aa r X i v : . [ m a t h . P R ] N ov Front. Math. China 2011, 6(6): 1025-1043
General estimate of the first eigenvalue on manifolds
Mu-Fa Chen(Beijing Normal University, Beijing 100875, China)December 30, 2010
Abstract
Ten sharp lower estimates of the first non-trivial eigenvalue of Lapla-cian on compact Riemannian manifolds are reviewed and compared. Animproved variational formula, a general common estimate, and a newsharp one are added. The best lower estimates are now updated. Thenew estimates provide a global picture of what one can expect by ourapproach.
Let M be a compact, connected Riemannian manifold, without or with convexboundary ∂M . When ∂M = ∅ , we adopt Neumann boundary condition. Next,let Ric M > K for some K ∈ R . Denote by d and D , respectively, the dimensionand diameter of M . We are interested in the estimate of the first non-trivialeigenvalue λ of Laplacian. On this topic, there is a great deal of publications(see for instance [5], Schoen and Yau [15], Wang [16], and references therein).Throughout this paper, we use the quantity α = α ( K, d, D ) = D r − Kd − , which involves all of the three geometric quantities: d , D and K . Clearly, | α | = α ( | K | , d, D ). By the Myers theorem, we have | α | π/ K > α isgeometrically meaningful. We adopt the convention: α = 0 if d = 1. With the Mathematics Subject Classifications . 58C40, 35P15.
Key words and phases . First non-trivial eigenvalue, sharp estimate, Riemannian man-ifold.Research supported in part by the National Natural Science Foundation of China (Nos.10721091, 11131003), by the “985” project from the Ministry of Education in China.
Mu-Fa Chen necessary notation in mind, the main result Theorem 1.4 and its illustratingfigures 7–9 should be readable now. One may have a look before going further.We are going to recall the sharp lower estimates, only the related part ofthem for saving space. First, for nonnegative curvature, the following sharplower bounds are perhaps well known. • Lichnerowicz (1958): dd − K = 4 dD | α | , d > , K > . (1) • B´erard, Besson and Gallot (1985): d (cid:26) R π/ cos d − t d t R D/ cos d − t d t (cid:27) /d = d (cid:26) R π/ cos d − t d t R | α | cos d − t d t (cid:27) /d , K = d − > . (2)Here and in what follows, cos k t = (cos t ) k . • Chen and Wang (1997): dK ( d − − cos d | α | ) = 4 d | α | D (1 − cos d | α | ) , d > , K > . (3) • Zhong and Yang (1984): π (cid:14) D , K > . (4)It is clear that (2) improves (1) by the Myers theorem. Even though it is notso obvious but it is true that (3) improves (2). The first three results indicatea long period for the improvements of (1) step by step. All of them are sharpfor the unit sphere in two or higher dimensions but fail for the unit circle( K = 0). To which, the sharp estimate is given by the last result (4). Notethat when | α | ↓
0, the limit of (3) equals 8 D − ∈ (0 , π D − ).Secondly, consider the non-positive curvature in which case, the problembecomes harder. Here are the main known sharp lower bounds. • Yang (1990), Jia (1991), Chen and Wang (1994): π D e − ( d − α , K . (5) • Chen and Wang (1997):1 D p π + 8( d − α cosh − d α, d > , K . (6) eneral estimate of the first eigenvalue on manifolds • [2; 1994, Theorem 6.6](corrected version):1 D (cid:0) ( d − α tanh α sech θ (cid:1) , d > , K , (7)where θ is obtained in the following way. Let θ = 2 − ( d − α tanh α, θ n = θ tanh θ n − , n > , then θ n ↓ θ .The first two results have the same decay rate but (6) is better than (5) ingeneral. They are sharp for the unit circle but not the last result which isdesigned for large α . The last two estimates are not comparable. For fixed d ,(6) is better than (7) for smaller α but the inverse assertion happens for large α (cf. Fig. 8 below).Thirdly, consider the optimal linear approximation of the lower estimateswith respect to the curvature K . In other words, one looks for a good combi-nation of the optimal estimates (1) and (4). Many authors have contributedto the result (8) below. It was proved by Zhao (1999) under the restriction − π / (3 D ) K K
