Gradient estimates for eigenfunctions on compact Riemannian manifolds with boundary

Transcription

1 Gradient estimates for eigenfunctions on compact Riemannian manifolds with boundary Xiangjin Xu Department of athematics Johns Hopkins University Baltimore, D Abstract The purpose of this paper is to prove the L gradient estimates and L gradient estimates for the unit spectral projection operators of the Dirichlet Laplacian and Neumann (or more general, Ψ 1 -Robin) Laplacian on compact Riemannian manifolds (, g) of dimension n 2 with C 2 boundary. And we also get an upper bounds for normal derivatives of the unit spectral projection operators of the Dirichlet Laplacian from L 2 () to L 2 ( ). 1 Introduction In this paper we discuss the L gradient estimates for the eigenfunctions of the Dirichlet Laplacian and Neumann (or more general, Ψ 1 -Robin) Laplacian on compact Riemannian manifolds (, g) of dimension n 2 with C 2 boundary. First we consider the Dirichlet eigenvalue problem ( + λ 2 )u(x) = 0, x, u(x) = 0, x, (1) with = g being the Laplace-Beltrami operator associated to the Riemannian metric g. Recall that the spectrum of (1) is discrete and tends to infinity. Let 0 < λ 2 1 λ 2 2 λ 2 3 denote the eigenvalues, and let {e j (x)} be an associated real orthogonal normalized basis in L 2 (), and define e j (f)(x) = e j (x) f(y)e j (y)dy, and the unit band spectral projection operators, χ λ f = e j (f), xxu@math.jhu.edu 1

2 In [13], Sogge proved that for a fixed compact Riemannian manifold (, g) with boundary, there is a uniform constant C so that which is equivalent to χ λ f Cλ (n 1)/2 f 2, λ 1, e j (x) 2 Cλ n 1, x. Using the above estimates, we obtain the L estimates for χ λ f, Theorem 1.1 Fix a compact Riemannian manifold (, g) with C 2 boundary, there is a uniform constant C so that Remark 1.1 Note that χ λ f Cλ (n+1)/2 f 2, λ 1. χ λ f(x) = e j (x)e j (y)f(y)dy, therefore, by the converse to Schwarz s inequality and orthogonality, one has the bounds at a given point x if and only if χ λ f(x) Cλ (n+1)/2 f 2, e j (x) 2 Cλ n+1. In the proof at section 2, we shall prove either one of these two kind gradient estimates. Remark 1.2 In [7], for a bounded region in R n with smooth boundary, by studying the heat kernel of the Laplacian on, Ozawa proved λ j λ e j(x) ν 2 = Cλ n+2 + o(λ n+2 ), as λ, for every x, where ν is the outward normal derivative at x. From our proof in Section 2, where we use standard gradient estimates for second order elliptic operators, we essentially prove e j (x) 2 = C(x)(λ 2 + O(λ)) e j (x) 2 2

3 for interior points with dist(x, ) ɛλ 1 and boundary points, where the constant C(x) is independent on λ. and e j (x) 2 C(λ 2 + O(λ)) e j (x) 2 for the points in the boundary strip {x : dist(x, ) ɛλ 1 }. Hence from the asymptotic estimates on we have {j : λ j λ} = λ j λ e j (x) 2 dx = C 1 λ n + C 2 λ n 1 + o(λ n 1 ), λ j λ λ 2 j = λ j λ e j (x) 2 = Cλ n+2 + O(λ n+1 ), as λ. We prove the above gradient estimates according to three different cases: for the interior points with dist(x, ) ɛλ 1, for the boundary points, and for the points on the strip near boundary with dist(x, ) ɛλ 1. For interior points, we use the idea of gradient estimates for poisson equations, and apply this idea to manifolds without boundary, we obtain the gradient estimates directly. For boundary points, we show the gradient estimates by a perturbation from a constant coefficients differential operators. For the points in the λ 1 strip near boundary, we follow the idea as used in [13] and use the maximum principle to show the gradient estimates. Next we consider the eigenvalue problem on with Neumann or Ψ 1 -Robin boundary conditions ( + λ 2 )u(x) = 0, x, n u(x) = K(x)u(x), x, (2) where n is the outward normal derivative on, K C ( ) as in [4], when K(x) = 0, (2) become Neumann boundary problem. As for Dirichlet Laplacian (1), the spectrum of (2) is also discrete and tends to infinity. Let 0 λ 2 1 λ 2 2 λ 2 3 denote the eigenvalues, and let {e j (x)} be an associated orthogonal normalized basis in L 2 (), and define e j (f)(x) and the unit band spectral projection operators, χ λ f, as for Dirichlet Laplacian. In [2], Grieser proved the L estimates e j (f) L Cλ (n 1)/2 j f L 2 for the Neumann Laplacian. Following the idea in [13], where the L estimates on χ λ is proved for Dirichlet Laplacian, and our proof for the L estimates on χ λ for Dirichlet Laplacian, we have the L estimates on χ λ and χ λ, 3

