Finite element approximation of the first eigenvalue of a nonlinear problem for some special domains

Gabriella Bognár Mathematical Institute, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary

October 26, 2000

Abstract

In this paper we present a method for the numerical approximation of the smallest eigenvalue of a nonlinear eigenvalue problem using the finite element method. Numerical results are presented for some special domains when the domain is bounded by a square, a ”circle”, a ”semicircle”, or a quadrant of a ”circle”. We compare the exact solutions with the approximate solutions when the exact solutions are known. We show a connection among the first eigenvalues related to different domains.

1 Introduction

Consider the eigenvalue problem

−Q_{p} = λ |u|^{p−1} u in Ω,  (1)
u = 0 on ∂Ω,

where Ω is a convex domain in R^{2} and Q_{p} is the nonlinear operator defined by

Q_{p} = \frac{∂}{∂x} (|\frac{∂u}{∂x}|^{p−1} \frac{∂u}{∂x}) + \frac{∂}{∂y} (|\frac{∂u}{∂y}|^{p−1} \frac{∂u}{∂y}) for 0 < p < ∞.

It is known ( see [2]) that (1) has a sequence of weak solutions (λ_{k}(p), u_{k}(p)) in R×W_{0}^{1,p+1} (Ω) , where 0 < λ_{k}(p) < λ_{k+1}(p), k = 1, 2, ... .The first eigenvalue can be defined by the variational principle

λ_{1}(p) = \begin{matrix}inf \\ u ∈ W_{0}^{1,p+1} (Ω) \end{matrix} \frac{∫_{Ω} (|u_{x}|^{p+1} + |u_{y}|^{p+1}) dx}{∫_{Ω} |u|^{p+1} dx} =

= inf ; \begin{matrix}u ∈ W_{0}^{1,p+1} (Ω) \\ ∥u∥_{p+1} = 1 \end{matrix} ∫_{Ω} (|u_{x}|^{p+1} + |u_{y}|^{p+1}) dx.  (2)

The weak formulation of (1) is

∫_{Ω} (|u_{x}|^{p−1} u_{x}v_{x} + |u_{y}|^{p−1} u_{y}v_{y)}dx = λ_{1}(p)∫_{Ω} |u|^{p−1} uvdx  (3)

for any v ∈ W_{0}^{1,p+1} (Ω) . The unique solution (λ_{1}(p), u_{1}(p)) of (3) which saisfies ∥u∥_{p+1} = 1 is called the first eigenpair of (3) . It is known that u_{1}(p) is positive [1]. From (3) we have got

λ_{1}(p) = ∫_{Ω} (|(u_{1}(p))_{x}|^{p+1} + |(u_{1}(p))_{y}|^{p+1}) dx.

The knowledge of the first eigenvalue is of considerable importance for the designer of safe and efficient structures. Such structures may appear in microphones, bridges, ships, or space vehicles.

The quantity λ_{1}(p) arises in a variety of situations where it is of interest to know its value more precisely. For example λ_{1}(p) is the principal frequency of Ω. We conceive Ω as the equilibrium position of a stretched membrane fixed along the boundary ∂Ω of Ω. The frequency of the gravest proper tone of this membrane is \sqrt[p+1]{λ_{1}(p)}. In the linear case, λ_{1}(1) is the classical principal eigenvalue of the Poisson equation Δu + λu = 0. The quantity λ_{1}(p) depends on shape and size of the domain Ω. This principal frequency of vibration has been calculated for various domains. These are: circle, square, quadrant of circle, sector of a circle 60^{o}, rectangle, equilateral triangle, semicircle [9], [10].

Lord Rayleigh stated that of all clamped membranes with a given area A, the circle has the minimal principal frequency [10]. This property can be expressed by the inequality

λ_{1}(1) ≥ \frac{πj_{0}^{2}}{A}  (4)

with equality only for the circle and where j_{0} is the first positive zero of the Bessel function of the first kind J_{0}(x). It was a conjecture, however Rayleigh supported it not only by numerical evidence, but also by computing the principal frequency of almost circular membranes. G. Faber [5] and E. Krahn [7] found independently the same proof of Rayleigh’s conjecture. Another proof was given by G. Polya and G. Szego [9] by using the Steiner symmetrization.

