Two new Weyl-type bounds for the Dirichlet Laplacian

In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following {\em lower} bounds for its counting function. For $\la\ge \la_1$, one has N(\la)>\dfrac{2}{n+2} \dfrac{1}{H_n} (\la-\la_1)^{n/2} \la_1^{-n/2}, and N(\la)>(\dfrac{n+2}{n+4})^{n/2} \dfrac{1}{H_n} (\la-(1+4/n) \la_1)^{n/2} \la_1^{-n/2}, where H_n=\dfrac{2 n}{j_{n/2-1,1}^2 J_{n/2}^2(j_{n/2-1,1})} is a constant which depends on $n$, the dimension of the underlying space, and Bessel functions and their zeros.


LOTFI HERMI
Abstract.In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian.As a consequence, we obtain the following lower bounds for its counting function.For λ ≥ λ 1 , one has , where Hn = 2 n j 2 n/2−1,1 J 2 n/2 (j n/2−1,1 ) is a constant which depends on n, the dimension of the underlying space, and Bessel functions and their zeros.

Four New Estimates
Let Ω ⊂ R n be a bounded domain with piecewise smooth boundary.We are interested in bounds for the eigenvalues of the fixed and free membrane whose shape is assumed by Ω.The first problem (also called the Dirichlet problem) is described by the equation, u = 0 on ∂Ω.
Theorem 1.1.For k ≥ 1, we have and the sharper, average-type, inequality k j=1 where . (1.5) As corollaries to these two inequalities, we prove (1.7) and (1.8).Here J n (x) and j n,p denote, respectively, the Bessel function of order n, and the pth zero of this function (see [1]).The proof of this theorem is offered in Section 3.That (1.4) is sharper than (1.3) follows from left Riemann sum considerations (See Fig. 1), namely To see this, we apply (1.3) Inequality (1.4) is tighter since In fact, we have the following corollary.
Corollary 1.2.For k ≥ 1, the eigenvalues of the Dirichlet problem satisfy the estimate This of course follows from the Cauchy-Schwarz inequality and inequality (1.4).Another consequence of (1.4) and H. C. Yang's [52] (see also [3] [4], [15], [16], [17]) inequality is the bound These types of inequalities follow the spirit of Weyl's asymptotic law, which states that where = volume of the unit n−ball, and |Ω| denotes the volume of Ω.These formulas were proved by Weyl [51] in 1910.
There is a beautiful exposé of the history of this problem in Kac's paper [30] (see also the equally entertaining paper [46]).Baltes and Hilf [18] trace the history of this type of asymptotics to Pockels (1891) (who proved the discreteness of the specturm of the Dirichlet Laplacian [19]), Lord Rayleigh (1905), Sommerfeld (1910), and Lorentz (1910).Many asymptotics of this type were developed by Courant and Hilbert [25], Pleijel, and Minakshisundaram (see [18] for further insight and references).
In 1992, Kröger [33] produced the Neumann parallels to the Li-Yau inequalities.For k ≥ 1, he proved and Notice that (1.15) implies that Again, we have used the left Riemann sum comparison (1.6) in these two inequalities.Kröger's inequality (1.14) is tighter, since Thus, the "averaged" version (namely (1.14)) of Kröger's two inequalities is sharper.
Our bounds (1.3) and (1.8) are also related to the result of Ashbaugh and Benguria [9] who proved, for m ≥ 1, Of course, one cannot expect to fare better in the case of m = 1 since this is another conjecture by Payne, Pólya, and Weinberger [40] [41] (herein referred to as PPW) which was settled by Ashbaugh and Benguria [5] (see also [6]) in 1991, namely . (1.17) The ratio on the RHS of (1.17) is that for the two first eigenvalues of an n-ball.It has the asymptotic expansion [10] where c 1 ≈ 1.8557571 and c 2 ≈ 1.033150 (see [1]).Payne, Pólya, and Weinberger [40] [41] (see also [9] [49]) proved the weaker form from which one can infer that Note that (1.16) can be put in the form where [x] stands for the integer part of x.This bound can be thought of as one of the form By virtue of the expansion (1.18), the power (1.23) Thus, while tight at the bottom of the spectrum (viz.(1.17)), (1.16) does not capture the expected Weyl behavior of k 2/n .Inequalities (1.3) and (1.8) remedy this.
The key to the new results is an observation by Ashbaugh and Benguria-the extent and limitations of which are discussed on p. 561 of [11].If one identifies µ k+1 with λ k+1 − λ 1 and |Ω| −2/n with λ 1 , then the RHS of the PPW inequality (1.17) can be seen as maximizing the ratio λ 2 /λ 1 in the same vein as the quantity C 2/n n p 2 n/2,1 maximizes |Ω| −2/n µ 2 for any domain Ω (p ν,k denotes the kth positive zero of the derivative of x 1−ν J ν (x) and C n is as defined above, i.e. the volume of the unit n-ball).The latter is a result of Szegő in 2 dimensions and Weinberger in n dimensions.The maximum for both is assumed when Ω is an n-ball.The strategy of proof for both is similar though the first is considerably more involved [5] [6].This loose analogy can also be seen in comparing the methods of proof and results for n j=1 and both of which were proved by Ashbaugh and Benguria in [10] and [11].Inequality (1.24) is the extension to n dimensions of a result in [41].(Note that (1.25) was proved with the further restriction that Ω is invariant with respect to 90 o rotations in the coordinate planes.)Our new inequality (1.4) can be viewed as an extension for k = n of (1.24).(See Section 4 for a comparison with existing results.) The loose correspondence can also be traced in the analogy between and . (1.27) Both of these bounds are also results found in [10] and [11] (with (1.27) also true under rotational symmetry of the base domain Ω).Inequality (1.26) is an extension and improvement of earlier results of Chiti [24].The n 2 /4λ .
Our new inequality (1.7) can be viewed as an extension, for k = n, of the Ashbaugh-Benguria-Chiti inequality (1.26).
We complete this section by giving the asymptotic expansions for the coefficients appearing in (1.3), and (1.8) (see [10] and [34] for similar estimates).