0. The case of
K > K in Chen, Scacciatelli and Yao (2001) (where some refined es-timates are included). A more direct analytic proof with some improvementwas given by Shi and Zhang (2007). Recently, the result (8) below has beenreproved by Ling (2006, 2007) using a different approach. • The following lower bound is studied/proved in the papers just men-tioned: π (cid:14) D + K/ , K ∈ R . (8) • More precisely, the lower bound given by Shi and Zhang (2007) is asfollows: sup s ∈ (0 , s (cid:20) − s ) π (cid:14) D + K (cid:21) = (cid:18) πD + KD π (cid:19) , − π KD π K, KD ∈ (cid:0) π , ( d − π (cid:3) , KD < − π (9) • A refined lower bound given by Chen, Scacciatelli and Yao (2001) is thefollowing: π D + K (cid:0) − π (cid:1) K D , | K | D . (10) Mu-Fa Chen
To see that (9) improves (8), simply set s = 1 /
2. On the region { ( K, D ) : | K | D } , it is obvious that (10) is better than (9). Unlike the results (1)–(7), the boundsgiven in (8)—(10) are independent of the dimension d .Before moving further, let us make some remarks on (10). Since one isseeking for the dimension-free estimate and( d − α tanh( αr ) x − KD r/ − βr as d ↑ ∞ Figures 1–3
The first eigenvalue λ corresponding to { β n } n =2 (the topcurve), { β n } n =20 (the middle curve), and { β n } n =2 (the bottom curve),respectively.(cf. proof of Theorem 1.2 below), the study on λ can be reduced to studythe first eigenvalue (say λ , for a moment) of the operator d / d r − βr d / d r on (0 ,
1) (cf. [7; Lemma 2.3] in which the constant α is replaced by β here).This now becomes a one-parameter problem and the required estimate (10)becomes a product of 4 D − and π / β + (10 − π ) β , | β | / λ from below. For a sequence of β , say { β n } ∞ n =2 with β = 1 / β n ↓ n → ∞ , the problem is numerically solvable and moreover, we eneral estimate of the first eigenvalue on manifolds { β n } . Actually, the correspondingeigenfunctions are all polynomials (cf. [7; Lemma 2.3]. Now, (10) is designedto be exact at β = 0 and β = 1 / K to be 1 / λ just mentioned corresponding to { β n } is shown by threepictures for different region of n (Figures 1–3), and then the difference betweenthe eigenvalue λ and its lower estimate (11) is shown by two pictures (Figures4 and 5). In Fig.3, even though the whole sequence { β n } n =2 is computed, butthe output is restricted to a smaller interval near zero. To see the other partof the interval, one needs two more figures (1 and 2). ´ - ´ - ´ - ´ - ´ - ´ - ´ - ´ - ´ - ´ - ´ - Figure 4–5
The difference of the first eigenvalue λ and its lower estimate(11) corresponding to { β n } n =2 (the curve on right) and { β n } n =2 (the curveon left), respectively.The difficulty is that for each n >
2, one has to find a root of a polynomialhaving order n −
1. Here, instead of an analytic proof, we have used five figuresto show carefully that the lower bound (10) is rather sharp and the coefficient1 / {− β n } ∞ n =1 (cf. [7; Lemma 2.4]). Therefore, (10) holds for allreal K with | K | D K = 0), some of them can be very poor in some region.For instance, (1)—(3) are meaningless when K = 0 and (8)—(10) can benegative or zero for sufficiently large − K . The question now is the existenceof a universal estimate. Fortunately, the answer is affirmative as shown byProposition 1.1 below. Its first assertion implies the lower bounds (1)–(10), interms of ¯ λ . Mu-Fa Chen