E. Krahn [8] showed that the Rayleigh inequality (4) can be extended to R^{N} in the form

λ_{1}(1) ≥ [\frac{V}{A_{N}} ]^{\frac{2}{N}} j2_{\frac{N−2}{2}} ,

where V is the volume of the unit ball in R^{N}, A_{N} is the volume of Ω ∈ R^{N} and j2_{\frac{N−2}{2}} is the first zero of the Bessel function J_{\frac{N−2}{2}} .

In [1] a lower bound was given for the first eigenvalue of the nonlinear elliptic eigenvalue problem (1) :

λ_{1}(p) ≥ [\frac{P j_{0}^{2}}{A} ]^{\frac{p+1}{2}} ,  P = 2\frac{p}{p + 1}B (\frac{p}{p + 1}, \frac{p}{p + 1}) ,  (5)

where A is the area of Ω ∈ R^{2}, and j_{0} is the first positive zero of the generalized nonlinear Bessel function J_{0}(x). In (5) there is equality if and only if ∂Ω is the curve defined by the equation

|x|^{\frac{1}{p} +1} + |y|^{\frac{1}{p} +1} = r^{\frac{1}{p} +1},  (6)

with suitable r ∈ R^{+} (Figure.1 with r = 1)

This curve is called isoperimetrix which plays the same role in case of nonlinear problem (1) as the circle in case of the Poisson equation. Therefore later we recall this curve as ”circle”. In the case p = 1 the inequality (5) is equivalent to (4) and P = π. Moreover we obtained that for the simply connected convex domain Ω ∈ R^{2} the inequality

λ_{1}(p) ≥ [\frac{A + σ}{(p + 1) ϱA}]^{p+1}

holds, where ϱ is the radius of the greatest inscribed isoperimetrix of Ω, and σ is the area of the isoperimetrix of radius ϱ [3].

The smallest first eigenvalue λ_{1}(p) is evaluated for the so called ”circle” in case of nonlinear problem (1). For the linear case (p = 1) the connection among the first eigenvalues regarding to some special domains is given by Rayleigh [10].

In this paper we examine the first eigenvalues for different domains. The exact solution is known only for a special domain (rectangular domain).It is not known that the problem has classical solution on other domains except rectangle. On such a domain we can obtain approximate solutions.We shall present a method for numerical approximation of λ_{1}(p) based on the finite element approximation to u_{1}(p). Some numerical results will be presented for different special domains when Ω is bounded by a square, a ”circle”, a ”semicircle”, a quadrant of a ”circle”. We compare the exact solutions with the approximate solutions when the exact solutions are known. We show that the membranes of the same area the same connection is valid for the first eigenvalues of the nonlinear problem as of the linear problem (p = 1).

2 Preliminary results

For the Dirichlet eigenvalue problem (1) we can find classical solutions when D is bounded by the rectangle

D = {(x, y) : 0 ≤ x ≤ a, 0 ≤ y ≤ b} .

The eigenvalues and eigenfunctions are

λ_{k,l} = pπ̃^{p+1} (\frac{k^{p+1}}{a^{p+1}} + \frac{l^{p+1}}{b^{p+1}} ) ,  (7)

u_{k,l} = A_{k,l}S_{p} (\frac{kπ̃}{a} x) S_{p} (\frac{lπ̃}{b} y) , k, l = 1, 2, ...,  (8)

where

π̃ = \frac{2\frac{π}{p+1}}{sin \frac{π}{p+1}} ,

A_{k,l} = const., and the function S_{p} is the solution of the differential equation

S_{p}” |S′_{p}|^{p−1} + |S_{p}|^{p−1} S_{p} = 0

under conditions

S_{p} (0) = 0, S_{p} \binom{}{π} = 0.

The function S_{p} is the generalized sine function which plays the same role in case of nonlinear problem (1) as the sine function in case of Poisson equation. For p = 1

S_{1}(x) = sin x, π̃ = π.

If the domain D is bounded by a unit square, as a corollary of the above, we get the smallest eigenvalue λ_{1}(p) and the corresponding eigenfunction u_{1}(p) for the Dirichlet eigenvalue problem of (1) if we put k = l = 1 in the expressions of λ_{k,l} and u_{k,l}:

λ_{1}(p) = 2pπ̃^{p+1},  (9)

u_{1}(p) = A_{1,1}S_{p}(π̃x)S_{p}(π̃y).

The exact value of the first eigenvalue for the linear eigenvalue problem (p = 1) was given by Rayleigh [10]:

λ_{1}(1) = 2π^{2} ≈ 19.7392.

