Locting the first nodl set in higher dimensions Sunhi Choi, Dvid Jerison nd Inwon Kim My 5, 26 Abstrct We extend the two dimensionl results of Jerison [J1] on the loction of the nodl set of the first Neumnn eigenfunction of convex domin to higher dimensions. If convex domin is contined in long nd thin cylinder [, N] B ɛ () with nonempty intersections with {x 1 = } nd {x 1 = N}, then the first nonzero eigenvlue is well pproximted by the eigenvlue of n ordinry differentil eqution, by bound proportionl to ɛ, whose coefficients re expressed in terms of the volume of the cross sections of the domin. Also, the first nodl set is locted within distnce comprble to ɛ ner the zero of the corresponding ordinry differentil eqution. 1 introduction Let be bounded convex domin in IR n. Let u be n eigenfunction for ssocited with the smllest nonzero eigenvlue λ of the Neumnn problem for, tht is, u = λu in, u ν = on (1) where u ν = ν u nd ν denotes the outer norml unit vector t ech point on. The purpose of this pper is to locte the first nodl set Λ = {u = } nd to estimte the first nonzero eigenvlue λ. We show tht Λ is ner the unique zero of the first nonconstnt eigenfunction of certin ordinry differentil eqution to within distnce comprble to the smllest dimeter of projections of onto hyperplnes. Also we estimte the difference between λ nd the first nonzero eigenvlue of the ordinry differentil eqution by bound proportionl to the rtio of the smllest dimeter of projection of the domin to the dimeter of the domin. This pper extends the twodimensionl results of Jerison [J1] to higher dimensions, but the methods presented in this pper re different from the methods used in [J1]. 1
In 2-dimensions, the loction of the first nodl set is estimted in [J1] by constructing suitble coordinte system on nd the corresponding ODE whose coefficients re expressed in terms of the width of the domin. From the bound on the loction of the nodl set, the bound on the difference between the first nonzero eigenvlues is obtined for the originl PDE nd for the corresponding ODE. In this pper, we first estimte the difference between the first nonzero eigenvlues, using shrp results of Kröger nd Kdlec which provide bounds on the first nd second derivtives of eigenfunctions of. Bsed on these estimtes we then re ble to find the bound on the width of the first nodl set. Once we prove tht the nodl set is thin, we re ble to locte the nodl set ner the zero of the corresponding ordinry differentil eqution. Assume {(x 1, x ) : x 1 [, N], x < ɛ} IR n, is convex (2) nd suppose further tht (s) := {x 1 = s} is nonempty for < s < N. (3) Let u be the first nonconstnt eigenfunction for. Denote by φ 1 the first nonconstnt eigenfunction with the smllest nonzero eigenvlue µ 1 for the Neumnn problem on [, N] given by (wφ 1) = µ 1 wφ 1 on (, N) ; w(s)φ 1(s) s s + or s N (4) where w(s) is the (n 1)-dimensionl volume of (s). Let s 1 (, N) be the unique zero of φ 1, i.e., φ 1 (s 1 ) =. The min results of this pper re s follows. Theorem 1.1. If u is the first nonconstnt Neumnn eigenfunction of nd stisfies (2) nd (3), then there is dimensionl constnt C such tht () u(x 1,..., x n ) = implies x 1 s 1 < Cɛ (b) (1 Cɛ/N)µ 1 λ µ 1. where s 1 nd µ 1 re given in (4). Recll tht u minimizes the Dirichlet integrl J(v) = v 2 (5) 2