4 Theorem 1.2 Fix a compact Riemannian manifold (, g) with C 2 boundary, for Neumann Laplacian on, there is a uniform constant C so that χ λ f Cλ (n 1)/2 f 2, λ 1, χ λ f Cλ (n+1)/2 f 2, λ 1. Here We follow the idea in [2] and [13] to show the L estimates on χ λ for Ψ 1 -Robin Laplacian. Notice that since the gradient estimates for boundary points is trivial for Ψ 1 - Robin Laplacian, we need only do the gradient estimates for interior points and for points on the λ 1 strip near the boundary, as did in Section 2, to get the L gradient estimates for Neumann Laplacian, after we prove the L estimates on χ λ. Finally we study the upper bounds for normal derivatives of the unit spectral projection operators Laplacian on L 2 (). Let 0 < λ 2 1 λ 2 2 λ 2 3 denote the eigenvalues for the Dirichlet Laplacian, and {e j (x)} be an associated real orthogonal normalized basis in L 2 (), and define e j (f)(x) and the unit band spectral projection operators, χ λ f, as above. In [3], Hassell and Tao proved the following inequality for single Dirichlet eigenfunctions cλ j e j ν L 2 ( ) Cλ j, where the upper bound holds for some constant C independent of j, and the lower bound holds provided that can be embedded in the interior of a compact manifold with boundary, N, of the same dimension, such that every geodesic in eventually meets the boundary of N. Using the idea to show the Rellich-type estimates in [3] and the orthogonality of the eigenfunctions, we have the following results: Theorem 1.3 Let be a smooth compact Riemannian manifold with boundary, and function f L 2 (), then χ λ(f) L ν 2 ( ) Cλ 3/2 f L 2 (), holds for some constant C independent of λ. Remark 1.3 From the Weyl s law {j : λ j λ} C λ n as λ with C = (2π) n ω n V ol(), where ω is the volume of the unit ball in R n, we know 4

5 {j : λ j [λ, λ + 1)} nc λ n 1 as λ Here in the proof, we make use of the orthogonality of spectral projections {e j (f)}, which is one key observation to get our results. Here is the outline of this paper. In Section 2, we prove Theorem 1.1 according to three cases, and obtain the same gradient estimates for χ λ on compact manifolds without boundary. In Section 3, we deal with the L estimates on χ λ and χ λ for the Neumann or Ψ 1 -Robin boundary problem, and we prove Theorem 1.2. In Section 4, we deal with the upper bounds for normal derivatives of the unit spectral projection operators for Dirichlet eigenfunctions in L 2 sense, and we prove Theorem 1.3. In what follows we shall use the convention that C will denote a constant that is not necessarily the same at each occurrence. It is a pleasure to thank my advisor, Professor C.D. Sogge, brings the problem of L gradient estimates on χ λ to me and a number helpful conversations on this problems and my research. And I also thank Professor J. Spruck shows me the proofs of Lemma 2.2 and Lemma 2.3 for single eigenfunctions, and Professor S. Zelditch brings the reference [7] to me. 2 Gradient estimates For Dirichlet Laplacian In this section, study the eigenfunctions for the Dirichlet Laplacian, and we shall prove Theorem 1.1. by using maximum principle according to three different cases: for the interior points with dist(x, ) ɛλ 1, for the boundary points, and for the points on the strip near boundary with dist(x, ) ɛλ 1. And for the eigenfunctions for the Dirichlet Laplacian on compact boundless manifolds, we also get the gradient estimates on χ λ f applying the gradient estimates for interior points. First we show the gradient estimate for the interior points with dist(x, ) ɛ(λ + 1) 1, Lemma 2.1 For the Riemannian manifold (, g) with boundary, we have the gradient estimate χ λ f(x) C ɛ λ (n+1)/2 χ λ f 2, for dist(x, ) ɛ(λ + 1) 1 Proof. We shall show this Lemma following the ideas in [1], where for the Poisson s equation u = f, there are gradient estimates for the interior point x 0 as u(x 0 ) C d sup B u + Cd sup f, B 5