3 Finite element formulation of the problem

Let Ω be a given convex open subset of R^{2} and let W^{1,p+1} (Ω) denotes the space of all functions which together with their derivatives u_{x}, u_{y} belong to L^{p+1} (Ω) . We define the norm in W^{1,p+1} (Ω) by

∥u∥_{W^{1,p+1}(Ω)} = (∫_{Ω} (|u|^{p+1} + |u_{x}|^{p+1} + |u_{y}|^{p+1}) dx)^{\frac{1}{p+1}}
for all u ∈ W^{1,p+1} (Ω) .

As usually, the symbol W_{0}^{1,p+1} (Ω) stands for the subspace of W^{1,p+1} (Ω) obtained by closing the set of all C^{∞}-functions with compact support in Ω. On the Sobolev space W_{0}^{1,p+1} (Ω) another norm can be defined by

∥u∥_{1,p+1} = (∫_{Ω} (|u_{x}|^{p+1} + |u_{y}|^{p+1}) dx)^{\frac{1}{p+1}}

which is equivalent to the norm ∥u∥_{W^{1,p+1}(Ω)} . The usual norm in L^{p+1} (Ω) is

denoted by ∥u∥_{p+1} = (∫_{Ω} |u|^{p+1} dx)^{\frac{1}{p+1}} .

Now we form the the finite element approximation of the problem (1) and (2). Let V_{h} (Ω) be the space of continuous linear functions based on regular triangulation of Ω and the subspace of W_{0}^{1,p+1} (Ω) as in [4]. The weak formulation of the problem (1) was formulated in (3). The corresponding minimization problem consists of finding the eigenpair (λ_{1h}(p), u_{h}) such that

λ_{1h}(p) = inf_{v_{h}∈V_{h}(Ω)} \frac{∫_{Ω} (|v_{hx}|^{p+1} + |v_{hy}|^{p+1}) dx}{∫_{Ω} |v_{h}|^{p+1} dx},  (10)

and

∫_{Ω} (|u_{hx}|^{p−1} u_{hx}v_{hx} + |u_{hy}|^{p−1} u_{hy}v_{hy)}dx = λ∫_{Ω} |u_{h}|^{p−1} u_{h}v_{h}dx  (11)
for any v_{h} ∈ V_{h} (Ω) .

The domain triangulation depends on the shape of the domain. For square equal mesh sizes is used. For other domain the triangulation has to be generated for each value of p and for each shape separately (Figure 2. the triangulation for the quadrant of the so called circle is given when p = 3).

4 Error estimation

Let the Rayleigh-Ritz projection of u ∈ W_{0}^{1,p+1} (Ω) is denoted by P u ∈ V_{h} (Ω) and P u is the unique finite element solution of

\begin{matrix}∫_{Ω}(|P u_{x}|^{p−1} P u_{x}v_{hx} + |P u_{y}|^{p−1} P u_{y}v_{hy})dx = \\ = ∫_{Ω}(|u_{x}|^{p−1} u_{x}v_{hx} + |u_{y}|^{p−1} u_{y}v_{hy})dx \end{matrix}  (12)

for given u ∈ W_{0}^{1,p+1} (Ω) and for any v_{h} ∈ V_{h} (Ω) . From Theorem 5.3.2 [4] follows that

lim_{h→0} ∥u − P u∥_{p+1} = 0.  (13)

Lemma 1 Let (λ_{1}, u_{1}) be the first eigenpair of (1) and let

ε_{1h} = |∫_{Ω} |u_{1}|^{p+1} dx − ∫_{Ω} |P u_{1}|^{p+1} dx| .

Then

λ_{1h}(p) ≤ \frac{λ_{1}}{1 − ε_{1}^{h}}

provided ε_{1h} < 1.

Proof. We suppose that P u_{1} ≠ 0. If P u_{1} = 0, then

ε_{1h} = |∫_{Ω} |u_{1}|^{p+1} dx − ∫_{Ω} |P u_{1}|^{p+1} dx| = ∫_{Ω} |P u_{1}|^{p+1} dx = 1 and condition ε_{1h} <

1 is not satisfied. Applying (10) we get

λ_{1h}(p) = inf_{v_{h}∈V_{h}(Ω)} \frac{∫_{Ω} (|v_{hx}|^{p+1} + |v_{hy}|^{p+1}) dx}{∫_{Ω} |v_{h}|^{p+1} dx} ≤ \frac{∫_{Ω} (|P u_{1x}|^{p+1} + |P u_{1y}|^{p+1}) dx}{∫_{Ω} |P u_{1}|^{p+1} dx}