mong ll functions v on stisfying v 2 = 1, v =. (6) The minimum vlue of J is the eigenvlue λ. If we consider functions of x 1 lone, i.e., v(x 1,..., x n ) = φ(x 1 ), then nd the constrints (6) re J(v) = I(φ) := N φ (s) 2 w(s)ds (7) N φ(s) 2 w(s)ds = 1, N φ(s)w(s)ds =. (8) As in [J1] we observe tht the minimizer of (7) under the constrints (8) is the first nonzero eigenfunction φ 1 given in (4) nd I(φ 1 ) = µ 1. Hence λ µ 1. (9) Remrk 1. If we normlize N = 1, then by [L] nd [ZY], C 1 λ for some bsolute constnt C 1 >, nd by plugging in the test function φ(x) = sin πx 1, we get λ C 2 for some dimensionl constnt C 2. Remrk 2. In the cse of Dirichlet problem on plnr convex domin, Jerison [J2] obtined results corresponding to Theorem 1.1. Lter Grieser nd Jerison ([GJ1], [GJ2]) showed tht the nodl line is in n x 1 -intervl of much shorter length Cɛ/N (possibly t distnce Cɛ from s 1 ). We expect tht there is n nlogous bound in the Neumnn problem. Remrk 3. An nlogous pproch to the method in [J1] for higher dimensions, by modifying the methods in [J1]-[J2], leds to result weker thn Theorem 1.1, i.e., with ɛ log(ɛ/n) insted of ɛ. 2 Preliminry results nd bsic lemms The key ingredients in the proof of Theorem 1.1 re Kröger s comprison theorem on the grdient of eigenfunction u nd Kdlec s inequlity on the L 2 bound of second order derivtive of u. Theorem 2.1 (Kröger, See [K1)., [K2], [BQ] ] Let be convex domin in IR n with smooth boundry. Let u be n eigenfunction for the Lplce 3
opertor on with n ssocited eigenvlue λ >, under Neumnn boundry conditions. Let v be solution on some intervl (, b) of the differentil eqution v (x) + n 1 x v (x) = λv(x) on (, b) v () = v (b) = (1) such tht v on (, b) nd [min u, mx u] [min v, mx v]. Then (v 1 u) 1. Corollry 2.2. Let nd u be given s in Theorem 1.1. Suppose tht N = 1 nd mx u = 1, then u C for some dimensionl constnt C. Proof. First, we clim tht if < < b then the solution v to (1) stisfies mx v λ v(). (11) To see this, multiply v to both sides of (1) to obtin v v + λv v = 1 2 ((v ) 2 + λv 2 ) = n 1 x (v ) 2 for x >. Since, b > it follows tht (v ) 2 (x) + λv 2 (x) (v ) 2 () + λv 2 () = λv 2 (). Hence the clim follows. Now, for sufficiently lrge c >, consider n intervl (, b) (c, ), which hs λ s first nonzero eigenvlue for the Neumnn problem (1). Let v be the corresponding eigenfunction. If c is lrge enough, then we cn normlize v so tht 1 min v 2 nd 1 mx v 2. Then by, Theorem 2.1, we get sup u sup v C where the second inequlity follows from Remrk 1 nd the bove clim. Theorem 2.3 (Kdlec, See?? ). Let be bounded convex domin in IR n. Let f L 2 () nd let u be solution to u = f in, u ν = on. Then 2 u L 2 () nd 2 u 2 4 f 2. 4