6 by using maximum principle in a cube B centered at x 0 with length d. Now we fix ɛ and x 0 with dist(x 0, ) ɛ(λ + 1) 1. We shall use maximum principle in the cube centered at x 0 with length d = ɛ(λ + 1) 1 / n to prove the same gradient estimates for χ λ f as above for Poisson s equation. Define the geodesic coordinates x = (x 1,, x n ) centered at point x 0 as following, fixed an orthonormal basis {v i } n T x0, identity x = (x 1,, x n ) R n with the point exp( n x i v i ). In small neighborhood of x 0 we take the metric with the form and the Laplacian can be written as g = n n g ij (x)dx i dx j, 2 g ij (x) + x i x j n b i (x) x i, Where (g ij (x)) 1 i,j n is the inverse matrix of (g ij (x)) 1 i,j n, and b i (x) are in C. Now define the cube Q = {x = (x 1,, x n ) R n x i < d, i = 1,, n}, where we can choose d = ɛ(λ + 1) 1 / n dist(x 0, )/ n. Denote u(x; f) = χ λ f(x), we have u C 2 (Q) C 0 ( Q), and g u(x; f) = λ 2 je j (f) := h(x; f). From the L estimate in [13], and Cauchy-Schwarz inequality, we have h(x; f) 2 = ( (λ + 1) 4 ( (λ 2 je j (x))( λ 4 je 2 j(x) ( C(λ + 1) n+3 χ λ f 2 L 2 () e j (y)f(y)dy)) 2 e 2 j(x)) χ λ f 2 L 2 () e j (y)f(y)dy) 2 We estimate D n u(0; f) = x n u(0; f) first, the same estimate holds for D i u(0; f) with i = 1,, n 1 also. Now in the half-cube Q = {x = (x 1,, x n ) R n x i < d, i = 1,, n 1, 0 < x n < d.}, 6

7 Consider the function ϕ(x, x n ; f) = 1 2 [u(x, x n ; f) u(x, x n ; f)], where we write x = (x, x n ) = (x 1,, x n 1, x n ). One see that ϕ(x, 0; f) = 0, sup Q ϕ sup Q u := A, and g ϕ sup Q h := N in Q. Consider also the function ψ(x, x n ) = A d 2 [ x 2 + αx n (nd (n 1)x n )] + βnx n (d x n ) defined on the half-cube Q and α 1 and β 1 will be determined below. Obviously ψ(x, x n ) 0 on x n = 0 and ψ(x, x n ) A in the remaining portion of Q. g ψ(x) = A n d 2 [2tr(gij (x)) (2nα 2α + 1) + 2 b i (x)x i +b n (x)(nαd (2nα 2α + 1)x n )] + Nβ[ 2g nn (x) + b n (x)(d 2x n )] Since in, tr(g ij (x)) and b i (x) are bounded uniformly and g nn (x) is positive, then for a large α, we can make 2tr(g ij (x)) (2nα 2α + 1) 1, Fix such a α, since d = ɛ(λ + 1) 1 / n, for large λ, we have n 2 b i (x)x i + b n (x)(nαd (2nα 2α + 1)x n ) < 1. Then the first term is negative. For second term, let β large enough, we have β[ 2g nn (x) + b n (x)(d 2x n )] < 1. Hence we have g ψ(x) N in Q. Now we have g (ψ ± ϕ) 0 in Q and ψ ± ϕ 0 on Q, from which it follows by the maximum principle that ϕ(x, x n ; f) ψ(x, x n ) in Q. Letting x = 0 in the expressions for ψ and ϕ, then dividing by x n and letting x n tend to zero, we obtain D n u(0; f) = lim ϕ(0, x n; f) αna xn 0 d x n + βdn. Note that d = ɛ(λ + 1) 1 / n, A C(λ + 1) (n 1)/2, and N (λ + 1) (n+3)/2, then we have the estimate D n u(0; f) C ɛ (λ + 1) (n+1)/2. The same estimate holds for D i u(0), i = 1,, n 1. Hence we have u(0) C ɛ (λ + 1) (n+1)/2. Since the estimate is for any x 0 with dist(x 0, ) ɛ(λ + 1) 1, the Lemma is proved. Q.E.D. 7

