Lobachevskii Journal of Mathematics Vol. 9, 2001, 3 – 5 \begin{matrix}c \\ ◯ \end{matrix} F.G. Avhadiev, R.G.Salahutdinov

F.G. Avhadiev, R.G.Salahutdinov

BILATERAL ISOPERIMETRIC INEQUALITIES FOR BOUNDARY MOMENTS OF PLANE DOMAINS

Abstract. In this paper we prove some analogs of the St Venant conjecture on the torsional rigidity. One of them is connected with Leavitt and Ungar’s inequality (Leavitt, J. & Ungar, P., Comment. Pure Appl. Math., 15 (1962), 35–37).

1 INTRODUCTION

Let Ω be a simply connected domain in the plane C. Consider the functional of Ω

I_{n}(∂Ω) = ∫ ∫_{Ω} dist^{n}(z, ∂Ω)dxdy,

where z = x + iy ∈ Ω, and dist(z, ∂Ω) is the distance from z to the boundary ∂Ω of Ω. It is clear that I_{n}(∂Ω) is the n–order moment of Ω about its boundary.

In [1] the first author proves the following inequalities

I_{2}(∂Ω) ≤ P (Ω) ≤ 64I_{2}(∂Ω),

where P (Ω) is the torsional rigidity of Ω.

Let A = A(Ω) be the area of Ω. The basic inequality for P (Ω) is the famous St Venant conjecture

P (Ω) ≤ \frac{A^{2}(Ω)}{2π}  (1)

that was proved by Pólya in 1948(see [2], [3], [4], [5], [6]).

In this paper we prove the following analog of (1)

I_{n}(∂Ω) ≤ \frac{2A^{1+n/2}(Ω)}{(n + 1)(n + 2)π^{n/2}} .  (2)

Equalities in (2) for any n > 0 hold if and only if Ω is a disk.

1991 Mathematical Subject Classification. Primary 30A10, 30C75..

Key words and phrases. Isoperimetric inequality, Moment of domain.

Supported by the Russian Fundation of Basic Research, Grant N 99–01– 00366.

We obtain (2) as corollaries of more general sharp estimates for the functional

I(∂Ω, Φ) = ∫ ∫_{Ω} Φ(dist(z, ∂Ω))dxdy,  (3)

where dΦ(t) ≥ 0 in [0, +∞).

2 MAIN THEOREM AND COROLLARIES

Let D be a disk with the area A(D) = A(Ω). Denote by L(∂Ω) the lenght of ∂Ω. Theorem 1 Let Φ(t) is a continuous non decreasing function in [0, +∞) and L(∂Ω) < +∞, then

0 ≤ I(∂D, Φ) − I(∂Ω, Φ) ≤ (L − 2\sqrt{πA}) ∫_{0}^{√\overline{A/π}} tdΦ(t),  (4)

where A = A(D) = A(Ω), L = L(∂Ω). Equalities occur if and only if Ω is a disk. Corollary 1.1 For each p > 0

∫ ∫_{Ω} dist^{p}(z, ∂Ω)dxdy ≤ \frac{2A^{1+p/2}(Ω)}{(p + 1)(p + 2)π^{p/2}} .  (5)

Equality holds only for a disk.

In the case p = 1 the inequality (5) was proved by Leavitt and Ungar [7]. Corollary 1.2 Suppose A = A(Ω) and L = L(∂Ω) are bounded, then

0 ≤ 1 − \frac{6πI_{2}(∂Ω)}{A^{2}} ≤ 8 (\frac{L}{2\sqrt{πA}} − 1) .  (6)

Both equalities hold only for a disk.

3 PROOF OF MAIN RESULTS

Proof of Theorem 1 Consider the domains (see [4])

Ω(t) = {z ∈ Ω : dist(z, ∂Ω) > t}

and the parameters

A(t, Ω) = ∫ ∫_{Ω(t)} dxdy, d(Ω) = sup_{z∈Ω} dist(z, ∂Ω).

In just the same way we define D(t), A(t, D) and d(D) for the disk D. Remind that A(Ω) = A(D) = A.

It is obvious that

A(0, Ω) = A(0, D) = A, A(d(Ω), Ω) = 0

and

d(Ω) ≤ d(D) = \sqrt{A/π}.

Suppose A(t, Ω) = 0 for t ∈ [d(Ω), d(D)]. Using the definition (3), we have

I(∂Ω, Φ) = − ∫_{0}^{d(Ω)} Φ(t)dA(t, Ω) = Φ(0)A + ∫_{0}^{d(Ω)} A(t, Ω)dΦ(t).

BILATERAL ISOPERIMETRIC INEQUALITIES 5

Using the similar formula for I(∂D, Φ) we get

I(∂D, Φ) − I(∂Ω, Φ) = ∫_{0}^{d(D)}[A(t, D) − A(t, Ω)]dΦ(t).  (7)

To complete the proof we need the following lemma. Lemma 1 (Steiner, Schmidt, see [3], p.8) For each simply connected domain Ω

2\sqrt{πA}t ≤ A − A(t, Ω) + πt^{2} ≤ Lt.  (8)

Equalities occur if and only if Ω is a disk.

It follows from (8) that

0 ≤ A(t, D) − A(t, Ω) ≤ (L − 2\sqrt{πA})t.  (9)

From (7) and (9) we have (4). Proof of Corollary 1.1 Put Φ(t) = t^{p}. The inequality (5) follows from the left-hand side of (4). Proof of Corollary 1.2 To prove (6) it is enough to choose Φ(t) = t^{2}.

References

[1] Avhadiev, F.G., Solution of generalizated St Venant problem, Matem. Sborn. 189 (1998), no. 12, 3–12. (Russian)

[2] Pólya, G. & Szegö, G., Isoperimetric Inequalities in Mathematical Physics, Princeton University Press , Princeton, N.J., 1951.

[3] Bandle, C., Isoperimetric Inequalities and Applications, Pitman Advanced Publishing Program , Boston, 1980.

[4] Payne, L.E. Isoperimetric inequalities and their applications, SIAM Review 9(1967),453–488.

[5] Osserman, R., Isoperimetric inequalities, Bull. Amer. Math. Soc. 84(1978), 1182–1238.

[6] Salahudinov, R.G., Isoperimetric inequality for torsional rigidity in the complex plane, J. of Inequal.& Appl.(to appear).

[7] Leavitt, J. & Ungar, P., Circle supports the largest sandpile, Comment. Pure Appl. Math. 15,(1962), 35–37.

Research Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya Str. 17, Kazan, 420008 Russia E-mail address: Farit.Avhadiev@@ksu.ru E-mail address: Rustem.Salahudinov@@ksu.ru

Received July, 17, 2001