Recall that even though α is an imaginary number when K >
0, thequantities tanh( αr ) and cosh( αr ) used below are still meaningful: cosh( iθ ) =cos θ , tanh( iθ ) = i tan θ for real θ . Proposition 1.1
We have λ > λ/D with the following estimates: δ δ ∧ δ ∗ ¯ λ δ ′ ∨ δ ∗ ′ δ , where δ = sup r ∈ (0 , [ ϕψ ]( r ) ,δ = sup r ∈ (0 , (cid:26) p ϕ ( r ) Z r Cϕ / + p ϕ ( r ) Z r Cϕ / (cid:27) ,δ ′ = sup r ∈ (0 , (cid:26) ϕ ( r ) Z r Cϕ + [ ϕψ ]( r ) (cid:27) ,δ ∗ = sup r ∈ (0 , (cid:26) p ψ ( r ) Z r C − ψ / + p ψ ( r ) Z r C − ψ / (cid:27) ,δ ∗ ′ = sup r ∈ (0 , (cid:26) ψ ( r ) Z r C − ψ + [ ϕψ ]( r ) (cid:27) (here the Lebesgue measure “ d u ” is omitted) with C ( s ) = cosh d − ( αs ) , ϕ ( r ) = Z r C ( u ) − d u, ψ ( r ) = Z r C ( u )d u. All of the quantities used here depend on d and α . To illustrate the power of Proposition 1.1, consider the simplest case that K = 0. Then δ = 1 / δ = δ ∗ = 5 / / ≈ . δ ′ = δ ∗ ′ = 3 /
8, and so δ δ ′ = δ ∗ δ ∗ ′ = 5 / (cid:30) ≈ . . The sharp estimate for λ is π /D and then ¯ λ − = 4 /π ≈ . δ = 0 . < δ ′ = δ ∗ ′ = 0 . < ¯ λ − ≈ . < δ = δ ∗ ≈ . < δ = 1 . For this example, the results that δ = δ ∗ and δ ′ = δ ∗ ′ are quite naturalby symmetry. The not so obvious fact is δ > δ ∗ in the most cases and theinequality can happen, as shown by numerical computations.Actually, Proposition 1.1 is deduced from the next result which is an im-provement of the main variational formula [9; Theorem 1]. eneral estimate of the first eigenvalue on manifolds Theorem 1.2
We have λ > λ/D and two variational formulas: ¯ λ = sup f ∈ F inf r ∈ (0 , f ( r ) R r C ( s ) − d s R s C ( u ) f ( u )d u = sup f ∈ F inf r ∈ (0 , f ( r ) R r C ( s )d s R s C ( u ) − f ( u )d u , where F = (cid:8) f ∈ C [0 ,
1] : f | (0 , > (cid:9) . Recall that the estimates given in (8)—(10) are all dimension-free. Thenext result is an improvement of (9) which depends on dimensions.
Proposition 1.3
For λ , we have lower bound (9) replacing K by KM α , where M α = π Z (1 − y ) cos πy ( αy )d y regarding α as the constant α if K > and | α | ∈ ( | α | , π/ , where | α | (depending on d ) is the first positive root of (cid:18) π √ d − | α | + √ d − | α | π (cid:19) cos | α | = 1 . (12)In Proposition 1.3, the improvement of (9) is due to the fact that M α < K < M α > K >
0. The proposition reduces to (9) once d → ∞ since then α →
0, and furthermore, M = 1. Here is the picture of M α (Figure6). - - - - - Figure 6
The curve of M α with α = p − sgn( x ) | x | , x ∈ ( − , π/ K < x <
Mu-Fa Chen
As mentioned in the earlier publication (cf. [5; Chapter 3] for instance) orin Theorem 1.2, our study on λ consists of two steps. The first one is reducingthe higher dimensions to dimension one as shown by the first assertion ofProposition 1.1 or Theorem 1.2. The second step is estimating ¯ λ . This wasstarted by [9], continued by several papers mentioned before (8), and is alsothe main aim of the present paper to justify the power of our one-dimensionalresults [3], [4] and [6]. The lower bound δ − ∨ δ ∗ − of ¯ λ provided by Proposition1.1 is universal in the sense that the upper and lower bounds of ¯ λ are thesame up to a factor 4. Actually, all of a large number of examples we haveever computed, as well as Figures 7–9 below, show that the ratio δ /δ ′ (and δ ∗ /δ ∗ ′ ) is no more than 2. It is somehow unexpected that the lower bound4 D − δ ∗ − of λ is better than the others except in two cases (cf. Figures 7–9below). In the case that K = 0, we have shown after Proposition 1.1 by anexample that 4 D − δ ∗ − is not sharp. When α closes to π/