Lemm 2.4. Let be bounded convex domin in IR n given s in (2) nd (3). Let v be bounded C 1 function in such tht 1 v 2 C, mx v C. (12) Let L be line joining point on () to point on (N). Let φ be function in such tht Then φ = v on L, φ(z) = φ( z) for z, z (s), < s < N. for some C 1 depending on n nd C. 1 φ 2 1 v 2 + C 1 ɛ (13) Proof. Without loss of generlity, we my ssume C = 1. To simplify nottion, ssume tht L = {(x 1,,..., ) : x 1 N}. For x = (x 1, x 2,..., x n ), let γ x be the line segment with endpoints (x 1,,..., ) nd x. For h >, let γ x,h be the line segment with endpoints x nd (x 1, Ax 2,..., Ax n ), where A is chosen so tht A > 1 nd the length γ x,h = h. Let ψ(x) = γ x v. Then v φ ψ nd v 2 φ 2 = (v+φ)(v φ) 2ψ. Hence it suffices to prove ψ Cɛ. For given x = (x 2,..., x n ) IR n 1, consider line segment I = {(x 1, x ) : x 1 b} contined in. Let S x,,b,θ be n (n 1)-dimensionl hypersurfce in, obtined by rotting I round L by smll ngle θ. For smll h (, x ), let B h,x,,b,θ = {γ x,h : x S x,,b,θ}. Define the set nd denote B = {B h,x,,b,θ}, height of B h,x,,b,θ = h, bottom of B h,x,,b,θ = S x,,b,θ. Suppose x nd v(x). Since v C 1, if we tke sufficiently smll b >, θ > nd h >, we cn find B B such tht x B nd mx B v min B v 2. 5
Decompose { v } L into disjoint nd countble union of B i s in B such tht = ( 1 B i ) { v = } L, C i v 2C i in B i. (14) Denote by h i, the height of B i nd by S i, the bottom of B i. Let B i be the n-dimensionl volume of B i nd let S i be the (n 1)-dimensionl volume of S i. Observe tht the first inequlity of (12) nd (14) imply Ci 2 B i C. (15) Hence ψ(x)dx = i v dsdx 2 γ x 2 i γ x B i C i h i S i ɛ C i h i dx 2ɛ( C 2 i h i S i ) 1/2 ( i Cɛ h i S i ) 1/2 where the lst inequlity follows from (15) nd h i S i C B i. In the following lemm, we prove Hrnck inequlity for Neumnn eigenfunctions in convex domins. Note tht B r (x) in the hypothesis of Lemm 2.5 is not necessrily contined in +. Lemm 2.5. Let nd u be given s in Theorem 1.1. If x + := {u > } nd B r (x) := B r (x) {u < } =, then for some dimensionl constnt C >, sup u C B r/2 (x) inf u. B r/2 (x) An nlogous result holds with in plce of +. Proof. We will deduce the lemm from the counterprt for prbolic equtions (See Theorem 1.2, [LY]). Let U(x, t) = e λt u(x), then U stisfies the het eqution ( t )U =. Define F (x, t) = t( log U 2 2(log U) t ) in +, then s in [LY], F nd F/ ν on + for outwrd unit norml vector ν, since is convex. 6
Let p + nd let B r (p) =. Let φ be nonnegtive, strictly decresing function on [, r] such tht φ(r) =. Using convexity of, we get φ( x p )/ ν < for x. Hence (φ( x p ) F (x, t)) ν = φ ν F + φ F ν < on + B r (p). Since φ(r) =, φ F ttins its mximum t some interior point in B r (p) +. Now we follow the proof of Theorem 1.2. in [LY] nd obtin u 2 u 2 + 2λ C r 2 on B r/2 (p), from which the lemm follows. Corollry 2.6. Let nd u be given s in Theorem 1.1. Suppose N = 1 nd sup u = 1. Then there exist dimensionl constnt c 1 > nd c 2 > such tht c 2 u(x) 1 for x 1 [, c 1 ] [1 c 1, 1]. Proof. Without loss of generlity, we my ssume mx u = u( x 1, x ) = 1. By Corollry 2.2, there exists dimensionl constnt c > such tht u(x 1, x ) > if x 1 x 1 < c. Hence we my ssume tht u > in {x 1 > x 1 c }. Then by Lemm 2.5, there exists dimensionl