Gradient estimate of a Dirichlet eigenfunction on a compact manifold with boundary
aa r X i v : . [ m a t h . A P ] F e b Gradient estimate of a Dirichlet eigenfunction ona compact manifold with boundary
Yiqian Shi ∗ and Bin Xu ∗ † Abstract.
Let e λ ( x ) be an eigenfun tion with respe t to the Diri hlet Lapla ian ∆ N on a ompa tRiemannian manifold N with boundary: ∆ N e λ = λ e λ in the interior of N and e λ = on theboundary of N . We show the following gradient estimate of e λ : for every λ ≥ , there holds λ k e λ k ∞ /C ≤ k (cid:209) e λ k ∞ ≤ Cλ k e λ k ∞ , where C is a positive onstant depending only on N . In theproof, we use a basi geometri al property of nodal sets of eigenfun tions and ellipti aprioriestimates. Mathematics Subject Classification (2000):
Primary 35P20; Se ondary 35J05
Key Words:
Diri hlet Lapla ian, eigenfun tion, gradient estimate
Let ( N, g ) be an n -dimensional ompa t smooth Riemannian manifold with smoothboundary ∂N and ∆ N the positive Diri hlet Lapla ian on N . Let L ( N ) be the spa eof square integrable fun tions on N with respe t to the Riemannian density dv ( N ) = p g ( x ) dx := p det ( g ij ) dx . Let e ( x ) , e ( x ) , · · · be a omplete orthonormal basis in L ( N ) for the Diri hlet eigenfun tions of ∆ N su h that < λ ≤ λ ≤ · · · for the orrespondingeigenvalues, where e j ( x ) ( j =
1, 2, . . . ) are real valued smooth fun tion on N and λ j arepositive numbers. Also, let e j denote the proje tion of L ( N ) onto the 1-dimensionalspa e C e j . Thus , an L fun tion f an be written as f = P ∞ j = e j ( f ) , where the partial sum onverges in the L norm. Let λ be a positive real number ≥ . We de(cid:12)ne the spe tralfun tion and the unit band spe tral proje tion operator χ λ as follows: e ( x, y, λ ) := X λ j ≤ λ e j ( x ) e j ( y ) ,χ λ f := X λ j ∈ ( λ,λ + ] e j ( f ) . ∗ Department of Mathemati s, University of S ien e and Te hnology of China, Hefei 230026 China. † E-mail of the orrespondent author: bxuust .edu. n1rieser [5℄ and Sogge [13℄ proved the L ∞ estimate of χ λ , || χ λ f || ∞ ≤ Cλ ( n − ) /2 || f || (1)where || f || r ( ≤ r ≤ ∞ ) means the L r norm of the fun tion f on N . In the whole of this paper C denotes a positive onstant whi h depends only on N and may take di(cid:11)erent values atdi(cid:11)erent pla es, if there is no otherwise stated. The idea of Grieser and Sogge is to use thestandard wave kernel method outside a boundary layer of width Cλ − and a maximumprin iple argument inside that layer. By using the maximum prin iple argument and theestimate (1), Xu [15℄ proved the gradient estimate of χ λ || (cid:209) χ λ f || ∞ ≤ Cλ ( n + ) /2 || f || . (2)Here (cid:209) is the Levi-Civita onne tion on N . In parti ular, (cid:209) f = P j g ij ∂f/∂x j is the gradientve tor (cid:12)eld of a C fun tion f , the square of whose length equals P i,j g ij ( ∂f/∂x i )( ∂f/∂x j ) .One of his motivation is to prove the H(cid:127)ormander multiplier theorem on ompa t manifoldswith boundary. Seeger and Sogge [11℄ (cid:12)rstly proved that theorem by using the parametrixof the wave kernel on manifolds without boundary. All the results mentioned in the in-trodu tion have their analog on ompa t manifolds without boundary. See the details inthe introdu tion of Shi-Xu [12℄ and the referen es therein. In general, the method usedin manifolds without boundary is not valid for the problems on manifolds with boundary.In parti ular, on manifolds with boundary the H(cid:127)ormander multiplier theorem annot beobtained by the standard pseudo-di(cid:11)erential operator al ulus as done on manifolds with-out boundary, sin e the square root of the Diri hlet Laplai an is not a pseudo-di(cid:11)erentialoperator any more and one annot obtain the L ∞ bounds for χ λ and (cid:209) χ λ only by usingthe Hadamard parametrix of the wave kernel.In the paper, by res aling χ λ f at the s ale of λ − both outside and inside the boundarylayer of width Cλ − , we obtain by ellipti apriori estimates a slightly stronger estimatethan (2) as follows: Theorem 1.1.