The Counting Function
One can motivate these inequalities in terms of the counting function, Our Theorem 1.1 can then be restated.
In fact, for λ ≥ λ 2 , Ashbaugh and Benguria have the sharper bound [9] which, in view of the above considerations, reads as .

Proof of Theorem 1.1
We begin with the Rayleigh-Ritz estimate for λ k+1 , where B r = is a ball of radius r ≥ r 0 , and This characterization is suggested by considerations similar to [33].In fact, the bulk of the arguments follow steps described there.The test function φ is required to satisfy It is chosen to be of the form The orthogonality conditions lead to a j (z) = Ω u 1 u j e ix•z dx.We calculate since Ω u j u ℓ = δ jℓ .One has Similarly, Therefore, Orthogonality makes Ω u j φ = 0.
We note that is the volume of the unit n−ball).Moreover, the constant H n defined in (1.5) is given by Remark.Safarov obtained (2.3) using the following result of E. B. Davies [26] ess sup|u 1 | ≤ e 1/8π λ n/4 1 . (3.10) Chiti's statement (3.8) is an isoperimetric inequality.It saturates when Ω is an n−ball.Note that e 1/8π ≈ 1.04059, while the constant in (3.8) takes the values listed in Table 2.
We now prove by induction the following lemma.
Proof.We first note that These two facts reduce (3.7) to For k = 1 It then obtains by virtue of (3.9) that, for r ≥ r 0 (1) = H 1/n n 2 1/n √ λ 1 (this condition guarantees the denominator is positive) as desired.Suppose now that, for r ≥ r 0 (k Then, this is also true for r ≥ r 0 (k) as well (since r 0 (k) > r 0 (k − 1)).This implies (We have used the equivalence α Hence, by virtue of (3.14), and for (3.15)For simplicity we let By (3.12), (3.17) This choice amounts to setting 1 .

HERMI
If we drop the sum in (3.11) and let r = r(k) = we are led to (1.3), namely, .18)This choice amounts to making (Note that r(k) ≥ r 0 (k) since n ≥ 2.) Remark.The case k = 1 in Lemma 3.1 provides a class of bounds for λ 2 − λ 1 for r ≥ r 0 (1).The function  (3.20) are not expected to fare better than the Ashbaugh-Benguria inequality (1.17)-the best constant of its type (see Table 3).In fact, the first has the asymptotic expansion Expanding the second, it obtains (see (1.28) above)

Comparison with Existing Results
Consider the convex function In [35], Laptev proved (see Theo. Combining this with the isoperimetric inequality of Chiti (3.8) gives (3.20).Note that Chiti's inequality (3.8) can be put in the form For λ ≥ λ 1 , Laptev's result (4.1) can also be interpreted as (see Cor. 4.4 of [35]) Again, bounding ũ1 using Chiti's isoperimetric inequality (4.3) yields the statement (1.3).In fact, one can write (4.1) in the form  [36]) that where [p] designates the integer part of p.The Legendre transform of the right hand side of (4.1) is given by   Now, we turn to comparing these bounds.We claim that (1.8) is sharper than (1.3).To see this, we take the limit of the ratio of bounds as k → ∞.This limit is equal to It is strictly less than 1 since 1 + 4/n < 1 + n/2 < (1 + n/2) 1+2/n , for n ≥ 3.This limit is equal to 3/4 at n = 2.That both (1.3) and (1.8) are sharper than (1.21) (in the form (1.22)) follows from Krahn's second inequality [32] (see in particular ineq.(22)  (see also (1.23) above).This is clearly displayed in Fig. 2 where the Ashbaugh-Benguria bound fares better to about k = 20.The "averaged" bound (1.8) takes over and-at a latter stage-so does (1.3).Three tables are included in this paper which display this fact as well (see Tables 3-5).The new inequalities-both of which disguised in earlier work of Laptev-cannot be expected to improve on existing bounds in the case of λ 2 /λ 1 .There is a competition (see Table 4) in the case of λ 32 /λ 1 between (1.3) and (1.16) (already (1.8) is better than both for n ≥ 3).In the case of λ 128 /λ 1 , both new bounds show considerable improvement (see Table 5).

Figure 2 .
Figure 2. Comparison of New and Old Bounds.
1term in(1.26)iswhat corresponds to (1.24) via the "usual Cauchy-Schwarz connection" (viz.the proof of Cor.1.2).On the other hand, there is also a conjectured inequality, from which, if proved, (1.26) would follow via the Cauchy-Schwarz argument.That inequality would be (1.24) but with its RHS replaced by

Table 2 .
Values of the constant in Chiti's bound (3.8) as a function of the dimension n.Lemma 3.1.For r ≥ r 0

Table 5 .
Bound forλ 128 λ 1as a function of the dimension n.