On the Inner Radius of Nodal Domains

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1 arxiv:math/ v1 [math.sp] 14 Nov 2005 On the Inner Radius of Nodal Domains Dan Mangoubi Abstract Let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λ α (log λ) β. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincaré type inequality proved by Maz ya. Sharp lower bounds are known only in dimension two. We give an account of this case too. 1 Introduction and Main Results Let (M,g) be a closed Riemannian manifold of dimension n. Let be the Laplace Beltrami operator on M. Let 0 < λ 1 λ 2... be the eigenvalues of. Let ϕ λ be an eigenfunction of with eigenvalue λ. A nodal domain is a connected component of {ϕ λ 0}. We are interested in the asymptotic geometry of the nodal domains. In particular, in this paper we consider the inner radius of nodal domains. Let r λ be the inner radius of the λ-nodal domain U λ. Let C 1,C 2,... denote constants which depend only on (M,g). We prove Theorem 1. Let M be a closed Riemannian manifold of dimension n 3. Then C 1 C 2 r λ λ λ k(n) (log λ) n 2, where k(n) = n 2 7n/4 1. In dimension two we have the following sharp bound Theorem 2. Let Σ be a closed Riemannian surface. Then C 3 λ r λ C 4 λ. 1

2 1.1 Upper Bound We observe that λ = λ 1 (U λ ). This is true since the λ-eigenfunction does not vanish in U λ ([Cha84], ch. I.5). Therefore, the existence of the upper bound in Theorems 1 and 2 follows from the following general upper bound on λ 1 of domains Ω M. Theorem 3. λ 1 (Ω) C 5 inrad(ω) 2. The proof of this theorem is given in Lower Bound For the lower bound on the inner radius in dimensions 3, we give a proof in 2.1 which is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman (Theorem 4). The same proof gives in dimension two the bound C/ λlog λ. In order to get rid of the factor log λ in dimension two, we treat this case separately in 2.2. The proof for this case can basically be found in [EK96], and we bring it here for the sake of clarity and completeness. Also, for the dimension two case we bring a new proof in 3. This proof is due to F. Nazarov, L. Polterovich and M. Sodin and is based on complex analytic methods. 1.3 Recent Works We would like to mention the recent work of B. Xu [Xu05], in which he obtains a sharp lower bound on the inner radius for at least two nodal domains, and the work of V. Maz ya and M. Shubin [MS05], in which they give sharp bounds on the inner capacity radius of a nodal domain. 1.4 Acknowledgements I am grateful to Leonid Polterovich for introducing me the problem and for fruitful discussions. I would like to thank Joseph Bernstein, Lavi Karp and Mikhail Sodin for enlightening discussions. I am thankful to Sven Gnutzmann for showing me the nice nodal domains pictures he generated with his computer program and for nice discussions. I 2

3 owe my gratitude also to Moshe Marcus, Yehuda Pinchover and Itai Shafrir for explaining to me the subtleties of Sobolev spaces. I would like to thank Leonid Polterovich, Mikhail Sodin and Fëdor Nazarov for explaining their proof in dimension two to me, and for letting me publish it in 3. 2 The Lower Bound on the Inner Radius In this section we prove the existence of the lower bounds on the inner radius given in Theorems 1 and Lower Bound in Dimension 3 In this section we prove the existence of the lower bound in Theorem 1. The proof also gives a bound in the case where dimm = 2, namely r λ C/ λlog λ, but in the next section we treat this case separately to get rid of the log λ factor. Let {σ i } be a finite cellulation of M by cubes, such that for each i we can put a Euclidean metric e i on σ i, which satisfies e i /4 g 4e i. Let r λ,i be the inner radius of U λ,i = U λ σ i, and r λ,i,e be the Euclidean inner radius of U λ,i. Notice that r λ,i,e 2r λ,i 2r λ. (1) Step 1. (See Fig. 1). We consider σ i as a compact cube in R n. We cover σ i by non-overlapping small cubes with edges of size 4h, where r λ,i,e < h < 2r λ,i,e. Let be a copy of one of these small cubes. Let be a concentric cube with parallel edges of size 2h. Step 2. We note that each copy of contains a point p \ U λ. Otherwise, we would have r λ,i,e h, which would contradict the definition of h. Step 3. Denote by hole(p) the connected component of \U λ which contains p. We claim Vol e (hole(p)) Vol e () C 1 λ α(n) (log λ) 2n, (2) where α(n) = 2n 2 + n/2, and Vol e denotes the Euclidean volume. We will denote the right hand side term of (2) by γ(λ). 3