2, we are near theunit sphere and so 4 D − δ ∗ − can not be better than (3). Thus, we may regard4 D − δ ∗ − (be careful to distinguish δ ∗ and δ ) as our general common lowerbound, and regard (3), (10) and Proposition 1.3 as an addition. We can nowsummarize the main result of the paper as follows. Theorem 1.4
In general, we have the following lower estimate: λ > D (cid:26) δ ∗ _ sup s ∈ (0 , s (cid:20) (1 − s ) π − ( d − α M α (cid:21)_(cid:20) { K> } d | α | − cos d | α | (cid:21)(cid:27) , where x ∨ y = max { x, y } , δ ∗ and M α are given in Propositions 1.1 and 1.3,respectively, with a restriction on | α | for the middle term in the case of K > :regarding α as the constant α if | α | ∈ ( | α | , π/ , where | α | is the first positiveroot of (12). Besides, we also have the dimension-free lower bound (10). Themiddle estimate is better than (10) if d , and conversely if d > . For the convenience of computation, according to (9), we express the mid-dle term in Theorem 1.4 as follows.sup s ∈ (0 , s (cid:2) (1 − s ) π − ( d − α M α (cid:3) = (cid:0) π/ − ( d − α M α / (2 π ) (cid:1) , − π − ( d − α M α π KM α , − ( d − α M α ∈ (cid:0) π , ( d − π / (cid:3) , − ( d − α M α < − π . However, when
K > | α | is essentially restricted to the subinterval (0 , | α | ),where | α | is the smallest positive root of equation (12).Fig. 7 illustrates the meaning of Theorem 1.4 ignoring the common factor4 D − . Here, we consider only K > d = 2. Clearly, ¯ λ is located betweenthe two dotted curves and the ratio of the upper and lower bounds (cid:0) δ ∗ ′ − and eneral estimate of the first eigenvalue on manifolds δ ∗ − (cid:1) of ¯ λ is obviously less than 2. Note that here we use ∗ twice. The curve,say Curve 1, corresponding to the last term in Theorem 1.4 is sharp at π/ | α | is close to π/
2, which is impossible since Curve 1 is sharp at π/ | α | for Curve 2 is necessaryand what we adopted is | α | .
97 (that is ignoring the dashed part of Curve2). Here for Curve 2, we omit the constant line starting from the endpoint ofthe non-dashed part of the curve to π/ , π/ δ > δ ∗ . The exceptional is that here δ /δ ∗ ∈ (0 . , | α | < .
87. Since this part is covered by Curve 2 and so δ is ignored inTheorem 1.4. Figure 7
The estimates of ¯ λ in the case of K > d = 2, | α | π/ K d = 2. Again, the upper andlower bounds (cid:0) δ ′ − (but not δ ∗ ′ − ) and δ ∗ − (cid:1) of ¯ λ are given by the two of topdotted curves. The solid curve is determined by the middle term of Theorem1.4. The figure shows that the bounds δ ′ − and δ ∗ − are rather good, evencoincide each other for larger α . In a small interval around 0, they are lesssharper than the solid curve. The dashed curve corresponds to (6) and dottedpart of the triangle corresponds to (7). They are not comparable, and are lesspowerful than at least one of the others and so are disappeared in Theorem1.4. For d >
2, the picture is similar but each curve decays fast (cf. Fig. 9).Theorem 1.4 is stated unified in K ∈ R . Fig. 9 is the result for bothnegative and positive K , where α = p − sgn( x ) | x | with x varies from − . π/ d = 5. Note that the axes in these figures have different scales. Themeaning of each curve should be clear. The top dotted curve is δ ′ − for x < δ ∗ ′ − for x >
0. The part of the curves near π/ Mu-Fa Chen missed part is very much imaginable, up to 14.75 high.
Figure 8
The estimates of ¯ λ in the case of K d = 2, 0 α - - Figure 9
The estimates of ¯ λ when d = 5 with α = p − sgn( x ) | x | , x ∈ ( − . , π/ Remark 1.5
Figures 7–9 show that the ratio δ ∗ (cid:14) δ ∗ ′ ∈ [1 ,
4] (as well as δ ∗ ′− − δ ∗ − ) is controlled by its value at the endpoint | α | = π/
2. The ratio increasesquickly from 1 . .
27 and then slowly to 1 .