constnt c 2 > such tht u c 2 if x 1 1 c /2 x 1 c /2. On the other hnd, since u = + u +, Lemm 2.2 implies min u c 3 for some dimensionl constnt c 3 >. Hence we obtin Corollry 2.6 by similr resoning for u s in u +. Corollry 2.6 shows tht the first nodl set is locted in the middle prt of. Lter in section 4, Corollry 2.6 nd Theorems 2.1, 2.3 will be used to estimte the first nonzero eigenvlue λ. Bsed on the bound on λ, the width nd loction of the nodl set re derived, gin by using Theorems 2.1 nd 2.3. 3 ODE eigenvlue estimtes In this section we prove severl lemms on ODE eigenvlue estimtes, which will be used in section 4. In prticulr Lemm 3.6 yields the bound on the width of nodl set, nd Lemm 3.5 will be used to locte the nodl set in section 4. Lemms 3.1, 3.2, 3.3 nd 3.4 re prllel to those in [J1], nd the only difference in the proofs from the prllel lemms in [J1] is tht insted of 7
the concvity of the height of the cross-section h(x) in two dimensionl cse, here we hve the concvity of w 1/n 1 (x) by the Brunn-Minkowski inequlity for the volume of cross-section w(x). In the proof of those lemms we will follow the corresponding proofs in [J1], only pointing out the differences in the proof. Throughout the rest of the pper, we will normlize N = 1. Lemm 3.1. Let 1/2, then inf {φ:φ()=} φ (x) 2 w(x)dx φ(x)2 w(x)dx 1 2 n 2 2. Proof. The proof is the sme s tht of Lemm 4.2 of [J1], using the fct tht w(x) 2 n 1 w(t) for x t. Lemm 3.2. There exists constnt C > depending on n such tht inf {φ:φ()=} b φ (x) 2 w(x)dx b φ(x)2 w(x)dx C (b ) 2. Proof. Tke the test function φ(x) = x, then we need to show tht w(x)dx C x 2 w(x)dx, where C depends on n. Multiply w by constnt so tht w1/n 1 (t)dt = 1. Due to the normliztion nd the concvity of w 1/n 1, the rguments in the proof of Lemm 4.3 in [J1] yields tht By Hölder inequlity, ( x x w 1/n 1 (t)dt (1 x) 2 for x 1. w(t)dt) 1/n 1 (1 x) n 2/n 1 x w 1/n 1 (t)dt (1 x) 2, nd thus W (x) := x w(t)dt (1 x)n. Therefore by integrtion by prts, x 2 w(x)dx = x 2 W (x)dx = 8 2xW (x)dx 2x(1 x) n dx = 2/(n + 1)(n + 2),
where the second inequlity holds becuse W (1) =. On the other hnd, since w 1/n 1 (t) is concve with w 1/n 1 (), w 1/n 1 (1), the grph of w 1/n 1 (t), t 1 is bove the tringle generted by (, ), (1, ) nd (t, w 1/n 1 (t )) where w(t ) =mx w. It follows tht w(t)dt w(t ) (2 nd our lemm holds with C = C(n) = w 1/n 1 (t)dt) n 1 = 2 n 1 2 n (n + 1)(n + 2). Lemm 3.3. Let s 1 be the zero point of φ 1 given in (4). Then there exist constnts c 1 > nd c 2 > depending on n such tht c 1 < s 1 < c 2. Proof. It follows from Lemms 3.1, 3.2 nd the proof of Lemm 4.4 in [J1]. For [, 1], define E[, 1] = inf {φ:φ()=} φ (x) 2 w(x)dx φ(x)2 w(x)dx. Lemm 3.4. Suppose tht c 1 c for some c >. Then there exists constnt C > depending on n nd c such tht Proof. Normlize w 1/n 1 so tht ( / )E[, 1] C. sup w 1/n 1 = 1. (16) Then by concvity, w 1/n 1 (c ) c. Let φ be the unique nonnegtive minimizer for E[, 1] with the normliztion φ(x) 2 w(x)dx = 1. (17) Following the proof of Lemm 3.4 in [J1], we only need to prove tht φ(x) CE 2 (18) 9
