A reverse Holder inequality for extremal Sobolev

functions

Tom Carroll∗ and Jesse Ratzkin†

April 6, 2014

Abstract

Let n ≥ 2, let Ω ⊂ Rn be a bounded domain with C1 boundary, and let 1 ≤ p < 2nn−2

(simply p ≥ 1 if n = 2). The well-known Sobolev imbedding theorem and Rellichcompactness implies that

Cp(Ω) = inf

∫Ω|∇f |2dm(∫

Ω|f |pdm

)2/p : f ∈W 1,20 (Ω), f 6≡ 0

is a finite, positive number, and the infimum is achieved by a nontrivial extremalfunction u, which one can assume is positive inside Ω. We prove that for everyq > p there exists K = K(n, p, q, Cp(Ω)) > 0 such that ‖u‖Lp(Ω) ≥ K‖u‖Lq(Ω). Thisinequality, which reverses the classical Holder inequality, mirrors results of G. Chiti[7] for the first Dirichlet eigenfunction of the Laplacian and of M. van den Berg [1]for the torsion function.

1 Introduction and statement of results

In 1972, Payne and Rayner [12] proved a reverse Holder inequality for the eigenfunction φof the Dirichlet Laplacian corresponding to the first eigenvalue λ(D) of a bounded planardomain D. Specifically they proved that∫

Dφ2 dm ≤ λ(D)

4π

(∫Dφdm

)2

,

with dm being Lebesgue measure. The inequality is isoperimetric in the sense thatequality holds if and only if D is a disk.

It proved not to be entirely straightforward to extend this inequality to regions inhigher dimensions. Payne and Rayner [13] obtained an isoperimetric extension to higherdimensions that they themselves described as not ‘entirely satisfactory’, since their in-equality became trivial for regions of given volume but large eigenvalue. Kohler-Jobin

∗School of Mathematical Sciences, University College Cork, t.carroll@ucc.ie†Department of Mathematics and Applied Mathematics, University of Cape Town,

jesse.ratzkin@uct.ac.za

1

[10] obtained an isoperimetric comparison between the L2 and the L1 norms of the eigen-function that did not suffer from the defects of that of Payne and Rayner. Her inequalityis ∫

Ωφ2 dm ≤ λ(Ω)n/2

2nωn jn−2n/2−1

(∫Ωφdm

)2

,

only leaving unanswered whether equality could hold for regions other than balls. HereΩ is a bounded domain in Rn, the volume of the unit ball in Rn is denoted by ωn, andjm denotes the first positive zero of the Bessel function Jm. Kohler-Jobin obtained [11]further isoperimetric inequalities between the L1 and L∞-norms and between the Lp andL1-norms of φ over the region Ω. Subsequently, Chiti [6, 7] obtained, for 0 < r < s, aninequality of the form(∫

Ωφs dm

)1/s

≤ K(n, r, s

)λ(Ω)

n2

( 1r− 1s

)

(∫Ωφr dm

)1/r

(1.1)

where Ω is a bounded region in Rn. The inequality is isoperimetric in that equality holdsif and only if Ω is a ball.

The torsion function u of an open set Ω in Rn is the solution of the problem ∆u = −2in Ω with zero Dirichlet boundary conditions. For 1 ≤ r < s < ∞, van den Berg [1,Theorem 1] obtains, as part of his work, an upper bound for the Ls(Ω)-norm of thetorsion function u in terms of its Lr(Ω)-norm, from which he concludes that if u is inLr(Ω) then it belongs to Ls(Ω) for all s > r. (A precursor of this result is Corollary 2 of[2].)

We aim herein to obtain a reverse Holder inequality for the extremal Sobolev functionthat is similar to that of Chiti for the eigenfunction of the Laplacian. For a boundeddomain Ω with C1 boundary and, for admissible values of p, we consider

Cp(Ω) = inf

Φp(f) : f ∈W 1,20 (Ω), f 6≡ 0

where Φp(f) =

∫Ω |∇f |

2 dm(∫Ω |f |p dm

)2/p . (1.2)

Here the allowable range of exponents is 1 ≤ p < 2nn−2 if n ≥ 3 and p ≥ 1 if n = 2. We

can see, using the change of variables y = x/r, that Cp obeys the scaling law

Cp(rΩ) = rn−2− 2n

p Cp(Ω) = rαn,pCp(Ω), (1.3)

and observe that, within the allowable range of exponents, αn,p < 0. By Rellich com-pactness and the Sobolev embedding theorem, Cp(Ω) is a finite, positive number, and is

realized by a nontrivial function u = up ∈ W 1,20 (Ω), which we can take to be positive

inside Ω. The significance of Cp(Ω) derives, in part, from the fact that it gives the sharp

constant in this Sobolev embedding of W 1,20 (Ω) in Lp(Ω) with

‖u‖Lp(Ω) ≤ Sp ‖∇u‖L2(Ω), u ∈W 1,20 (Ω).

