. This discovery was due to numerical evidence and a computation

of the principal frequency of almost circular membranes. Mathematically, this

property was known as the Rayleigh conjecture. Faber (1923) and Krahn

(1925) found essentially the same proof for Rayleigh’s conjecture.

If Ω = B1the principal frequency is the ﬁrst positive root of the Bessel

function

Λ=2.4048 . . . .

arXiv:2210.14806v1 [math.AP] 26 Oct 2022

2 EMANUEL INDREI

Nevertheless, the principal frequency of a triangle is generally inaccessible via

known formulas. In 1947 1P´olya proved that of all triangular membranes T

with a given area, the equilateral triangle has the lowest principal frequency

[P´

48]. A formula exists for the principal frequency of an equilateral triangle

and therefore a lower bound on Λ(T) is accessible. In the same article, P´olya

proved: of all quadrilaterals with a given area, the square has the smallest

principal frequency. An essential tool in his proof is Steiner symmetrization:

the set is transformed into a set of the same area with at least a line of

symmetry. However, if one considers polygons with n≥5 sides, the technique

in general increases the number of sides. P´olya mentions in the article (p.

277) that “It is natural to suspect that the propositions just proved about the

equilateral triangle and square can be extended to regular polygons with more

than four sides.”

The classical 1951 book “Isoperimetric Inequalities In Mathematical Physics”

written by P´olya and Szeg¨o mentions the problem in a less conﬁdent way [PS51,

p. 159]: “to prove (or disprove) the analogous theorems for regular polygons

with more than four sides is a challenging task.” Afterwards, the problem was

known as the P´olya-Szeg¨o conjecture. In the book, there are several related

problems. One of them is to prove the analog for the logarithmic capacity and

this was solved in 2004 by Solynin and Zalgaller [SZ04]. In 2006, Antunes and

Freitas [AF06] write about the principal frequency problem that “no progress

whatsoever has been made on this problem over the last forty years.”

In addition, Henrot includes the principal frequency problem as Open prob-

lem 2, subsection 3.3.3 A challenging open problem in the book “Extremum

Problems for Eigenvalues of Elliptic Operators” [Hen06] where he writes (p.

51) “a beautiful (and hard) challenge is to solve the Open problem 2.”

Recently, Bogosel and Bucur [BB22] showed that the local minimality of

the convex regular polygon can be reduced to a single certiﬁed numerical

computation. For n= 5,6,7,8 they performed this computation and certiﬁed

the numerical approximation by ﬁnite elements up to machine errors. However,

since the formulation of the problem no theorem asserting minimality was

proved.

My main theorem in this paper explicitly constructs sets for all suﬃciently

large nsuch that the principal frequency is minimized by the regular convex

n-gon in the collection of n-gons with the same area having vertices in these

sets mod rotations and translations.

1The article was received Dec. 20, 1947 and published in 1948.

THE REGULAR POLYGON 3

Theorem 1.1 (A local set for which Pnis the minimizer).There exists a

computable N≥5such that for all n≥N, if Pnis the convex regular n-gon

having vertices {wi}n

i=1 there exist explicit An,i ⊂Ban(wi),an>0, such that

the minimizer of the principal frequency among n-gons with ﬁxed area having

vertices in Nn

i=1 An,i (mod rigid motions) is Pn.

Corollary 1.2 (A global set for which Pnis the minimizer).There exists

N≥5and a modulus ω(0+)=0such that for all n≥N, if Pnis the convex

regular n-gon with |Pn|=π,

Ωn=nP0∈ M(n, π) : ω(|P0∆(B1+a)|)≥λ(Pn)−λ(B1)for an a∈R2o,

Qn=nP0∈ M(n, π) : vertices(P0)∈

i=1 An,i(mod a rigid motion)o,

the minimizer of the principal frequency among n-gons with ﬁxed area in An=

Ωn∪Qnis Pn.

The method of proof involves tensor calculus and a partial (Steiner) sym-

metrization. For large n, since the area is ﬁxed, if one considers a perturbation

and a set of initial vertices {v1, v2, v3}in a clockwise-consecutive arrangement,

the generated triangle is symmetrized with respect to the line intersecting the

mid-point of the line-segment between v1and v3in a perpendicular fashion.

Thus |v∗

2−v1|=|v3−v∗

2|, where v27→ v∗

2. In order to investigate how the

eigenvalue is aﬀected, a ﬂow t7→ Ptis generated via the symmetrization.

Therefore, the calculus which encodes the theory of (singular) moving surfaces

is utilized. The process is iterated which yields a series depending on the ﬁrst

and second derivatives of the eigenvalues.

Remark 1.3. Observe that assuming P∈ M(n, α) is also convex, {Pk}

generated in the proof always converges to an equilateral P∞∈ M(n, α) and

for n≥3. One may generate many explicit examples via Mn(T(n)) ⊂ An

in the main proof. Let a∈N& choose n≥Nsuﬃciently large such that

T(n)≥a; hence if the convergence is in aiterations or fewer, then λn(P)≥

λn(P∞). In particular, the minimization encodes the equilateral polygons,

and if P∞=Pn, the minimality is true. Therefore letting for instance a=

202220222022 implies lots of examples. The general minimization improvement to

equilateral polygons in (36) is with fewer assumptions; more precisely, a bound

via the rate explicit in Qnand the localization, thus, combined with the global

enhancement in Corollary 1.2 the minimization improvement generates a large

collection of polygons. Interestingly, one may prove the rate for triangles §2.9

and the localization is simple to generalize thanks to comparison arguments

with rectangles, hence the space constructed when nis large is natural and

4 EMANUEL INDREI

the optimal assumptions for the rate may have formulations in terms of angle

restrictions.