where E = E[, 1] nd C is constnt depending on n nd c. Observe tht since φ stisfies (wφ ) = Ewφ nd φ (1) =, wφ (x). E( x x Ew(t)φ(t)dt (19) φ 2 (t)w(t)dt) 1/2 ( x w(t)dt) 1/2 In prticulr, (16) nd (17) imply tht wφ E for x < 1. Since φ() =, we hve t ds φ(t) E w(s). (2) On the other hnd, by concvity of w 1/n 1 (t), for s t 1, w 1/n 1 (s) is bove the line l(s) = αs+β connecting (c, c ) nd (t, w 1/n 1 (t)). Without loss of generlity, we my ssume tht c > w 1/n 1 (t) nd α <. (Otherwise, (2) would imply (18).) Thus for n 3 t ds w(s) t ds (αs + β) n 1 C (αt + β) n 2 = Hence by (19), (2) nd (21) φ E (x) w(x) x w(t)φ(t)dt C w (n 2)/n 1 (t). (21) E2 w(x) x w(t)( t 1 w(s) ds)dt CE2 w(x) x w1/n 1 (t)dt (22) CE 2 (1 x) w (n 2)/n 1 (x) CE 2 1 w n 3/n 1 (x), where the fourth nd fifth inequlities follow from w 1/n 1 (t) C(n)w 1/n 1 (x) for x t 1 (23) nd w 1/n 1 (x) min{1 x, c } for x 1 (24) 1
which re due to the concvity of w 1/n 1 nd the normliztion (16). (For (23), see Remrk 4.1 (b) in [J1].) It follows tht, by (22) nd (23), φ(x) x φ (t) dt C 1 E 2 w n 3/n 1 (x) (25) where C 1 is constnt depending on n nd c. We go bck to the first inequlity of (22) nd pply (25) nd then (23) nd (24), then we obtin φ (x) C 1E 2 w(x) x w(t) 2/n 1 dt C 2 E 2 w(x) n 4/n 1 where C 2 is dimensionl constnt. Now by similr resoning s in (25), we hve the improved estimte on φ : φ(x) C 3 E 2 w(x) n 4/n 1. We repet the bove process (n 4) more times to obtin φ(x) C(n)E 2. Lemm 3.5. Let φ 1 nd s 1 be given s in (4) nd suppose N = 1. If φ is function on (, 1) such tht φ(s 1 ) = nd s 1(φ ) 2 s φ 2 1 (1 + Mɛ) s 1 (φ 1 )2 s 1 φ 2 1, then s 1 s 1 + Cɛ for some constnt C depending on n nd M. Proof. The lemm follows from Lemms 3.3, 3.4 nd µ 1 = s 1 (φ 1 )2 s 1 φ 2 1. In the following lemm we show tht if the energy of φ is bounded by (1 + Mɛ)µ 1, then sup φ is bigger thn ɛ on ny intervl of length Cɛ. 11
Lemm 3.6. Let µ 1 be given in (4) nd suppose N = 1. Suppose φ is function on (, 1) such tht φ =, sup φ = 1, sup φ C 1 nd µ 1 (φ ) 2 φ2 (1 + Mɛ)µ 1. (26) If for some < < b < 1 nd C 2 >, sup φ ɛ nd [,b] φ C 2 then b < Cɛ for some C depending on C 1, C 2 nd M. Proof. Let φ 1 nd s 1 be given s in (4). Suppose tht sup [,b] φ ɛ for some nd b = + Cɛ where C > is sufficiently lrge. If < s 1 < b, then s 1 Cɛ/2 or b s 1 Cɛ/2. In cse b s 1 Cɛ/2, replce with (s 1 + b)/2, nd in cse s 1 Cɛ/2, replce b with (s 1 + )/2. Then we get s 1 + Cɛ/4 or b s 1 Cɛ/4. Also when < b < s 1 or s 1 < < b, by replcing or b with ( + b)/2, we get s 1 + Cɛ/4 or b s 1 Cɛ/4. Without loss of generlity we my ssume tht s 1 +Cɛ nd b = +Cɛ where C is sufficiently lrge. We lso suppose tht φ = φ > C 2. Define A φ = φ2 φ2, B φ = φ2 φ2, A φ + B φ = 1 (27) then by hypothesis, (φ ) 2 φ2 = A (φ ) 2 φ φ2 + B (φ ) 2 φ φ2 (1 + Mɛ)µ 1. If C is sufficiently lrge, then Lemm 3.4 nd bove inequlity imply tht (φ ) 2 φ2 µ 1 (φ ) 2 φ2. (28) We will construct test function ψ such tht ψ = nd (1 + Mɛ) (ψ ) 2 ψ2 < (φ ) 2 φ2, (29) 12