4 4h σ i 2h p hole(p) Figure 1: Proof of Lower Bound on Inner Radius Indeed, hole(p) is a connected component of U λ for some λ- nodal domain U λ. Hence, we can apply the following Local Courant s Nodal Domain Theorem. Theorem 4 ([DF90, CM91]). Let B M be a fixed ball. Let B be a concentric ball of half the radius of B. Let U λ be a λ-nodal domain which intersects B. Let B λ be a connected component of B U λ. Then where α(n) = 2n 2 + n/2. Vol(B λ )/Vol(B) C 2 λ α(n), (3) (log λ) 2n We remark that in our case (3) is true also for the quotient of Euclidean volumes, since the Euclidean metric on σ i is comparable with the metric coming from M. Step 4. We let ϕ λ = χ(u λ )ϕ λ, where χ(u λ ) is the characteristic function of U λ, and similarly, ϕ λ,i = χ(u λ σ i )ϕ λ. Then we have the inequality ϕ λ,i 2 d(vol) β(λ)h 2 ϕ λ,i 2 d(vol), (4) where β(λ) = { C3 log(1/γ(λ)), n = 2, C 4 /(γ(λ)) (n 2)/n, n 3. Proof. Observe that ϕ λ,i vanishes on hole(p). We will use the following Poincaré type inequality due to Maz ya. We discuss it in

5 Theorem 5. Let R n be a cube whose edge is of length a. Let 0 < γ < 1. Then, u 2 d(vol) βa 2 u 2 d(vol) for all Lipschitz functions u on, which vanish on a set of measure γa n, and where { C5 log(1/γ), n = 2, β = C 6 /γ (n 2)/n, n 3. From (2) and Theorem 5 applied to ϕ λ,i, it follows ϕ λ,i 2 d(vol e ) β(λ)h 2 e ϕ λ,i 2 e d(vol e ). Since the metric on σ i is comparable to the Euclidean metric, we have also inequality (4). Step 5. σ i ϕ λ 2 d(vol) 16β(λ)r 2 λ σ i ϕ λ 2 d(vol). (5) This is obtained by summing up inequalities (4) over all cubes which cover σ i, and recalling that h < 2r λ,i,e 4r λ. Step 6. We sum up (5) over all cubical cells σ i to obtain a global inequality. ϕ λ 2 d(vol) = ϕ λ 2 d(vol) = ϕ λ 2 d(vol) U λ M i σ i 16β(λ)rλ 2 ϕ λ 2 d(vol) = 16β(λ)rλ 2 ϕ λ 2 d(vol) i σ i M = 16β(λ)rλ 2 ϕ λ 2 d(vol) (6) U λ Step 7. r λ { C7 / λlog λ, n = 2, C 8 /λ n2 7n/4 1 (log λ) n 2, n 3. Indeed, by (6) λ = U λ ϕ λ 2 d(vol) U λ ϕ λ 2 d(vol) 1 16β(λ)rλ 2. 5