334 ( <
2) when d varies from 2to 5 and then to 63. eneral estimate of the first eigenvalue on manifolds Remark 1.6
Consider the convex mean η z = γ z δ ∗ ′ − + (cid:0) − γ z (cid:1) δ ∗ − with γ = 5 / − · − π / − · − ≈ .
39 or γ π/ = 4 − dπ − δ ∗ − δ ∗ ′ − − δ ∗ − (cid:12)(cid:12)(cid:12)(cid:12) | α | = π/ . The latter one depends on d but the former one does not. Here η is designedto be sharp (= π /
4) at α = 0 and so is η π/ (= d π /
4) at | α | = π/ K > γ π/ ≈ .
35 if d = 63 . Numerical computations (illustratingfigures are given in the author’s homepage) exhibit the following unexpectednice conclusion:¯ λ − . η π/ ¯ λ η ¯ λ + 1 . , d , ∀ α. Thus, one may regard η and η π/ (for each fixed d ) as upper and lower boundsof ¯ λ , respectively, but they are almost the same since ¯ λ ≈
155 when d = 63.This illustrates the power of Proposition 1.1, and is independent of (1)–(10). Proof of Theorem 1.2 . (a) Consider first the case that ¯ λ is defined by itsfirst equality given in the theorem. The first assertion is a comparison of λ with the principal eigenvalue λ of the operator L = d d r + ( d − α tanh( αr ) dd r on (0 ,
1) with Dirichlet boundary at 0 and Neumann boundary at 1. This wasdone in [8], as explicitly pointed out by [9; Remark d) after Theorem 1.1].Actually, the result came out by using the coupling method to a computationof some distance which is certainly valued in the half-line. This reduces thehigher dimensions to dimension one. Then it was proved in [9; Theorem 1.1]that λ > ¯ λ . The equality here holds because of [4; Theorem 1.1] noting thatone allows the test function f to be positive at origin not necessarily zero (cf.[4; (1.3)]).(b) Next, let λ ∗ be the principal eigenvalue of the dual operator L ∗ = d d r − ( d − α tanh( αr ) dd r with Neumann boundary at 0 and Dirichlet boundary at 1. Here and in whatfollows, the notation “ ∗ ” is used for dual quantity. Then, we have not only λ = λ ∗ but also the second equality for ¯ λ given in the theorem, as an analogof [6; § (cid:3) Proof of Proposition 1.1 . The first assertion comes from Theorem 1.2.The result is very helpful since the parameter D is separated out, and thenthe three parameters d , D , and K are now reduced to two: d and α .2 Mu-Fa Chen
The “ δ part” of the assertion was presented in [5; Corollary 1.4], comesoriginally from [3; Theorem 2.2] with a change of the variable: r → r/D reduc-ing the interval (0 , D ) to (0 , δ and δ ′ parts” have not yetpublished in the geometric context. However, it is indeed the first step of ageneral approximating procedure for the first eigenvalue, given by [4; Theorem2.2]. Here we state the procedure in the present context. Define f = √ ϕ , f n +1 ( r ) = Z r C ( s ) − d s Z s C ( u ) f n ( u )d u, n > δ n = sup r ∈ (0 , f n +1 ( r ) /f n ( r ). Then¯ λ > . . . > δ − n > δ − n − > . . . > δ − > (4 δ ) − . Next, fix r ∈ (0 ,