The minimum of the generalised Rayleigh-Ritz quotient Cp and the best Sobolev constantSp are related by

Cp(D) = S−2p (D).

2

The extremal Sobolev function up satisfies (see, for example, [3])

∆u+ Λup−1 = 0, u|∂Ω = 0, Λ = Cp(Ω)

(∫Ωupdm

) 2−pp

. (1.4)

The significance of Cp also derives from the fact that Φ2(f) is the classical Rayleigh-Ritzquotient for f , so that C2(Ω) = λ(Ω) and the extremal Sobolev function u2 is the firstDirichlet eigenfunction φ of the Laplacian in this case. Furthermore, the torsional rigidityP (Ω) of the region Ω is 4/C1(Ω) and is given by 2

∫Ω u1 dm where ∆u = −2 in Ω and u

vanishes on the boundary of Ω.We previously generalized the Payne-Rayner theorems to extremal Sobolev functions

in two dimensions [4] and in higher dimensions [5]. However, our inequality in dimensionthree or more suffers the same flaw as that of Payne and Rayner in that our result in [5]becomes trivial for regions for which Cp is large compared to its volume. This deficiencyis rectified in the following result.

Theorem 1. Let 1 ≤ p ≤ 2 and let q ≥ p. There is a positive constant K(n, p, q),depending only on the dimension n and the exponents p and q, such that the extremalSobolev function satisfies(∫

Ωuq dm

)1/q

≤ K(n, p, q) Cp(Ω)− nαn,p

(1p− 1q

)(∫Ωup dm

)1/p

. (1.5)

The constant K(n, p, q) is given by

K(n, p, q) = Cp(B)n

αn,p

(1p− 1q

) (∫B ψ

q dm)1/q(∫

B ψp dm

)1/p (1.6)

in terms of the extremal Sobolev function ψ of the unit ball B. Equality can only occurif Ω is a round ball, except for a set of measure zero.

The restriction 1 ≤ p ≤ 2 is necessary for our proof, but we expect a similar inequalityto hold for all admissible values of p. There is no restriction on the range of q whichcan exceed the sharp Sobolev constant 2n/(n− 2). Observe that, in the case p = 2, thedependence on C2(Ω) in (1.5) is the same as the dependence on λ(D) in Chiti’s inequality(1.1), as it must be. In fact, αn,p = n − 2 − 2n/p reduces to −2 when p = 2. It shouldbe noted that, while Chiti’s reverse Holder inequality holds for any pair of exponents,the inequality (1.5) in the case p = 2 of the first Dirichlet eigenfunction of the Laplacianreturns only the special case r = 2 of Chiti’s inequality (1.1). In the special case p = 1,we may take u to be the torsion function which satisfies ∆u = −2 in Ω with Dirichletboundary conditions. Then, since

∫Ω u dm = 2/C1(Ω), the inequality (1.5) becomes, for

q > 1, (∫Ωuq dm

)1/q

≤ 2K(n, 1, q) C1(Ω)− 1n+2

(2+n

q

).

This inequality for integral means of the torsion function appears to be new. The constantK(n, 1, q) is reasonably explicit since the torsion function of the unit ball B is given

3

explicitly as u(x) =(1−|x|2

)/n. The inequality improves some cases of [1, Theorem 1(2)]

and can be written in terms of the torsional rigidity P (Ω) since P (Ω) = 4/C1(Ω).Our technique is inspired by the rearrangements of Talenti [15] and of Chiti [6, 7].

However, a key difference is that, since Chiti works with a linear and homogeneous partialdifferential equation, he may scale solutions arbitrarily, choosing either the Lq-norm ofu to be 1 for some arbitrary q, or the sup-norm of u to be 1. In our case, due tothe nonlinearity and the inhomogeneity, the only natural normalization is that in which∫

Ω up dm = 1. With a careful analysis we find that some parts of Chiti’s proof carry

over, and some parts do not. This nonlinearity and inhomogeneity is the main reason weonly obtain estimates for integrals of certain powers of u, rather than the full range ofexponents as Chiti does in [7].