Let f be a square integrable fun tion on N . Then, for every λ ≥ ,there holds k (cid:209) χ λ f k ∞ ≤ C (cid:16) λ k χ λ f k ∞ + λ − k ∆ N χ λ f k ∞ (cid:17) . (3) Remark 1.1.
Putting f ( · ) = P λ j ∈ ( λ,λ + ] e j ( x ) e j ( · ) in (1), we obtain the uniform estimateof eigenfun tions for all x ∈ N , X λ j ∈ ( λ,λ + ] | e j ( x ) | ≤ Cλ n − . x ∈ N , X λ j ∈ ( λ,λ + ] | (cid:209) e j ( x ) | ≤ Cλ n − . (4) Remark 1.2.
By the (cid:12)nite propagation speed of the wave equation, the asymptoti formula of derivatives of the spe tral fun tion e ( x, y, λ ) in Theorem 1 [14℄ whi h is provedby the standard wave kernel method, also holds for ea h interior point x of N . Mu h moregeneral asymptoti formulae are given in Theorems 1.8.5 and 1.8.7 of Safarov-Vassiliev[10℄. In parti ular, we have the following asymptoti formula that as λ → ∞X λ j ≤ λ | (cid:209) e j ( x ) | = n λ n + ( ) n/2 Γ ( + n2 ) + O x ( λ n + ) , where the onstant in the reminder term O x ( λ n + ) depends on the distan e of x to theboundary of N . Hen e, the exponents of λ in estimates (2) and (4) are sharp at x as λ → ∞ . For ea h point z on the boundary of N , Ozawa [9℄ used the heat kernel methodto show the asymptoti formula that as λ → ∞ , X λ j ≤ λ (cid:12)(cid:12)(cid:12)(cid:12) ∂e j ∂ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = λ n + ( ) n/2 Γ ( + n2 ) + o ( λ n + ) , where ν is the unit outer normal ve tor (cid:12)eld on the boundary of N . Hen e, (2) and (4)are also sharp on the boundary. Corollary 1.1.
Let e λ ( x ) be an eigenfun tion with respe t to the positive Diri hletLapla ian ∆ N on N : ∆ M e λ = λ e λ in the interior of N and e λ = on the boundary of N . Then, for every λ ≥ , there holds the upper bound estimate of (cid:209) e λ : k (cid:209) e λ k ∞ ≤ Cλ k e λ k ∞ . Proof.
Putting f = e λ in the estimate (3), we obtain the orollary. (cid:3) A tually, by the basi geometri property of nodal sets of an eigenfun tion, we an(cid:12)nd a omplete pi ture for the L ∞ norm of (cid:209) e λ in the following: Theorem 1.2.
Let e λ ( x ) be an eigenfun tion with respe t to the Diri hlet Lapla ianoperator ∆ N on N without boundary: ∆ N e λ = λ e λ . Then, for every λ ≥ , there holds λ k e λ k ∞ /C ≤ k (cid:209) e λ k ∞ ≤ Cλ k e λ k ∞ . (5)3 emark 1.3. The authors [12℄ proved the analog of Theorems 1 and 2 on ompa t man-ifolds without boundary. The proof in this paper is more ompli ated than there be ausewe need to do analysis at points near the boundary. We believe that there also hold theanalog for k - ovariant derivatives (cid:209) k χ λ f and (cid:209) k e λ on N . We plan to dis uss this questionin a future paper. Remark 1.4.
Let ψ j be the normal derivative of e j at the boundary ∂N of N . The lowerbound estimate k ψ j k L ∞ ( ∂N ) ≥ C k e j k L ∞ ( N ) does not hold in general. Using Examples 3-5 in Hassell-Tao [6℄ and doing a little bitmore omputations, we an see that the above estimate does not hold on the (cid:13)at ylinder,the hemisphere and the spheri al ylinder. We hope to (cid:12)nd a suÆ ient ondition for thelower bound estimate in a future work.We on lude the introdu tion by explaining the organization of this paper. In Se tion2, we show the lower bound of the gradient (cid:209) e λ by the basi geometri al property of thenodal set of eigenfun tions. In Se tion 3 we use the res aling method and the H(cid:127)olderestimate about ellipti PDEs to show (3) and the upper bound part of (5). The point isto do the res aling both outside and inside the boundary layer of width C λ − . (cid:209) e λ The nodal set of an eigenfun tion e λ of ∆ N is the zero set Z e λ := { x ∈ N : e λ ( x ) = } . A onne ted omponent of the open set N \ Z e λ is alled a nodal domain of the eigenfun -tion e λ . We have the same de(cid:12)nition for manifolds without boundary. Lemma 2.1. (Br(cid:127)uning [1℄) Let M be a ompa t Riemannian manifold without bound-ary. Let λ ≥ and e λ be an eigenfun tion of the positive Lapla i an ∆ M : ∆ M e λ = λ e λ .Then there exists a onstant C only depending on M su h that ea h geodesi ball ofradius C/λ in M must interse t the nodal set Z e λ of e λ . Proof.