1) and define f ( r )1 = ϕ ( · ∧ r ), f ( r ) n +1 = Z •∧ r C ( s ) − d s Z s C ( u ) f ( r ) n ( u )d u, n > δ ′ n = sup r ∈ (0 , inf s ∈ (0 , f ( r ) n +1 ( s ) (cid:14) f ( r ) n ( s ). Then¯ λ . . . δ ′ n − δ ′ n − − . . . δ ′ − δ − . Besides, we also have ¯ λ ¯ δ − n for all n , where¯ δ n = sup r ∈ (0 , Z f ( r ) n ( s ) C ( s )d s (cid:30) Z (cid:0) f ( r ) n ( s ) (cid:1) ′ C ( s )d s, n > . Moreover, ¯ δ = δ ′ . All of these approximating procedure comes from somevariational formulas [4; Theorem 2.1]. In particular, the sequence { δ n } n > comes from the first variational formula given in Theorem 1.2.The “ δ ∗ and δ ∗ ′ parts” are parallel to the “ δ and δ ′ parts” based on thedual formula just mentioned, as an analog of [6; Theorem 3.2 and Corollary3.3]. The details are omitted here. We mention that the duality is studiedmore carefully in the author’s forthcoming paper entitled “Basic estimates ofstability rate for one-dimensional diffusions”.Clearly, the estimates given in Proposition 1.1 can be still improved byusing the above approximating procedure. (cid:3) To prove Proposition 1.3, we need some preparations. The following niceresult it due to [14; proof of Lemma 2.2], it is the key leading to (9). As usual,we use C m to denote the set of functions having continuous m -th derivative. eneral estimate of the first eigenvalue on manifolds Lemma 2.1
Let λ and f satisfy f ′′ + F f ′ = − λf on [0 , ℓ ] , ℓ < ∞ with boundary conditions f (0) = 0 and f ′ ( ℓ ) = 0 , where F ∈ C [0 , ℓ ] ∩ C (0 , ℓ ) having F (0) = 0 . Then for each s ∈ (0 , , the function g := ( f ′ ) − (1 − s ) − satisfy s (1 − s ) Z ℓ g ′ d r = Z ℓ ( λ + sF ′ ) g d r. Moreover, g satisfies the mixed boundary condition: g ′ (0) = 0 and g ( ℓ ) = 0 . The proof of Lemma 2.1 goes as follows. Making derivatives of the originalequation, we get f ′′′ + F f ′′ = − ( λ + F ′ ) f ′ . Regarding f ′ as a new function h , one sees that this corresponds to a secondorder Schr¨odinger operator (since the appearance of F ′ ). This step is known asan application of the coupling technique (cf. [6; (10.4), (10.7)] and referencestherein). The next step we have used is to adopt the dual operator d / d x − F d / d x (cf. [6; (10.6)]) which is isospectral to the Schr¨odinger one. Refer to [6;proof of Theorem 10.2] for more details. However, here the idea is different.Set t = (1 − s ) − . Multiply the last equation by ( f ′ ) t − and then integrateeach term in the last equation over (0 , ℓ ). Based on the fact that F (0) = 0 andthen f ′′ (0) = 0, some careful computations lead to the required conclusion.Having Lemma 2.1 at hand, the assertion (9) is immediate. Simply replace λ + sF ′ by the constant λ + s sup r ∈ (0 ,ℓ ) F ′ ( r ) (which then does not depend onthe dimension d explicitly) and use the known exact inequality (cf. Proof ofProposition 1.2 below) Z ℓ g d r (cid:18) ℓπ (cid:19) Z ℓ g ′ d r. It is at this step, the original proof is improved. This leads to a use of thenext result.