2 Rearrangements

In this section we review properties of the distribution function µ associated to theextremal Sobolev function u, and two different symmeterizations of u.

Associated to up (and, indeed, any positive measurable function on Ω) is the distri-bution function, which measures the volume of sublevel sets. Let M = supx∈Ω u(x), andfor 0 ≤ t ≤M define

Ωt = x ∈ Ω : u(x) > t and µ(t) =∣∣Ωt

∣∣, (2.1)

where | · | denotes Lebesgue measure. By Sard’s theorem, Ωt has a smooth boundary∂Ωt = x ∈ Ω : u(x) = t for almost every t and hence, in particular, µ is defined almosteverywhere. We also observe that a.e. t we have

µ(t) =

∫ M

t

∫∂Ωτ

dσ

|∇u|dτ, so that

dµ

dt= −

∫∂Ωt

dσ

|∇u|< 0. (2.2)

(Here dσ denotes (n−1)-dimensional surface measure.) In particular, µ is non-increasing,so it has a (left) inverse function u∗(s) defined by

u∗(s) = inft ≥ 0 : µ(t) < s. (2.3)

It will later be useful to notice that, since u∗(µ(t)) = t,

du∗

ds=

1

dµ/dt, (2.4)

which again holds almost everywhere.We can now define the symmetric decreasing rearrangement u† of u. First let Ω∗ be

the ball centered at 0 with |Ω| = |Ω∗|, and then define

u†(x) = u∗(ωn|x|n). (2.5)

As we did with Ω, we define

Ω∗t = x ∈ Ω∗ : u†(x) > t

4

and see that, by construction, |Ωt| = |Ω∗t | for almost every t. In particular, u andu† (and u∗) have exactly the same function values. Moreover, these two functions areequimeasurable so that, for any q > 0 and t ∈ [0,M ], we have∫

u>t|u|q dm =

∫u†>t

|u†|q dm =

∫ µ(t)

0u∗(τ)q dτ. (2.6)

The symmetric decreasing rearrangement compares functions on Ω with functions onthe ball Ω∗ with the same volume as Ω. We can also compare with functions defined onthe ball B∗ with the same value of Cp as Ω, in that Cp(Ω) = Cp(B∗). We let φ = φp bethe extremal function for (1.2) on B∗, normalized so that ‖φ‖Lp(B∗) = 1. Then

∆φ+ Cp(B∗)φp−1 = ∆φ+ Cp(Ω)φp−1 = 0, φ|∂B∗ = 0. (2.7)

A theorem of Gidas, Ni, and Nirenberg [8] tells us φ is radial and dφdr < 0, so in particular

φ(0) = supφ(x) : x ∈ B∗. (2.8)

By the Faber-Krahn inequality (see, for example, [3, Theorem 4]), Cp(Ω∗) ≤ Cp(Ω). Then,B∗ ⊂ Ω∗, with equality if and only if Ω = Ω∗ = B∗. This observation plays a key role inthe proof.

3 Preliminary analysis

We first rewrite the extremal Sobolev function φ on B∗ as a function of volume. For0 ≤ s ≤ |B∗|, define φ∗(s) by

φ(x) = φ∗(ωn|x|n).

Lemma 1. The function φ∗ satisfies

(φ∗)′(s) = −Cp(Ω)n−2ω−2/nn s−2+ 2

n

∫ s

0φ∗(τ)p−1 dτ. (3.1)

Proof. The change of variables from radial distance r to volume s is given by s = ωnrn,

so thatdφ

dr=ds

dr

dφ∗

ds= nωnr

n−1(φ∗)′(s) = nω1/nn s

n−1n (φ∗)′(s).

The extremal function φ satisfies

−Cp(Ω)φp−1 = ∆φ = r1−n d

dr

(rn−1dφ

dr

),

which we can rewrite as

−Cp(Ω)φ∗(s)p−1 =

(s

ωn

) 1−nn

nω1/nn s

n−1n

d

ds

((s

ωn

)n−1n

nω1/nn s

n−1n (φ∗)′(s)

)

= n2ω2/nn

d

ds

(s2− 2

n (φ∗)′(s)).

5

Rearranging this last equation yields

− d

ds

(s2− 2

n (φ∗)′(s))

=Cp(Ω)

n2ω2/nn

φ∗(s)p−1,

which we can integrate to obtain

−s2− 2n (φ∗)′(s) =

Cp(Ω)

n2ω2/nn

∫ s

0φ∗(τ)p−1 dτ.