6 Thus, { 1 r λ 4 λβ(λ) = C9 / λlog(1/γ(λ)), n = 2, C 10 γ(λ) (n 2)/2n / λ, n 3. { C11 / λlog λ, n = 2, ) C 12 / (λ n2 7n/4 1 (log λ) n 2, n Lower Bound in Dimension = 2 We prove the existence of the lower bound on the inner radius in Theorem 2. The arguments below can basically be found in [EK96], Chapter 7. We begin the proof of Theorem 2 with Step 1 and Step 2 of 2.1. We proceed as follows. Step 3. If hole(p) does not touch Area e (hole(p)) Area e () C 1, where Area e denotes the Euclidean area. Proof. We recall the Faber-Krahn inequality in R n. Theorem 6. Let Ω R n be a bounded domain. Then λ 1 (Ω) C 2 /Vol(Ω) 2/n We apply Theorem 6 with Ω = hole(p). We emphasize that λ 1 (hole(p),g) C 3 λ 1 (hole(p),e), since the two metrics are comparable. Thus, we obtain λ = λ 1 (hole(p),g) C 4 Area e (hole(p)), or, written differently, Area e (hole(p)) C 4 /λ. On the other hand, Area e () = (4h) 2 64r 2 λ 64C 5/λ, where the last inequality is the upper bound on the inner radius in Theorem 2. So take C 1 = C 4 /(64C 5 ). 6

7 Step 4 (part a). There exists an edge of, on which the orthogonal projection of hole(p) is of Euclidean size γ 4h, where 0 < γ < 1 is independent of λ. Let us denote by pr(hole(p)) the maximal size of the projections of hole(p) on one of the edges of. If hole(p) touches, then pr(hole(p)) 4h/4 = h, and we can take γ = 1/4. Otherwise, by Step 3 So, we can take γ = C 1. Step 4 (part b). pr(hole(p)) Area e (hole(p)) C 1 (4h) 2 = 4 C 1 h. ϕ λ,i 2 dvol e C 6 h 2 ϕ λ,i 2 dvol e. (7) Notice that ϕ λ,i vanishes on hole(p). Hence, Step 4 (part a) permits us to apply the following Poincaré type inequality to ϕ λ,i. Its proof is given in 4.2. Theorem 7 ([EK96], ch. 7). Let R 2 be a cube whose edge is of length a. Let u be a Lipschitz function on which vanishes on a curve whose projection on one of the edges is of size γa. Then u 2 dx C γ a 2 u 2 dx, where C γ depends only on γ. Steps 5 7. To conclude we continue in the same way as in Steps 5 7 of A New Proof in Dimension Two This section is due to L. Polterovich, M. Sodin and F. Nazarov. In dimension two we give a proof based on the harmonic measure and the fact due to Nadirashvili that an eigenfunction on the scale comparable to the wavelength is almost harmonic in the following precise sense. Let D p Σ be a metric disk centered at p. Let f be a function defined on D. Let D denote the unit disk in C. 7

8 Definition 8. We say that f is (K, δ)-quasiharmonic if there exists a K-quasiconformal homeomorphism h : D D, a harmonic function u on D, and a function v on D with 1 δ v 1, such that f = (v u) h. (8) Remark. We will assume without loss of generality that h(p) = 0. Theorem 9 ([Nad91, NPS05]). There exist K,ε,δ > 0 such that for every eigenvalue λ and disk D Σ of radius ε/ λ, ϕ λ D is (K, δ)-quasiharmonic. We now choose a preferred system of conformal coordinates on (Σ,g). Lemma 10. There exist positive constants q +,q,ρ such that for each point p M, there exists a disk D p,ρ centered at p of radius ρ, a conformal map Ψ p : D D p,ρ with Ψ p (0) = p, and a positive function q(z) on D such that Ψ p (g) = q(z) dz 2, with q < q < q +. Let us take a point p, where ϕ λ admits its maximum on U λ. Let R = ε/ λq +. Let D p,r q+ D p,ρ be a disk of radius R q + centered at p. We now take the functions u,v defined on D which correspond to ϕ λ Dp,R q+ in Theorem 9. We observe that for all z D. Hence ϕ λ (p) = u(0)v(0) u(z)v(z) u(z)(1 δ), u(0) (1 δ)max u. (9) D Now we apply the harmonic measure technic. Let Uλ 0 D be the connected component of {u > 0}, which contains 0. Let E = D \ Uλ 0. Let ω be the harmonic measure of E in D. ω is a bounded harmonic function on Uλ 0, which tends to 1 on U0 λ Int(D) and to 0 on the interior points of Uλ 0 D. Let r 0 = inf{ z : z E}. By the Beurling-Nevanlinna theorem ([Ahl73], sec. 3-3), ω(0) 1 C 1 r0. (10) 8