A proof written in English is given by Zeldit h in pp. 579-580 of [16℄. (cid:3)
We need a manifold-with-boundary version of Lemma 1 as follows:4 emma 2.2.
Let λ > 0 and λ be greater than the smallest eigenvalue λ of theDiri hlet Lapla ian ∆ N . Let e λ be an eigenfun tion of ∆ N : ∆ N e λ = λ e λ in the inte-rior Int ( N ) of N and e λ = on the boundary ∂N of N . Then there exists a positive onstant D only depending on N su h that ea h geodesi ball of radius D/λ ontainedin Int ( N ) must interse t the nodal set Z e λ of e λ . Proof.
We here adapt the proof of Zeldit
h [16℄ with a slight modi(cid:12)
ation.Step 1 We show the following fa
t: There exists a
onstant C su
h that for ea
hinterior point p of N and ea
h positive number r > 0 satisfying that the distan
e d ( p, ∂N )
0, D/λ ] × S n − ( ) in the ball B ( x, D/λ ) . By themean value theorem, there exists a point z on the geodesi segment onne ting x and y su h that (cid:12)(cid:12)(cid:12)(cid:12) ∂e λ ∂r ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ λD | e λ ( x ) | = λD k e λ k ∞ . Case 2 Assume d ≤ D/λ . We may assume λ so large that there exists a uniquegeodesi γ : [
0, d ] → N of ar length parameter onne ting x and ∂N , γ ( ) = x, γ ( d ) ∈ ∂N. Sin e e λ ( γ ( d )) = , by the mean value theorem, there exists t in (
0, d ) su h that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) de λ (cid:0) γ ( t ) (cid:1) dt ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | e λ ( x ) | d ≥ λD k e λ k ∞ . (cid:3) (cid:209) χ λ f Re all the prin iple: On a small s ale omparable to the wavelength , the eigen-fun tion e λ behaves like a harmoni fun tion. It was developed in H. Donnelly andC. Fe(cid:11)erman [2℄ [3℄ and N. S. Nadirashvili [8℄ and was used extensively there. Re entlyMangoubi [7℄ applied this prin iple to studying the geometry of nodal domains of eigen-fun tions. In this se tion, for a square integrable fun tion f on N we give a modi(cid:12) ationof this prin iple, whi h an be applied to the Poisson equation ∆ N χ λ f = X λ j ∈ ( λ,λ + ] λ e j ( f ) in Int ( N ) with the Diri hlet boundary ondition χ λ f = on ∂N . In parti ular, in this subse tion, wedo the analysis outside the boundary layer L = { z ∈ N : d ( z, ∂N ) ≤ } of width .Take an arbitrary point p with d ( p, ∂N ) ≥ . We may assume that is suÆ- iently small su h that there exists a geodesi normal oordinate hart ( x , . . . , x n ) on thegeodesi ball B ( p, 2/λ ) in N . In this hart, we may identify the ball B ( p, 2/λ ) with the n -dimensional Eu lidean ball B ( ) entered at the origin , and think of the fun tion χ λ f in B ( p, 2/λ ) as a fun tion in B ( ) . Our aim in this subse tion is to show the inequality | ( (cid:209) χ λ f )( p ) | ≤ C (cid:18) λ k χ λ f k L ∞ (cid:0) B ( ) (cid:1) + λ − k ∆ N χ λ f k L ∞ (cid:0) B ( ) (cid:1)(cid:19) . (6)6or simpli ity of notions, we rewrite u = χ λ f and v = ∆ N χ λ f in what follows. The Poissonequation satis(cid:12)ed by u in B ( ) an be written as − √ g X i,j ∂ x i (cid:16) g ij √ g∂ x j u (cid:17) = v. Consider the res aled fun tions u λ ( y ) = u ( y/λ ) and v λ ( y ) = v ( y/λ ) in the ball B ( ) .The above estimate we want to prove is equivalent to its res aled version | ( (cid:209) u λ )( ) | ≤ C (cid:18) k u λ k L ∞ (cid:0) B ( ) (cid:1) + λ − k v λ k L ∞ (cid:0) B ( ) (cid:1)(cid:19) . (7)On the other hand, the res aled version of the Poisson equation has the expression, X i,j ∂ y i (cid:16) g ijλ √ g λ ∂ y j u λ (cid:17) = − λ − √ g λ v λ , (8)where g ij,λ ( y ) = g ij ( y/λ ) , g ijλ ( y ) = g ij ( y/λ ) and √ g r ( y ) = ( √ g )( y/λ ) .For ea h , there exists K > 0 su h that the C α norm of the oeÆ ients g ijλ √ g λ , √ g λ in B ( ) are bounded uniformly from above by K , and the smallest eigenvalue of the n × n matrix ( g ijλ √ g λ ) ij in B ( ) bounds from below by , for all λ ≥ . By Theorem8.32 in page 210 of Gilbarg-Trudinger [4℄, there exists onstant C = C ( n, α, K ) su h that k u λ k C (cid:0) B ( ) (cid:1) ≤ C (cid:18) k u λ k L ∞ (cid:0) B ( ) (cid:1) + λ − k v λ k L ∞ (cid:0) B ( ) (cid:1)(cid:19) , This is stronger than the estimate (7). Therefore, we omplete the proof of Theorem 1.1outside the boundary layer L λ . Using the notions in subse tion 3.1, We are going to prove the following estimate: k (cid:209) u k L ∞ ( L ) ≤ C (cid:16) λ k u k ∞ + λ − k v k ∞ (cid:17) , (9)with whi h ombining (6) ompletes the proof of Theorem 1.1.We may assume that λ is suÆ iently large so that there exists a geodesi normal oordinate hart ( z ′ , z n ) on the boundary layer L = { p ∈ N : d ( p, ∂N ) ≤ } with respe tto the boundary ∂N . Hen e, for ea h point ( z ′ , z n ) ∈ L , we have ≤ z n ≤ and d (cid:0) ( z ′ , z n ) , ∂N (cid:1) = z n . For ea h point q ∈ ∂N and r > 0 , denote by B + ( q, r ) the set of points of N with distan eless than r to q . Denote by B + ( r ) the upper half Eu lidean ball { x = ( x , . . . , x n ) ∈ R n : | x | < r, x n ≥ } r . Then, for ea h q ∈ ∂N , there exists a geodesi nor-mal hart on B + ( q, 3/λ ) su h that the exponential map exp q at q gives a di(cid:11)eomorphismfrom B + ( ) onto B + ( q, 3/λ ) .Sin e { B + ( q, 2/λ ) : q ∈ ∂N } forms an open over of L , the question an be redu ed toshowing the analog of (9) on B + ( q, 2/λ ) for ea h q . We only need to prove its equivalentres aled version, k (cid:209) u λ k L ∞ (cid:0) B + ( ) (cid:1) ≤ C (cid:18) k u λ k L ∞ (cid:0) B + ( ) (cid:1) + λ − k v λ k L ∞ (cid:0) B + ( ) (cid:1)(cid:19) , (10)where u λ and v λ are the the res aling fun tion of u an v , respe tively. Observe that u λ and v λ satisfy the Poisson equation ( ) in the upper half Eu lidean ball B ( ) and theDiri hlet boundary ondition, u λ = on the portion { x ∈ B ( ) : x n = } of the boundary of B + ( ) . For ea h , there exists K > 0 su h that the C α normof the oeÆ ients g ijλ √ g λ , √ g λ in B + ( ) are bounded uniformly from above by K , and thesmallest eigenvalue of the n × n matrix ( g ijλ √ g λ ) ij in B + ( ) bounds from below by ,for all λ ≥ . By Theorem 8.36 in page 212 of Gilbarg-Trudinger [4℄, there exists onstant C = C ( n, α, K ) su h that k u λ k C (cid:0) B + ( ) (cid:1) ≤ C (cid:18) k u λ k L ∞ (cid:0) B + ( ) (cid:1) + λ − k v λ k L ∞ (cid:0) B + ( ) (cid:1)(cid:19) . This is a stronger estimate than (10).
Acknowledgements
Yiqian Shi is supported in part by the National Natural S ien eFoundation of China (No. 10671096 and No. 10971104) and Bin Xu by the NationalNatural S ien e Foundation of China (No. 10601053 and No. 10871184).
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