Nevertheless, the unique minimizer of the polygonal isoperimetric inequality

in the class of convex n-gons is the regular, and in general, the convex regular.

Since an example of a regular polygon in a more general class is the pentagram,

the convexity is necessary. Thus when the polygon is convex a necessary and

suﬃcient condition is cyclicity and equilaterality. Observe that the sequence

{Pk}converges to an equilateral polygon P∞∈ M(n, α) when P1∈ M(n, α):

therefore the missing characteristic is cyclicity; but, if nis large, Pnis close

to a disk, thus the cyclicity shows up naturally & Corollary 1.2 localizes the

problem for nlarge to a neighborhood of Pn. Now since in Theorem 1.1, the

formula I proved (see the proof) is true without the rate in Qn, the remaining

parts to completely prove that the global minimizer is Pnare: (i) to estimate

the radius of the neighborhood and reduce the complete problem to a neigh-

borhood where one always has convexity via the non-degenerate convex Pn;

one way of investigating this is to explicitly identify the modulus ωwhich is

(modulo a non-explicit constant) quadratic [BDPV15]; nevertheless, for some

subsets, the modulus could be much better than quadratic; (ii) when localizing,

to prove that when nis large, many polygons in the neighborhood converge

via the partial symmetrization to the regular (and therefore convex) n-gon and

utilize the formula to then compare the eigenvalues; the formula is encoded

more generally in Theorem 1.4. The second derivative via the localization is

for non-trivial iterations always positive, therefore one has to investigate the

ﬁrst derivative carefully and exploit the symmetry to obtain a fast decay (I

used the rate, symmetry, W2,p/BMO theory, and the polygonal isoperimetric

stability to show this decay). The manifold M(n, α) has non-convex polygons

and the equilateral polygons near Pngenerate an (n−3)−dimensional sub-

manifold although there are non-convex equilateral n-gons which when taking

the radius of the neighborhood around Pnsmall, vanish via the non-degenerate

convexity of Pn. This then hints towards a complete solution for nlarge in

case the general minimization is as suggested by P´olya in his paper [P´

48].

Last, to remove the largeness assumption on n, the constants appearing in the

proof of Theorem 1.1 have a fundamental role via explicit estimates. In spite

of the preclusion of cyclicity for small values of n, when n= 4 for example,

the limit is a rhombus which after one Steiner symmetrization is changed into

a rectangle and then the eigenvalue is explicit and may be compared to the

eigenvalue of a square. Thus for low values of n, one has to understand a new

way of evolving P∞into a more cyclical polygon, see §2.10, §2.7.

THE REGULAR POLYGON 5

Theorem 1.4. Assume n≥3and P∈ M(n, α)is convex. Then there exists

an equilateral Peq ∈ M(n, α)such that

λ(P) = λ(Peq) + ∞

k=2

αktk−1+∞

k=2

βk

k−1

where P1=P,Pk= (Pk−1)∗represents the n-gon constructed from Pk−1as

in the proof of Theorem 1.1,

αk=dλ(Pk)

dt t=0 =Zξk

ξk|∇un,k(α, y−)|2dα −Zξk

ξk|∇un,k(α, y+)|2dα,

βk=d2λ(Pk)

dt2t=tek−1

=2(bk

2−tek−1)

ξk(ξ2

k+ (tek−1−bk

2)2)Zξk

α|∇un,tek−1(α, y+)|2dα

+2(bk

2+tek−1)

ξk(ξ2

k+ (tek−1+bk

2)2)Zξk

α|∇un,tek−1(α, y−)|2dα,

y±, ξk, bk, tk−1are calculated explicitly from Pk−1,tek−1∈[0, tk−1], and un,k

&un,tek−1denote the corresponding eigenfunctions. Furthermore, the set of

equilateral polygons with area α,E(n, α), is an (n−3)–dimensional submanifold

of the (2n−4)–dimensional manifold M(n, α)near Pn.

In 1951,P´olya & Szeg¨o [PS51, vii] discussed the problem of ﬁnding an

explicit formula for the eigenvalue of a triangle. The theorem addresses this:

e.g. via Corollary 1.5, Remark 1.6, Corollary 1.7, Corollary 1.8, & Remark

1.9.

Corollary 1.5. The principal frequency of a triangle Twith given area Ais

Λ(T) = v

4π2

A√3+∞

k=2

αk

k−1

where T1=T,Tk= (Tk−1)∗represents the triangle constructed from Tk−1as

in the proof of Theorem 1.1 (when n= 3),

αk=d2λ(Tk)

dt2t=tek−1

=2(bk

2−tek−1)

ξk(ξ2

k+ (tek−1−bk

2)2)Zξk

α|∇u3,tek−1(α, y+)|2dα

+2(bk

2+tek−1)

ξk(ξ2

k+ (tek−1+bk

2)2)Zξk

α|∇u3,tek−1(α, y−)|2dα,

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ONTHEFIRSTEIGENVALUEOFTHELAPLACIANFORPOLYGONSEMANUELINDREIAbstract.In1947,Polyaprovedthatifn=3;4theregularpolygonPnminimizestheprincipalfrequencyofann-gonwithgivenarea>0andsuggestedthatthesameholdswhenn5.In1951;Polya&Szegodiscussedthepossibilityofcounterexamplesinthebook\Isoperi-metricInequalit...

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