which would yield us to desired contrdiction. For α = C 2 /1, let φ(s) for s ψ(s) = φ(s) + α(s ) for s b = + Cɛ Then by clcultion, we cn show φ(s) + α(b ) for b s 1. ψ 2 ψ 2 (1 C Cɛ) φ 2 φ2 (3) for some C > depending on C 1 nd C 2. Let ψ(s) = φ(s) for s ψ(s) = β( ψ(s) φ()) + φ() for s 1 where β > is chosen so tht ψ =, i.e., ψ = β = 1 α b 1 φ + O(ɛ2 ). φ. Then Since we get φ φ 2, (31) ψ 2 (1 + 2α b 1 φ + C 3ɛ 2 ) φ 2. Hence ψ2 (1 + C 3 ɛ 2 ) φ2 nd B ψ (1 + C 3 ɛ 2 )B φ (32) where B ψ is defined s in (27) (Recll tht A ψ = 1 B ψ ). Therefore, 13
(ψ ) 2 ψ2 = A (ψ ) 2 ψ ψ2 + B (ψ ) 2 ψ ψ2 = A (φ ) 2 ψ φ2 + B (ψ ) 2 ψ ψ2 (1 + C 3 ɛ 2 )(A (φ ) 2 φ φ2 + B (ψ ) 2 φ ψ2 ) (1 + C 3 ɛ 2 )(A φ (φ ) 2 A φ (φ ) 2 φ2 + B φ(1 C Cɛ) φ2 + B φ(1 C Cɛ) (φ ) 2 φ2 B φc Cɛ (φ ) 2 φ2 ) (φ ) 2 φ2 + C 3ɛ 2 (1 + Mɛ)µ 1 (φ ) 2 φ2 + C 3ɛ 2 (1 + Mɛ)µ 1 where we obtin the first inequlity from A φ + B φ = A ψ + B ψ = 1, (32) nd (28), the second inequlity from (3), the third inequlity from (26). From the hypothesis φ C 2 nd (31), we observe tht B φ is bounded below by constnt depending on C 2. Hence if we choose sufficiently lrge C depending on C 1, C 2 nd M, then we obtin (29). In the next lemm, we show the nondegenercy of φ 1 ner the zero s 1. Lemm 3.7. Let φ 1 be the first nonzero eigenfunction of (4) with N = 1 nd let s 1 be the zero of φ 1 (Note tht c 1 s 1 c 2 by Lemm 3.3.) Normlize φ 1 such tht φ 1 > on (s 1, 1] nd mx φ 1 = 1. Then φ 1 on [, 1]. Moreover there exists dimensionl constnt c = c(n) such tht φ 1 c(n) on [s 1 c 1 /2, s 1 + (1 c 2 )/2]. Proof. By our normliztion φ 1 > on (s 1, 1] nd φ 1 < on [, s 1 ). Hence from (4) we observe tht wφ 1 hs its mximum t x = s 1 nd wφ 1 is incresing on [, s 1 ] nd decresing on [s 1, 1]. From the boundry condition it follows tht φ 1 on [, 1] nd thus φ 1(1) = 1. We will only prove tht φ 1 c(n) on [s 1, s 1 + (1 c 2 )/2]: prllel rgument leds to the rest of the clim. Observe tht from (4) we hve φ x 1(x) = µ 1w(t)φ 1 (t)dt. (33) w(x) 14
We will use bove equlity to find bounds on φ respectively ner t = 1 nd then ner t = s 1, using tht wφ hs its mximum t s 1. Note tht for c 1 x t 1, the concvity of w 1/n 1 (x) implies (1 t)w 1/n 1 (x) w 1/n 1 (t). (34) (See Remrk 4.1 () in [J1].) Then (23) nd (34) imply tht C 1 (1 x) n w(x) x w(t)dt C 2 (1 x)w(x) (35) where C 1 nd C 2 re dimensionl constnts. Let φ 1 (s 2 ) = 1/2 for s 2 [s 1, 1]. Then by (33) nd (35), it follows tht for x [s 2, 1], C 1 µ 1 (1 x) n+1 φ 1(x) C 2 µ 1 (1 x). (36) Then since φ 1 (1) = 1, φ 1 (x) 1 C 3 µ 1 (1 x) 1/2 if x [A, 1] where C 3 nd A re dimensionl constnts. Therefore s 2 A nd φ 1 (s 2) C 1 µ 1 (1 s 2 ) n+1 C 4, (37) where C 4 is dimensionl constnt. Now we hve, for s 1 t s 2, φ 1 (t) w(s 2) w(t) φ 1 (s 2) C 5 φ 1 (s 2) C 6 where the second inequlity is due to t s 2 < A nd the lst from (37). For t [s 1, s 1 + (1 c 2 )/2], (36) implies φ 1 (t) C 7 nd our clim is proved. 4 Proof of Theorem 1.1 Assume tht hs smooth boundry (We will consider the generl cse t the end of proof). Normlize u nd so tht sup u = 1 nd N = 1. First, we will estimte the difference between λ nd µ by bound Cɛ. From this bound we will show tht the nodl set is contined in n x 1 - intervl of length Cɛ nd will proceed to locte the nodl set ner the zero s 1 of the corresponding ordinry differentil eqution φ 1. By Corollry 2.2 sup u C. (38) Let L be line in given in Lemm 2.4 nd let φ be function of x 1 lone in such tht φ = u on L. Then by (38) nd Corollry 2.6, 1 Cɛ φ2 1 + Cɛ. (39) u2 15