Lemma 2.2
Let a > on (0 , and a ∈ C (0 , . Next, let ˆ λ be the principaleigenvalue of the operator b L = a ( x ) − d (cid:14) d x on (0 , with Neumann boundaryat and Dirichlet boundary at . Then ˆ λ obeys the following estimates inf x ∈ (0 , h ( x ) − ˆ λ sup x ∈ (0 , h ( x ) − , where h ( x ) = (cid:18) π (cid:19) (cid:26) a ( x ) − sec πx (cid:20) (1 − x ) a ′ (0) + (1 − x ) Z x a ′′ ( y ) cos πy y + Z x (cid:2) − a ′ ( y ) + (1 − y ) a ′′ ( y ) (cid:3) cos πy y (cid:21)(cid:27) . Mu-Fa Chen
Proof . We are now in the case which is the continuous analog of [6; § a ( x ) is a constant, the assertion of the lemma is exact. Inthis case, the eigenfunction is cosine and so our approximation should start atthe function cosine, because we are looking for such estimates which are sharpwhen a ( x ) is a constant. Define f ( x ) = cos πx and f n ( x ) = Z x d y Z y a ( u ) f n − ( u )d u, x ∈ (0 , , n > . Here, the proof of the lemma is mainly for simplifying f /f and so is ratherelementary. Exchanging the integrals, we get f ( x ) = Z x d y Z y a ( u ) cos πu u = Z (1 − x ∨ u ) a ( u ) cos πu u = (1 − x ) Z x a ( y ) cos πy y + Z x (1 − y ) a ( y ) cos πy y. As an application of the integration by parts formu1a, we have f ( x ) = 2 π (cid:20) (1 − x ) Z x a ( y ) d sin πy Z x (1 − y ) a ( y ) d sin πy (cid:21) = − π (cid:20) (1 − x ) Z x a ′ ( y ) sin πy y + Z x (cid:2) − a ( y ) + (1 − y ) a ′ ( y ) (cid:3) sin πy y (cid:21) . Using the integration by parts formu1a again, we get f ( x ) = (cid:18) π (cid:19) (cid:26) a ( x ) cos πx − (1 − x ) a ′ (0) − (1 − x ) Z x a ′′ ( y ) cos πy y − Z x (cid:2) − a ′ ( y ) + (1 − y ) a ′′ ( y ) (cid:3) cos πy y (cid:27) . Unlike the original one, no double integral appears in the last formula. Obvi-ously, f ( x ) /f ( x ) = h ( x ). The required assertion now follows by [4; Theorem1.1]. (cid:3) Proof of Proposition 1.3 . Applying Lemma 2.1 to L and ¯ λ defined in theproof of Theorem 1.2 with F ( r ) = ( d − α tanh( αr ) , we obtain4 s (1 − s ) Z g ′ d r = Z (cid:2) ¯ λ + s ( d − α sech ( αr ) (cid:3) g d r = ¯ λ Z g d r + s ( d − α Z sech ( αr ) g d r. (13) eneral estimate of the first eigenvalue on manifolds K a ( r ) = sech ( αr ). Then a > ,
1] and a ′ ( r ) = − α sech ( α r ) tanh( α r ) ,a ′′ ( r ) = − α sech ( α r ) [2 − cosh(2 α r )] . Applying Lemma 2.2 to this a ( r ), we obtain, replacing h by h α , that π h α ( x ) = sech ( αx ) + 2 α sec πx (cid:20) α (1 − x ) Z x q α ( y ) cos πy y + Z x (cid:2) − p α ( y ) + α (1 − y ) q α ( y ) (cid:3) cos πy y (cid:21) , where p α ( y ) = sech ( αy ) tanh( αy ) and q α ( y ) = sech ( αy ) (cid:2) − cosh(2 αy ) (cid:3) . Clearly, we have π h α (0) = 1 + 2 α Z (cid:2) − p α ( y ) + α (1 − y ) q α ( y ) (cid:3) cos πy y. (14)Note that α > K < x ∈ (0 , h α ( x ) = h α (0) (see part(c) of the proof below). Now, as an application of Lemma 2.2, we have Z sech ( αr ) g d r h α (0) Z g ′ d r, K . In particular, letting α ↓
0, it follows that Z g d r π Z g ′ d r. By Lemma 2.2 again, we have for all α > α Z sech ( αr ) g d r α h α (0) Z g ′ d r, and then by (13), π s (1 − s ) ¯ λ + s ( d − α π h α (0) = ¯ λ − s KD π h α (0) . (15)Noticing that among q α , p α , sech ( αr ), and tanh( αr ), the former comes fromthe derivative of the latter, starting from (14), by using the integral by partsformula three times, one finally sees that π h α (0) = π Z (cid:18) cos πy π − y ) sin πy (cid:19) tanh( αy ) α d y. Mu-Fa Chen