8 Remark 2.1 One can also use the parametrix for wave kernel to show local Weyl estimates for interior points, then obtain the gradient estimates as our Lemma directly by estimating the gradient of the parametrix for wave kernel. Here since we use the L estimates on χ λ f in [13], where have used the parametrix method already, we can use basic method as maximum principle to get gradient estimates here. From the proof, we can see that for interior points we essentially show which in turn implies χ λ f(x) 2 = C(x)(λ 2 + O(λ)) χ λ f 2 L, e j (x) 2 = C(x)(λ 2 + O(λ)) e j (x) 2. Next we show the following gradient estimates for the points on boundary. Lemma 2.2 Assume u(x) C 1 λ (n 1)/2 in and u(x) = 0 on C 2, and g u(x) C 2 λ (n+3)/2, then we have u(x) Cλ (n+1)/2, x. Proof. As did in proof of Proposition 2.2 in [13], we use the geodesic coordinates with respect to the boundary. We can find a small constant c > 0 so that the map (x, x n ) [0, c), sending (x, x n ) to the endpoint, x, of the geodesic of length x n which starts a x and is perpendicular to is a local diffeomorphism. In this local coordinates x = (x 1,, x n 1, x n ), the metric has the form n n g ij (x)dx i dx j = (dx n ) 2 + g ij (x)dx i dx j, and the Laplacian can be written as g = n 2 g ij (x) + x i x j n b i (x) x i, Where (g ij (x)) 1 i,j n is the inverse matrix of (g ij (x)) 1 i,j n, and g nn = 1, and g nk = g kn = 0 for k n, and b i (x) are in C and real valued. Fix a point x 0, choose a local coordinate so that x 0 = (0,, 0, 0). Without loss the generality, we may assume that at the point x 0, g (x 0 ) =, the Euclidean Laplacian, since we can transfer g (x 0 ) to by a suitable nonsingular linear transformation as did 8