9 By the majorization principle u(0)/max u 1 ω(0). (11) Combining inequalities (9), (10) and (11) gives us r 0 C 2. (12) In the final step we apply a distortion theorem proved by Mori for quasiconformal maps. Denote by D r C the disk { z < r}. Observe that Ψ p (D R ) D p,r q+. Hence, we can compose h = h Ψ p : D R D. h is a K-quasiconformal map. By Mori s Theorem ([Ahl66], Ch. III.C) it is 1 K-Hölder. Moreover, it satisfies an inequality ( ) h(z 1 ) h(z z1 z 2 1/K 2 ) M, (13) R with M depending only on K. Inequalities (12) and (13) imply that dist(p, (U λ D p,r )) R ( ) K C2 q. (14) M Hence, inrad(u λ ) ( ) K C2 q R = C 3 / λ, (15) M as desired. 4 A Review of Poincaré Type Inequalities We give an overview of several Poincaré type inequalities. In particular, we prove Theorem 5 and Theorem 7. 9

10 4.1 Poincaré Inequality and Capacity Theorem 5 is a direct corollary of the following two inequalities proved by Maz ya. Theorem 11 ( in [Maz85], [Maz03]). Let R n be a cube whose edge is of length a. Let F. Then u 2 C 1 a 2 d(vol) u 2 d(vol) cap(f, 2) for all Lipschitz functions u on which vanish on F. A few remarks: (a) 2 denotes a cube concentric with, with parallel edges of size twice as large. (b) If Ω R n is an open set, and F Ω, then cap(f,ω) denotes the L 2 -capacity of F in Ω, namely { } cap(f,ω) = inf u 2 dx, u F Ω where F = {u C (Ω), u 1 on F, supp(u) Ω}. (c) By Rademacher s Theorem ([Zie89]), a Lipschitz function is differentiable almost everywhere, and thus the right hand side has a meaning. The next theorem is a capacity volume inequality. Theorem 12 ( in [Maz85]). { C2 log(area(ω)/area(f)), n = 2, cap(f,ω) C 3 /(Vol(F) (n 2)/n Vol(Ω) (n 2)/n ), n 3. In particular, for n 3 we have cap(f,ω) C 3 Vol(F) (n 2)/n. 4.2 A Poincaré Inequality in Dimension Two In this section we prove Theorem 7. The proof can be found in chapter 7 of [EK96]. We bring it here for the sake of clarity. 10

11 Proof. Let the coordinates be such that = {0 x 1,x 2 a}. Let the given edge be {x 1 = 0}, and let pr denote the projection from onto this edge. Set E = pr 1 (pr(hole(p))). We claim u 2 dx a 2 u 2 dx. (16) E Indeed, let E t := E {x 2 = t}. We recall the following Poincaré type inequality in dimension one whose proof is given below. Lemma 13. b a b u 2 dx b a 2 u 2 dx (17) for all Lipschitz functions u on [a,b] which vanishes at a point of [a,b]. By this lemma u 2 dx 1 a 2 E t a E t 1 u(x 1,t) 2 dx 1. Integrating over t pr(hole(p)) gives us (16). Next we show u 2 dx C γ a 2 u 2 dx. (18) By the mean value theorem t 0 such that u 2 dx 1 1 u 2 dx. (19) E t0 γ a E In addition, we have u(x) 2 2 u(x 1,t 0 ) u(x) u(x 1,t 0 ) 2 2 u(x 1,t 0 ) u(x 1,t 0 ) a ( x2 t 0 a 0 2 u(x 1,s) ds ) 2 2 u(x 1,s) 2 ds. Integrating the last inequality over gives us u 2 dx 2 a u(x 1,t 0 ) 2 dx 1 + 2a 2 2 u 2 dx. (20) E t0 11