The right-hand side gives us M α in terms of the integration by parts formula(sin = − d cos). Applying this to (15), Proposition 1.3 now follows from thefirst assertion of Proposition 1.1 in the case of K K > a ( r ) = ¯ λ + s ( d − α sech ( αr ) = ¯ λ − s ( d − | α | sec ( | α | r ) . In other words, we do not separate this a into two parts as in (13). By (9), inorder that a > , (cid:18) π √ d − | α | + √ d − | α | π (cid:19) cos | α | > . Once the assertion is proved under this assumption, one may replace “ > ” herewith “ > ” by a limiting procedure. For the other part of α , simply regard α as the constant α (or α − ε if necessary) since then a ( r ) is upper boundeduniformly in r by a positive constant on that subinterval of α . We remarkthat even though the restriction here can be relaxed a little we do not do sosince the estimate is mainly essential for small K (or for small | α | ). ApplyingLemma 2.2 to this a , we obtainˆ λ − sup x ∈ (0 , h ( x ) : h ( x ) = 4¯ λ/π + s ( d − α h α ( x ) , where h α is the same as used in proof (a). Note that in the present case, wehave α < x ∈ (0 , h ( x ) = 4¯ λ/π + s ( d − α inf x ∈ (0 , h α ( x ) . This is the main different point to the previous case of K
0. Luckily, wethen have inf x ∈ (0 , h α ( x ) = h α (0) (see part (c) of the proof below). In thiscase, since α = i | α | , it may be more convenient to rewrite h α as π h α ( x ) = sec ( | α | x ) − | α | sec πx (cid:20) | α | (1 − x ) Z x q + α ( y ) cos πy y + Z x (cid:2) − p + α ( y ) + | α | (1 − y ) q + α ( y ) (cid:3) cos πy y (cid:21) , where p + α ( y ) = sec ( | α | y ) tan( | α | y ) and q + α ( y ) = sec ( | α | y ) (cid:2) − cos(2 | α | y ) (cid:3) . From this, one sees that h α is always real for any K ∈ R . The remainder ofthe proof is similar to proof (a) above.(c) To see that sup x ∈ (0 , h α ( x ) = h α (0) when K x ∈ (0 , h α ( x ) = h α (0) when K >
0, it is helpful to look at first three figures (Figures 10–12) eneral estimate of the first eigenvalue on manifolds x = 0 for each fixed | α | ,so two more figures (11 and 12) are included. The pictures in case of K (cid:3) Once again, as indicated in the proof of Lemma 2.2, Proposition 1.3 usesonly the first step of our general approximating procedure for the first eigen-value ˆ λ . Figure 10
When K >
0, the surface of π h α ( x ) / | α | ∈ (0 , . x ∈ (0 , Figure 11–12
The curve of π h α ( x ) / x ∈ (0 ,
1) when K > | α | = 0 . | α | = 1 .
57, respectively.8
Mu-Fa Chen
To conclude the paper, we make some remarks about the methods used inthe paper. Recall that all of the results (3), (5)–(8), and (10) are an applicationof a coupling to a carefully designed (case by case) distance (equivalent tothe Riemannian one). As mentioned in [2; Theorem 6.2], the method worksfor more general “cost” functions, not necessarily a distance. With couplingmethod in mind, the boundaries of the reduced process with operator L givenin the proof of Theorem 1.2 is natural. Then the “ δ ” (resp. “ δ ′ ”) part ofProposition 1.1 says that we do have a universal distance which is equivalentto the Riemannian one and provides us a universal lower bound 4 D − δ − .Thus, all of these results can be regarded as an application of the generalvariational formula given in [9]. However, for part “ δ ∗ ”, as a dual of L , it hasa different probabilistic meaning. The use of the dual technique is an essentialnew point of the present paper. Finally, in Lemma 2.1 (or Proposition 1.3), theoriginal boundary conditions are also dualled, at the same time, the operatoris changed. It is mainly a specific comparison result, one can not say that thetwo operators used in Lemma 2.1 have the same principal eigenvalue. Acknowledgments . The main results of the paper have been presented at “Inter-national Conference on Stochastic Partial Differential Equations and Related Topics(April 25 - 30, 2011; Chern Institute of Math.)” and at “Stochastic Analysis and Ap-plication to Financial Mathematics, in honor of Professor Jia-An Yan’s 70th Birthday(July 4–6, 2011; AMSS, CAS)”. The author acknowledges the organizing groups fortheir kind invitation and financial support.
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