9 in Chapter 6 at [1]. Since g ij (x) and b i (x) are C, we have a constant Λ > 0 such that g ij (x) < Λ and b i (x) < Λ hold for all x is the λ 1 strip of the boundary. We first assume n = dim 3. For n = 2, we need define another comparing function v(x). Now let R = λ 1, define a function v(x) = αλ (3 n)/2 1 ( R 1 n 2 r ) + n 2 βλ(n+3)/2 (R 2 r 2 ), on A R = B 2R (0) B R (0), where r = x (0,, 0, R) R n, B r (0) = {x = (x 1,, x n ) R n x (0,, 0, R) < r}, and α 1 and β 1 will be determined below. Here we assume B R (0), is tangent to at point x 0 from outside. We will compare v(x) and u(x) in A R. For any x A R, we have g v(x) = [ (n 2)tr(gij (x)) n(n 2) g ij (x)x i x j + r n r n+2 [tr(g ij (x)) + 2 b i (x)x i ]βλ (n+3)/2, (n 2) bi (x)x i ]αλ (3 n)/2 r n Since is a compact manifolds, we have 0 < θ < Θ < such that b i (x) < Θ, θ < tr(g ij (x)) < Θ and θ y 2 < g ij (x)y i y j < Θ y 2 hold for all x. And since x B R and R = λ 1, we have λ 1 < x < 2λ 1 and b i (x)x i < θ/4 for large λ. Then we have Now let g v(x) n(n + 1)Λ x x 0 αλ (3 n)/2 (θ 2 θ r n 4 )βλ(n+3)/2 3n(n + 1)ΛR αλ (3 n)/2 θ 2 βλ(n+3)/2 r n 3n(n + 1)Λαλ (n+1)/2 θ 2 βλ(n+3)/2 3n(n + 1)Λαλ 1 θ 2 β C 2, (3) where C 2 is the constant in the assumptions of this Lemma, then we have g v(x) u(x) in A R. Next we compare the values of v and u on (A R ). Case I, x (A R ) A R. v(x) = αλ (3 n)/2 1 ( R 1 n 2 (2R) ) + n 2 βλ(n+3)/2 (R 2 (2R) 2 ) ( α 2 3β)λ(n 1)/2, Now let 9

10 α 2 3β C 1, (4) then we have v(x) u(x) for x (A R ) A R. Case II, x (A R ). v(x) = αλ (3 n)/2 1 ( R 1 n 2 r ) + n 2 βλ(n+3)/2 (R 2 r 2 ) := h(r), Since B R (0) is tangent to at point x 0 from outside, we know the range of h(r) is [R, 2R] and f(r) = 0, Now let f (r) = (n 2)αλ (3 n)/2 1 r n 1 2βλ(n+3)/2 r (2 1 n (n 2)α 2β)λ (n+1)/ n (n 2)α 2β > 0, (5) Then f(r) f(r) = 0, and we have v(x) u(x) for x (A R ). Finally we need determine the values of α and β. Let β = 4C 2 /θ, then from (3), (4), (5), we have the range of α as max{ 2n β n 2, 6β + 2C 1} α C 2 λ 3n(n + 1)Λ. For large λ, the range of α is not empty set. Hence for large λ, we can find a function v(x) such that g (v(x)±u(x)) 0 in A R and v(x) ± u(x) 0 on (A R ), from which it follows by the maximum principle that v(x) ± u(x) 0 in A R. On the other hand, v(x 0 ) u(x 0 ) = 0. Hence we have u(x 0 ) v r (x 0) = [(n 2)α 2β]λ (n+1)/2 := C λ (n+1)/2 Since x 0 is an arbitrary point on, the Lemma is proved for n 3. For n = 2, we define the function v(x) = αλ 1/2 (lnr lnr) + βλ 5/2 (R 2 r 2 ), for x A R. By the same computation as above, we can show the Lemma holds also. Q.E.D. Note that from [13], we know χ λ f(x) satisfies the conditions of above Lemma, since 10

11 χ λ f(x) 2 = e j (f) 2 ( e j (x) 2 ) 2 χ λ f 2 L 2 Cλn 1 χ λ f 2 L 2, and g χ λ f(x) = (λ + 1) 2 ( λ 2 je j (f) Cλ (n+3)/2 χ λ f L 2 e j (x) 2 ) 1 2 χλ f L 2 Apply that above Lemma to χ λ f, we have the gradient estimate on the boundary points. Lemma 2.3 For the Riemannian manifold (, g) with boundary C 2, we have the gradient estimate χ λ f(x) Cλ (n+1)/2 χ λ f 2, x. Remark 2.2 One may also deal with the gradient estimate for boundary points as Lemma 2.1. Notice χ λ f(x, 0) = 0 for boundary points when we set the half cube on the boundary, then compare u(x, x n ) and ψ(x, x n ) we can get the gradient estimates for the boundary points. Now we need only to deal with the gradient estimate in the λ 1 strip of the boundary. Lemma 2.4 For the Riemannian manifold (, g) with boundary C 2, we have the gradient estimate χ λ f(x) C ɛ λ (n+1)/2 χ λ f 2, for 0 dist(x, ) ɛ(λ + 1) 1. where ɛ is the same as in Lemma 2.1 and we will determine its value here. Proof. Here we shall apply the maximum principle to λ l [λ,λ+1) e j (x) 2 on the λ 1 boundary strip as in [13], where dealt with the L estimates for χ λ f on the λ 1 boundary strip. As Lemma 2.2, we also use the geodesic coordinates with respect to the boundary. First we estimate 1 2 g λ l [λ,λ+1) e j (x) 2 = λ l [λ,λ+1) i,j,k=1 [ n g ij (x) 2 e l (x) x i x k 2 e l (x) x j x k +( e l (x), ( e l (x))) + Ric( e l (x), e l (x))] 11