12 Finally, we combine (16), (19) and (20) to get (18). u 2 dx 2 a 1 u 2 dx + 2 a 2 u 2 dx γa E ( ) 2 γ + 2 a 2 u 2 dx. 4.3 A Poincaré Inequality in Dimension One We prove Lemma 13. Proof. By scaling, it is enough to prove (17) for the segment [0,1]. Suppose u(x 0 ) = 0. Since a Lipschitz function is absolutely continuous, we have u(x) 2 x = u 2 1 (t)dt u (t) 2 dt. x 0 0 We integrate over [0,1] to get the desired inequality. 5 λ 1 and Inner Radius We prove Theorem 3, which relates the inner radius to λ 1. Proof. Let {V i } be a finite open cover of M, such that for each i one can put a Euclidean metric e i on V i, which satisfies e i /4 g 4e i. Let α be the Lebesgue number of the covering. Let r = min(inrad(ω),α) Let B Ω be a ball of radius r. We can assume that B V 1. Let B e B be a Euclidean ball of radius r/2. By monotonicity of λ 1, we know that λ 1 (B,g) λ 1 (B e,g) but since the Riemannian metric on B e is comparable to the Euclidean metric on it, it follows from the variational principle that λ 1 (B e,g) C 1 λ 1 (B e,e 1 ) = C 2 /r 2 C 2 /inrad(ω) 2. Remark. We would like to emphasize that there is no lower bound on λ 1 is terms of the inner radius. To see this, take Ω R n to be a ball B with arbitrary small balls centered at the points of an ε-net removed. λ 1 (Ω) is small, since it is approximately λ 1 (B), and the inner radius is also small. 12

13 References [Ahl66] Lars V. Ahlfors, Lectures on quasiconformal mappings, Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, [Ahl73], Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York, 1973, McGraw- Hill Series in Higher Mathematics. [Cha84] Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press Inc., Orlando, FL, 1984, Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk. [CM91] Sagun Chanillo and B. Muckenhoupt, Nodal geometry on Riemannian manifolds, J. Differential Geom. 34 (1991), no. 1, [DF90] [EK96] H. Donnelly and C. Fefferman, Growth and geometry of eigenfunctions of the Laplacian, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp Yuri Egorov and Vladimir Kondratiev, On spectral theory of elliptic operators, Operator Theory: Advances and Applications, vol. 89, Birkhäuser Verlag, Basel, [Maz85] Vladimir G. Maz ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, [Maz03] Vladimir Maz ya, Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp [MS05] V. Maz ya and M. Shubin, Can one see the fundamental frequency of a drum?, arxiv:math.sp/ , [Nad91] Nikolai S. Nadirashvili, Metric properties of eigenfunctions of the Laplace operator on manifolds, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 1, [NPS05] Fëdor Nazarov, Leonid Polterovich, and Mikhail Sodin, Sign and area in nodal geometry of Laplace eigenfunctions, Amer. J. Math. 127 (2005), no. 4,

14 [Xu05] [Zie89] Bin Xu, Asymptotic behavior of L 2 -normalized eigenfunctions of the Laplace-Beltrami operator on a closed Riemannian manifold, arxiv:math.sp/ , William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989, Sobolev spaces and functions of bounded variation. Dan Mangoubi, Department of Mathematics, The Technion, Haifa 32000, ISRAEL. address: mangoubi@techunix.technion.ac.il 14