The following is essentially equation (34) of [15], but we include its proof for thereader’s convenience.

Lemma 2. The function u∗ satisfies

−(u∗)′(s) ≤ Cp(Ω)n−2ω−2/nn s−2+ 2

n

∫ s

0u∗(τ)p−1 dτ. (3.2)

Proof. First use Gauss’s divergence theorem to see that, for almost every t, we have

Cp(Ω)

∫Ωt

up−1 dm = −∫

Ωt

∆u dm =

∫∂Ωt

⟨∇u, ∇u|∇u|

⟩dσ =

∫∂Ωt

|∇u| dσ. (3.3)

Combining (3.3), (2.2), and the Cauchy-Schwarz inequality, we see that

∣∣∂Ωt

∣∣ =

∫∂Ωt

dσ ≤[−µ′(t)

∫∂Ωt

|∇u| dσ]1/2

=

[−Cp(Ω)µ′(t)

∫Ωt

up−1 dm

]1/2

.

Now square this last inequality and apply the isoperimetric inequality to obtain

−Cp(Ω)µ′(t)

∫Ωt

up−1 dm ≥∣∣∂Ωt

∣∣2 ≥ n2ω2/nn |Ωt|

2n−2n = n2ω2/n

n µ(t)2− 2n . (3.4)

Finally, we make the change of variables s = µ(t) and recall that

dµ

dt=

1

du∗/ds,

∫Ωt

up−1 dm =

∫ s

0u∗(τ)p−1 dτ,

by which (3.4) becomes

− 1

(u∗)′(s)Cp(Ω)

∫ s

0u∗(τ)p−1 dτ ≥ n2ω2/n

n s2− 2n .

Finally, we quote a result due to Hardy, Littlewood, and Polya [9].

Lemma 3. Let M > 0, let 0 < q1 ≤ q2, and let f, g ∈ Lq2([0,M ]). Also let f∗ be thedecreasing rearrangement of f and g∗ the decreasing rearrangement of g. If∫ s

0(f∗(t))q1 dt ≤

∫ s

0(g∗(t))q1 dt for all 0 ≤ s ≤M

then ∫ M

0|f |q2ds ≤

∫ M

0|g|q2ds.

6

4 Proof of the main theorem

In this section we prove (1.5).If |Ω| = |B∗| then, by the Faber-Krahn inequality, we must have Ω = Ω∗ = B∗. In

this case, the inequality (1.5) holds with

K =

(∫B∗ φ

q dm)1/q(∫

B∗ φp dm

)1/p (4.1)

This will, in fact, turn out to be the optimal constant in the general case. If we denotethe radius of B∗ by ρ, then the change of variables ψ(r) = φ(ρr) produces an extremalfunction on the unit ball B. We find that

K =

(∫B∗ φ

q dm)1/q(∫

B∗ φp dm

)1/p =ρn/q

(∫B ψ

q dm)1/q

ρn/p(∫

B ψp dm

)1/p . (4.2)

Using the scaling law (1.3) we find that

ρ =

(Cp(B∗)Cp(B)

)1/αn,p

=

(Cp(Ω)

Cp(B)

)1/αn,p

. (4.3)

Combining (4.2) and (4.3) yields

K = Cp(B)n

αn,p

(1p− 1q

) (∫B ψ

q dm)1/q(∫

B ψp dm

)1/p Cp(Ω)n

αn,p

(1q− 1p

)(4.4)

= K(n, p, q) Cp(Ω)− nαn,p

(1p− 1q

),

in agreement with (1.6), where B is the unit ball and ψ is an extremal function for Cp(B),We may now assume B∗ is strictly contained in Ω∗. Since we have normalized the

extremal Sobolev function φ so that ‖φ‖Lp(B∗) = 1, we have

1 =

∫B∗φp dm =

∫ |B∗|0

(φ∗)p ds =

∫ |Ω|0

(u∗)pds >

∫ |B∗|0

(u∗)pds. (4.5)

Thus, we cannot have φ∗ ≤ u∗ on the entire interval [0, |B∗|]. On the other hand,φ∗(|B∗|) = 0 < u∗(|B∗|). Combining these two facts implies that the graphs of φ∗ andu∗ must cross. Define

s1 = infs ∈ [0, |B∗|] : φ∗(s) < u∗(s) for all s ∈ (s, |B∗|],

and observe that (by continuity) φ∗(s1) = u∗(s1), while φ∗ < u∗ on (s1, |B∗|]. The points1 is the first crossing point of the two graphs, as viewed from the right hand side.