On the other hnd, Kdlec s inequlity (Theorem 2.3) with f = λu gives 2 u 2 C. (4) Then by (38), (4) nd Lemm 2.4 with v = u/ x 1, (φ ) 2 u 2 + Cɛ (1 + Cɛ) u 2 (41) where the second inequlity is by Cλ u2 = C u 2. From (39) nd (41), we cn conclude tht (φ ) 2 (1 + Cɛ) u 2. (42) φ2 u2 On the other hnd, since u C nd u =, φ Cɛ. (43) Let z = (z 1,..., z n ) Λ L, then (43) implies tht there exists c such tht c < C for some dimensionl constnt C nd φ(x) := φ(x) + c ɛ(x 1 z 1 ) + stisfies φ =. Observe tht (42) lso holds for φ, nd hence µ 1 ( φ ) 2 φ 2 (1 + Cɛ) u 2 u2 = (1 + Cɛ)λ (44) which proves prt (b) of theorem 1.1. For the proof of prt (), first we will show tht the nodl set is contined in n x 1 -intervl of length Cɛ nd then we will estimte its loction. Let p be projection on the x 1 -xis. Then p( + ) nd p( ) re intervls becuse the Cournt nodl domin theorem [CH, p.452] implies + nd re connected. Hence p(λ) = [, b] for some b. By (38), sup [,b] u < Cɛ nd hence sup φ < Cɛ. (45) [,b] By Corollry 2.6, φ > C 2 for some dimensionl constnt C 2 >. Hence by Lemm 3.6, (44) nd (45), we get b < Cɛ for some dimensionl constnt C. 16
Now, we obtin z φ 2 1 z φ2 1 z (1 + Cɛ) 1 φ 2 z 1 φ 2 u (1 + Cɛ) 2 u 2 (1 + Cɛ) + u 2 + u 2 = (1 + Cɛ)λ (1 + Cɛ)µ 1 where = {z 1 x 1 1}, φ(z1 ) = φ(z 1 ) =, the second inequlity follows from similr rgument s in (42) nd the third inequlity follows from b < Cɛ, u 1 nd u C. Then by Lemm 3.5, z 1 s 1 + Cɛ. By similr rgument on the intervl [, z 1 ], we obtin s 1 Cɛ z 1. Since the length of the projection of Λ on the x 1 -xis is less thn Cɛ, prt () is proved. For generl domin, let k be n incresing sequence of smooth domins which converges to uniformly on ech cross sections k (x 1 ). Let u k be the corresponding first nonzero eigenfunctions of k with sup u k = 1. Then by Kröger s theorem (Theorem 2.1), sup k u k is uniformly bounded. Hence there exists subsequence u kj which converges uniformly to u nd λ kj converges to λ. On the other hnd, since the volume w k (x) of k (x) uniformly converges to w(x), we my let φ kj converge uniformly to φ 1 nd µ kj converge to µ. Now by the nondegenercy of φ kj (Lemm 3.7) nd the nondegenercy of u kj ner in the scle ɛ (Lemm 3.6), we obtin Theorem 1.1 for u nd λ. References [AJ] [BQ] V. Adolfsson, D. Jerison, L p -Integrbility of the second order derivtives for the Neumnn problem in convex domins Indin Univ. Mth. J., 43(1994), 1123-1138. D. Bkry, Z. Qin, Some new results on eigenvectors vi dimension, dimeter, nd Ricci curvture Adv. Mth., 155(2), 98-153. [CH] R. Cournt, D. Hilbert Methods of Mthemticl Physics vol. I, Interscience Publishers, New York (1953). [GJ1] D. Grieser, D. Jerison, Asymptotics of the first nodl line of convex domin Inventiones Mth., 125(1996), 197-219. 17
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