(14)

for P u_{1} ∈ V_{h} (Ω) . Making use of (??) with v_{h} = P u_{1}, the inequality [6] and the Hölder inequality we obtain

∫_{Ω} (|P u_{1x}|^{p+1} + |P u_{1y}|^{p+1}) dx = ∫_{Ω}(|u_{1x}|^{p−1} u_{1x}P u_{1x} + |u_{1y}|^{p−1} u_{1y}P u_{1y)}dx

≤ ∫_{Ω} [(|u_{1x}|^{p+1} + |u_{1y}|^{p+1})^{\frac{p}{p+1}} (|P u_{1x}|^{p+1} + |P u_{1y}|^{p+1})^{\frac{1}{p+1}} ] dx

≤ ∥u_{1}∥_{1,p+1}^{p} ∥P u_{1}∥_{1,p+1}

which leads to

∥P u_{1}∥_{1,p+1}^{p+1} ≤ ∥u_{1}∥_{1,p+1}^{p} ∥P u_{1}∥_{1,p+1} ,

∥P u_{1}∥_{1,p+1} ≤ ∥u_{1}∥_{1,p+1}^{p} .  (15)

Since

∫_{Ω} |P u_{1}|^{p+1} dx = ∫_{Ω} |u_{1}|^{p+1} dx + ∫_{Ω} ( |P u_{1}|^{p+1} − |u_{1}|^{p+1})dx ≥

≥ 1 − |∫_{Ω} (|P u_{1}|^{p+1} − |u_{1}|^{p+1})dx|
= 1 − ε_{1h}

we obtain the following estimate from (14) with (15)

λ_{1h}(p) ≤ \frac{∫_{Ω} (|u_{1x}|^{p+1} + |u_{1y}|^{p+1}) dx}{∫_{Ω} |P u_{1}|^{p+1} dx} ≤ \frac{λ_{1}}{1 − ε_{1h}}

which has to be proved. Lemma 2 There exists a constant α > 0 depending on p and u, such that

ε_{1h} ≤ α ∥u_{1}∥_{1,p+1}^{p} ∥u_{1} − P u_{1}∥_{p+1} .  (16)

Proof. Applying the inequality |x^{p+1} − y^{p+1}| ≤ (p + 1) |x − y| (|x|^{p−1} x − |y|^{p−1} y) for p > 0, x ≥ 0, y ≥ 0.

for ε_{1h} we get

\begin{matrix}ε_{1h} = |∫_{Ω} |u_{1}|^{p+1} dx − ∫_{Ω} |P u_{1}|^{p+1} dx| \\ ≤ (p + 1) ∥u_{1} − P u_{1}∥_{p+1} (∥u_{1}∥_{p+1}^{p} + ∥P u_{1}∥_{p+1}^{p}). \end{matrix}

By the Poincare inequality [1]

∫_{Ω} |P u_{1}|^{p+1} dx ≤ C_{1}∫_{Ω} (|P u_{1x}|^{p+1} + |P u_{1y}|^{p+1}) dx with C_{1} = const.

and (15) we have

ε_{1h} ≤ (p + 1) ∥u_{1} − P u_{1}∥_{p+1} (∥u_{1}∥_{p+1}^{p} + C_{1}^{p} ∥u_{1}∥_{p+1}^{p}) ≤
≤ α ∥u_{1}∥_{p+1}^{p} ∥u_{1} − P u_{1}∥_{p+1} .

Thus we can give error estimates for the first eigenvalue Theorem 3 Let (λ_{1}, u_{1}) be the first eigenpair of (1) and let (λ_{1h}(p), u_{1h}) is the counterpart of (λ_{1}, u_{1}) for (10) and (11) then

lim_{h→0}λ_{1h}(p) = λ_{1}.  (17)

Proof. We can choose h small enough so that ε_{1h} < \frac{1}{2}, then

\frac{1}{1 − ε_{1h}} ≤ 1 + 2ε_{1h}

and by Lemma1 we get

λ_{1} ≤ λ_{1h}(p) ≤ \frac{λ_{1}}{1 − ε_{1h}} ≤ λ_{1}(1 + 2ε_{1h}).

Using Lemma 2 we obtain that

λ_{1} ≤ λ_{1}(1 + 2ε_{1h}) ≤
≤ λ_{1} + 2λ_{1}α ∥u_{1}∥_{p+1}^{p} ∥u_{1} − P u_{1}∥_{p+1}

which gives (17) with (13).