Our next task is to prove that s1 is in fact the only crossing. It is useful to first

observe s1 > 0. Indeed, if s1 = 0 then∫ |B∗|

0 (u∗)pds ≥∫ |B∗|

0 (φ∗)pds, which contradicts(4.5).

Lemma 4. On the interval [0, s1] we have φ∗(s) ≥ u∗(s).

7

Figure 1: Schematic plot of φ∗ and u∗

φ∗

u∗

|Ω||B∗|s1s2s3

Proof. Suppose there exists s ∈ [0, s1) such that u∗(s) > φ∗(s), and set

s2 = sups < s1 : u∗(s) > φ∗(s).

Since we suppose that the set s ∈ [0, s1) : u∗(s) > φ∗(s) is nonempty, the continuity ofφ∗ and u∗ imply that s2 > 0. Next let

s3 = infs < s2 : u∗(s) > φ∗(s) for all s ∈ (s, s2).

Now we have φ∗ < u∗ in (s1, |B∗|], φ∗ ≥ u∗ in [s2, s1], and φ∗ < u∗ in (s3, s2) (seeFigure 1). Define the function

v∗(s) =

maxφ∗(s), u∗(s) 0 ≤ s ≤ s3

u∗(s) s3 ≤ s ≤ s2

φ∗(s) s2 ≤ s ≤ |B∗|.

By construction, we have v∗ > 0 on [0, |B∗|) and v∗(|B∗|) = 0. We claim that

−(v∗)′(s) ≤ Cp(Ω)n−2ω−2/nn s−2+ 2

n

∫ s

0v∗(τ)p−1 dτ, 0 < s < |B∗|. (4.6)

For 0 ≤ s ≤ s3 we have v∗(s) = maxφ∗(s), u∗(s). This is the maximum of twononincreasing functions, so it is nonincreasing itself, and therefore v∗ is differentiablealmost everywhere. If u∗(s) < φ∗(s) then, by Lemma 1,

−(v∗)′(s) = −(φ∗)′(s) = Cp(Ω)n−2ω−2/nn s−2+ 2

n

∫ s

0φ∗(τ)p−1 dτ

≤ Cp(Ω)n−2ω−2/nn s−2+ 2

n

∫ s

0v∗(τ)p−1 dτ,

Similarly, if u∗(s) > φ∗(s) then, by Lemma 2,

−(v∗)′(s) = −(u∗)′(s) ≤ Cp(Ω)n−2ω−2/nn s−2+ 2

n

∫ s

0u∗(τ)p−1 dτ

≤ Cp(Ω)n−2ω−2/nn s−2+ 2

n

∫ s

0v∗(τ)p−1 dτ.

8

If u∗(s) < φ∗(s) then (v∗)′(s) equals either (φ∗)′(s) or (u∗)′(s), and again (4.6) holds.We conclude that (4.6) holds for 0 ≤ s < s3. Nearly identical arguments show (4.6) alsoholds in the intervals (s3, s2) and (s2, |B∗|).

Associated to v∗ we have the radial test function v : B∗ → R defined by v(x) =v∗(ωn|x|n). Observe that∫

B∗vp dm =

∫ |B∗|0

v∗(s)p ds >

∫ |B∗|0

φ∗(s)p ds = 1 (4.7)

(the inequality follows from v∗ > φ∗ on the interval (s3, s2)) and also that∫B∗|∇v|2 dm = n2ω2/n

n

∫ |B∗|0

s2− 2n

(dv∗

ds

)2

ds

≤ Cp(Ω)

∫ |B∗|0

(−dv

∗

ds

)∫ s

0v∗(τ)p−1 dτ ds

= Cp(Ω)

∫ |B∗|0

v∗(τ)p−1

∫ |B∗|τ

(−dv

∗

ds

)ds dτ

= Cp(Ω)

∫ |B∗|0

v∗(τ)p dτ. (4.8)

Since v is a valid test function for Cp(B∗), and making use of (4.8), we find that

Cp(B∗) ≤∫B∗ |∇v|

2 dm(∫B∗ v

p dm)2/p ≤ Cp(Ω)

(∫B∗vp dm

) p−2p

= Cp(B∗)(∫

B∗vp dm

) p−2p

. (4.9)

However, with p ≤ 2 and, by (4.7),∫B∗ v

p dm > 1, the inequality (4.9) is impossible.