12 Since (g ij (x)) is metric tensor and is a compact manifolds, we have a constant θ > 0 such that g ij (x)y i y j θ y 2 holds for all y = (y 1,, y n ) R n and all x. And Ric > B for some positive constant B in whole. Then for large λ, we have B 2λ + 3. Hence we have 1 2 g λ l [λ,λ+1) e j (x) 2 λ l [λ,λ+1) (λ + 2) 2 [λ 2 l e j (x) 2 + Ric( e l (x), e l (x))] λ l [λ,λ+1) e j (x) 2 Now we define a function w(x) = 1 a(λ + 1) 2 x 2 n for the strip {x 0 x n ɛ(λ + 1) 1 }, and the constants a and ɛ will be determined below. We have aɛ2 w(x) 1, (6) and following the computation in proof of Proposition 2.2 in [13], we have w(x) = 2a(λ + 1) 2 2ab n (x)x n (λ + 1) 2 a(λ + 1) 2, for all point in the strip, here assuming that λ is large enough so that b n (x)x n 1/2. Define h(x) = λ l [λ,λ+1) e j (x) 2 /w(x), we have w(x) g h(x) + h(x) w(x) + 2( h(x), w(x)) = g e j (x) 2 2(λ + 2) 2 e j (x) 2 = 2(λ + 2) 2 w(x)h(x). λ l [λ,λ+1) λ l [λ,λ+1) Divide w(x) both sides, and apply the estimate of g w(x), we have h(x) + 2( h(x), w(x) w(x) ) + (4 a)(λ + 1)2 h(x) 0. If we assume a > 4, h(x) achieves its maximum on {x 0 x n ɛ(λ + 1) 1 }, and we have the gradient estimates both boundary point and interior points. Hence we have sup {x 0 x n ɛ(λ+1) 1 } λ l [λ,λ+1) e j (x) 2 Cλ n+1. 12

13 Finally we determine the constant a and ɛ. From proof, we need a > 4 and aɛ 2 1/2, which is easy to satisfy, for example, we may let a = 8 and ɛ = 1/4. Q.E.D. Combine above Lemmas, we get the gradient estimates for eigenfunctions of compact Riemannian manifolds with C 2 boundary. Since here our proof only involve the L estimate of the eigenfunctions and didn t use any geometric property. Notice that for compact Riemannian manifolds without boundary, by Lemma 2.1 and the L estimates on the unit spectral projection operators χ λ for a second-order elliptic differential operator on compact manifolds without boundary [12], we have the following gradient estimates Theorem 2.1 Fix a compact Riemannian manifold (, g) of dimension n 2 without boundary, there is a uniform constant C so that χ λ f Cλ (n+1)/2 f 2, λ 1. And the bounds are uniform if there is a uniformly bound on the norm of tr(g ij (x)) for a class metrics g on. For Riemannian manifolds without boundary, in [14], the authors proved that for generic metrics on any manifold one has the bounds e j L () = o(λ (n 1)/2 j ) for L 2 normalized eigenfunctions. As Lemma 2.1, we can show the L gradient estimates e j = o(λ (n+1)/2 j ) for generic metrics on any manifold. For Laplace-Beltrami operator g, there are eigenvalues { λ 2 j}, where 0 λ 2 0 λ 2 1 are counted with multiplicity. Let {e j (x)} be an associated orthogonal basis of L 2 normalized eigenfunctions. If λ 2 is in the spectrum of g, let V λ = {u g u = λ 2 u} denote the corresponding eigenspace. We define the eigenfunction growth rate in term of L (λ, g) = and the gradient growth rate in term of sup u L. u V λ ; u L 2 =1 L (, λ, g) = sup u L. u V λ ; u L 2 =1 In [14], Sogge and Zelditch proved the following results L (λ, g) = o(λ (n 1)/2 j ) for a generic metric on any manifold. Here we apply our Lemma 2.1, we have the gradient estimates 13