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 1, p. 7

5 Numerical results

The functions N_{i}, i = 1, 2, ..., m will denote a finite element basis of the m dimensional space V_{h}, thus

V_{h} = { m ; ∑_{i=1}x_{i}N_{i} |(x_{1}, x_{2}, ..., x_{m}) ∈ R^{m} } .

The finite element approximation of u_{1} is

u_{h} = m ; ∑_{i=1}x_{i}N_{i} = x^{T} N^{T} ,

∇u_{h} = m ; ∑_{i=1}∇(N_{i}x_{i}) = x^{T} B^{T} ,

and from the normalization formula ∫_{Ω} |u_{h}|^{p+1} dx = 1 we obtain

∫_{Ω} |N x|^{p+1} dx = 1.

For arbitrary ũ_{h} = N x̃ = N (xa), where a ≠ 0, we have got

∫_{Ω} \left|{ \atop u_{h}}\right|^{p+1} dx = ∫_{Ω} |N xa|^{p+1} dx = ∫_{Ω} |N x|^{p+1} |a|^{p+1} dx,

therefore

|a| = [∫_{Ω} \left|{ \atop u_{h}}\right|^{p+1} dx]^{\frac{1}{p+1}} , and x =\frac{x̃}{a}.

After normalization we obtain

λ_{1h}(p) = ∫_{Ω} |B x|^{p+1} dx.  (18)

From \frac{∂λ_{h}}{∂x_{i}} = 0 (i = 1, 2, ..., m) we get

0 = [∫_{Ω} B^{T} |B x|^{p} dx] [∫_{Ω} |N x|^{p+1} dx] − [∫_{Ω} |B x|^{p+1} dx] [∫_{Ω} N^{T} |N x|^{p} dx]

and using (18) we obtain

0 = ∫_{Ω} B^{T} |B x|^{p} dx − λ_{h}∫_{Ω} N^{T} |N x|^{p} dx.  (19)

Since (19) is nonlinear we get

0 = ∫_{Ω} B^{T} |B x_{j}|^{p} dx − λ_{h,j}∫_{Ω} N^{T} |N x_{j}|^{p} dx + p∫_{Ω} B^{T} |B x_{j}|^{p−1} B dx (x − x_{j})

(20)

by using the Newton’s method for given x_{j}. Applying the following notations

H_{j} = p∫_{Ω} B^{T} |B x_{j}|^{p−1} B dx,

f_{N} = ∫_{Ω} N^{T} |N x_{j}|^{p} dx,

f_{B} = ∫_{Ω} B^{T} |B x_{j}|^{p} dx,

Δx_{j} = (x − x_{j}) ,

and we can write (20) as follows

H_{j} Δx_{j} = λ_{1h,j}f_{N} − f_{B}.

In order to give initial values for the eigenfunction and eigenvalue we solve a linear eigenvalue problem (p = 1). Then the non-linearity is gradually imposed on the equations, that means, that p is increased or decreased up to the required value.

We briefly describe the algorithm below

1. Start with an initial approximation x_{1} (x_{1} ≠ 0).

2. For given x_{j}, calculate H_{j} Δx_{j} = λ_{1h,j}f_{N} − f_{B}, then Δx_{j}.

3. Evaluate x̃_{j+1} = x_{j} + Δx_{j}.

4. Evaluate x_{j+1} = _{\frac{x̃_{j+1}}{[∫_{Ω}|N x_{j+1}|^{p+1}dx]^{\frac{1}{p+1}}}} .

5. Calculate λ_{1h,j+1} = ∫_{Ω} |B x_{j+1}|^{p+1} dx.

6. Terminate when \frac{|λ_{1h,j+1}−λ_{1h,j}|}{λ_{1h,j+1}} is smaller then a predetermined tolerance.

An experimental computer code written in FORTRAN source language was used for the solution of the approximate eigenvalues. The eigenvalues are calculated by h-version of the finite element method.