In conclusion, we have shown that there exists s1 ∈ (0, |B∗|) such that φ∗ ≥ u∗ inthe interval [0, s1) and u∗ > φ∗ in the interval (s1, |B∗|]. Extend φ∗ to be zero on theinterval (|B∗|, |Ω|], so that the inequality u∗ > φ∗ continues to hold. That φ∗(s) ≥ u∗(s)for 0 ≤ s ≤ s1 immediately implies that∫ s

0φ∗(τ)p dτ ≥

∫ s

0u∗(τ)p dτ for all s ∈ [0, s1]. (4.10)

We claim (4.10) also holds for s ∈ [s1, |Ω|]. To see this, let

I(s) =

∫ s

0φ∗(τ)p dτ −

∫ s

0u∗(τ)p dτ,

and observeI(|Ω|) = I(0) = 0, I ′(s) = φ∗(s)p − u∗(s)p.

Therefore I is nondecreasing on the interval [0, s1] and decreasing on the interval [s1, |Ω|].In particular, if I(s) < 0 for some s ∈ (s1, |Ω|) then I(|Ω|) < 0 as well, which does notoccur. Thus (4.10) holds for all s ∈ [0, |Ω|].

Combining (4.10) with Lemma 3, we conclude that for all q ≥ p we have(∫Ωuq dm

)1/q

≤(∫

B∗φq dm

)1/q

= K

(∫B∗φp dm

)1/p

= K

(∫Ωup dm

)1/p

,

9

where K is given by (4.1) and we have used the normalization∫B∗φp dm =

∫Ωup dm.

Acknowledgements: Much of this research was completed while J. R. visited Univer-sity College Cork. He thanks thank UCC for its hospitality. J. R. is partially supportedby the National Research Foundation of South Africa.

References

[1] M. van den Berg, Estimates for the torsion function and Sobolev constants. Poten-tial Analysis, 36 (2012) 607–616.

[2] M. van den Berg and T. Carroll. Hardy inequality and Lp estimates for the torsionfunction. Bull. London Math. Soc. (2009), 980–986.

[3] T. Carroll and J. Ratzkin, Interpolating between torsional rigidity and principalfrequency. J. Math. Anal. Appl. 379 (2011), 818–826.

[4] T. Carroll and J. Ratzkin, Two isoperimetric inequalities for the Sobolev constant.Z. Angew. Math. Phys. 63 (2012), 855–863.

[5] T. Carroll and J. Ratzkin, An isoperimetric inequality for extremal Sobolev func-tions. RIMS Kokyuroku Bessatsu B43 (2013), 1–16.

[6] G. Chiti, An isomerimetric inequality for the eigenfunctions of linear second orderelliptic equations. Boll. Un. Mat. Ital. A 1 (1982), 145–151.

[7] G. Chiti, A reverse Holder inequality for the eigenfunctions of linear second orderelliptic operators. Z. Angew. Math. Phys. 33 (1982), 143–148.

[8] B. Gidas, W.N. Ni, and L. Nireberg, Symmetry and related properties via themaximum principle. Comm. Math. Phys. 68 (1979), 209–243.

[9] G. H. Hardy, J. E. Littlewood, and G. Polya. Some simple inequalities satisfied byconvex functions. Messenger Math. 58 (1929), 145–152.

[10] M.-T. Kohler-Jobin, Sur la premiere fonction propre d’une membrane: une exten-sion a N dimensions de l’inegalite isoperimetrique de Payne-Rayner. Z. Angew.Math. Phys. 28 (1977), 1137–1140.

[11] M.-T. Kohler-Jobin, Isoperimetric monotonicity and isoperimetric inequalities ofPayne-Rayner type for the first eigenfunction of the Helmholtz problem. Z. Angew.Math. Phys. 32 (1981), 625–646.

[12] L. Payne and M. Rayner, An isoperimetric inequality for the first eigenfunction inthe fixed membrane problem. Z. Angew. Math. Phys. 23 (1972), 13–15.

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[13] L. Payne and M. Rayner, Some isoperimetric norm bounds for solutions of theHelmholtz equation. Z. Angew. Math. Phys. 24 (1973), 105–110.

[14] G. Polya and G. Szego, Isoperimetric Inequalities in Mathematical Physics. Prince-ton University Press (1951).

[15] G. Talenti, Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. PisaCl. Sci. 3 (1976), 697–718.

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