In the second section we showed solutions of the nonlinear eigenvalue problem (1) for the unit square. Here the exact values of the first eigenvalues λ_{1} are known as a function of p in (9) . Using the finite element method we calculated the approximate values of λ_{1} for various values of p for different shape of domain Ω in twodimension. These domains are the unit square, the ”unit circle” defined in (6) , and the ”semicircle”, and the quadrant of the ”circle” with radius 1. The exact values for λ_{1} are known from [10] for the linear case (p = 1) . In the tables below we give the finite element approximations of the first eigenvalue for different number of nodal points n, when the mesh size is different. The first computation was made for the unit square. The values are reported in Table 1.

\begin{matrix}p\n & 9 & 16 & 64 & 256 \\ 0.5 & 14.1856 & 11.1788 & 10.7632 & 10.6683 \\ 0.75 & 18.5345 & 15.4269 & 14.8608 & 14.7285 \\ 1 & 24.0000 & 20.7733 & 19.9941 & 19.8027 \\ 2 & 64.0000 & 59.7528 & 57.5335 & 56.8420 \\ 3 & 164.5714 & 154.1444 & 148.9242 & 146.9674 \\ 5 & 1063.3840 & 890.3235 & 862.0190 & 850.2630 \end{matrix}

Table 1.

In case p = 1 the known value is λ_{1}(1) = 2π^{2} ≈ 19.7392. Our approximation differs from the known value by 0.32 %. In case of square domain we can evaluate the relative error for any value of p since we have the formula (9) . The relative error is less then 1 % when p ∈ [0.25, 10] . Similar formula is not known for the other domains. Thus we can compare the approximate solution with the exact solution only when p = 1. The approximate values are reported in Table 2 for the unit ”circle”.

\begin{matrix}p\n & 67 & 227 & 835 & 3203 \\ 0.5 & 4.1774 & 4.0573 & 4.0277 & 4.0203 \\ 0.75 & 5.0962 & 4.9366 & 4.8978 & 4.8882 \\ 1 & 6.0646 & 5.8522 & 5.8004 & 5.7875 \\ 2 & 10.5134 & 10.0104 & 9.8780 & 9.8435 \\ 3 & 15.9061 & 15.0216 & 14.7739 & 14.7095 \\ 5 & 29.4856 & 27.6169 & 27.0521 & 26.8875 \end{matrix}

Table 2.

In the linear case the exact solution is 5.7831 [10]. Here the difference between the exact and finite element solution is 0.08 %. For the ”semicircle” The calculated values are presented in Table 3.

\begin{matrix}p\n & 33 & 113 & 417 & 1601 \\ 0.5 & 8.2958 & 7.8851 & 7.7885 & 7.7645 \\ 0.75 & 11.5277 & 10.9784 & 10.8399 & 10.8066 \\ 1 & 15.7835 & 14.9546 & 14.7500 & 14.6989 \\ 2 & 50.7108 & 46.2403 & 44.9594 & 44.6189 \\ 3 & 150.3274 & 130.6471 & 124.0847 & 122.1544 \\ 5 & 1168.4845 & 921.3901 & 822.6043 & 788.0588 \end{matrix}

Table 3.

In the linear case the exact solution is given by 14.6842 [10]. The difference is here 0.1 %. The numerical results for λ_{1} when the domain is bounded by the quadrant of the ”circle” are given in Table 4.

\begin{matrix}p\n & 19 & 61 & 217 & 817 \\ 0.5 & 13.2191 & 12.1671 & 11.9362 & 11.8787 \\ 0.75 & 19.7591 & 18.3118 & 17.9715 & 17.8864 \\ 1 & 29.3712 & 27.1025 & 26.5555 & 26.4198 \\ 2 & 144.0884 & 120.3870 & 115.2347 & 113.9156 \\ 3 & 735.6182 & 504.7015 & 462.9082 & 451.2033 \\ 5 & 18610.8737 & 8298.0772 & 6753.8355 & 6269.8435 \end{matrix}

Table 4.

The known value is 26.3682 for p = 1, and the difference is 0.2 %. The following table gives the relative frequency in certain cases for the gravest

tone of membranes under similar mechanical conditions and of equal area (Table 5.). We can conclude that for any value p from [0.5, 5] the ”circular” membrane has the gravest mode similarly as in the linear case [10].

p  ”circle” square quadrant of ”circle” ”semicircle”
0.5 4.7525 4.8368 4.8929  5.2117
0.75 4.5075 4.6429 4.7297  5.0154
1  4.2629 4.4429 4.5552  4.8039
2  3.5461 3.8389 4.0105  4.1498
3  3.1221 3.4767 3.6738  3.7477
5  2.6546 3.0741 3.2931  3.2961

Table 5.

Acknowledgement 4 This work was supported by the National Research Foundation OTKA T026138. The author is grateful to the referee for his valuable comments and helpful

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