14 Theorem 2.2 L (, λ, g) = o(λ (n+1)/2 j ) for a generic metric on any manifold. And the bounds are uniform if there is a uniformly bound on the norm of tr(g ij (x)) for (, g). Proof. here the manifold has no boundary, we can apply our Lemma 2.1 to any point in. From Theorem 1.4 in [14], we have L (λ, g) = o(λ (n 1)/2 j ) for a generic metric on any manifold. Fix that metric on the manifold and a L 2 normalized eigenfunction u(x), apply our Lemma 2.1 to u(x) at each point x 0, we have u(x 0 ) αna + βdn. d where α and β are constants depending on the norm of tr(g ij (x)) at only, and A = sup Q u = o(λ (n 1)/2 j ), and N = sup Q λ 2 u = (λ (n+3)/2 j ), where the cube we choose d = (λ + 1) 1 / n. Hence we have Q = {x = (x 1,, x n ) R n x i < d, i = 1,, n}, u(x 0 ) = o(λ (n+1)/2 j ) holds for all L 2 normalized eigenfunction u(x) V λ, furthermore, the bounds are uniform when those metrics of (, g) have a uniformly bound on the norm of tr(g ij (x)) from the proof. Hence we have our Theorem. Q.E.D. 3 The Neumann or Ψ 1 -Robin Problem In this section we shall prove Theorem 1.2 for Neumann or Ψ 1 -Robin Laplacian following the ideas in [2], where the same L estimates were done for single eigenfunctions of Neumann Laplacian, and [4], where the parametrix construction for Ψ 1 -Robin condition is discussed, and in [13], where the L estimates were done for χ λ f of Dirichlet Laplacian. Here we use the geodesic coordinates with respect to the boundary. We can find a small constant c > 0 so that the map (x, x n ) [0, c), sending (x, x n ) to the endpoint, x, of the geodesic of length x n which starts a x and is perpendicular to is a local diffeomorphism. In this local coordinates x = (x 1,, x n 1, x n ), the metric has the form n n g ij (x)dx i dx j = (dx n ) 2 + g ij (x)dx i dx j, and the Laplacian can be written as g = n 2 g ij (x) + x i x j n b i (x) x i, 14

15 Where (g ij (x)) 1 i,j n is the inverse matrix of (g ij (x)) 1 i,j n, and g nn = 1, and g nk = g kn = 0 for k n. Also the b i (x) are C and real valued. We first show that one has the uniform bounds for Ψ 1 -Robin boundary problem Note that χ λ f(x) Cλ (n 1)/2 f L 2, λ 1. χ λ f(x) = e j (x)e j (y)f(y)dy, therefore, by the converse to Schwarz s inequality and orthogonality, one has the above bounds at one given point x if and only if (e j (x)) 2 C 2 λ n 1. Because of this, we need only to should the following two results. Proposition 3.1 Fix (, g) as Theorem 1.2. Then, for a given small constant ɛ > 0, there is a uniform constant C so that for λ 1 (e j (x)) 2 C ɛ λ n 1, for any interior point x satisfying dist(x, ) ɛλ 1. Proposition 3.2 If (, g) as Theorem 1.2, then for large λ we have max {x : dist(x, ) 1 2 (λ+1) 1 } (e j (x)) 2 4 max {x : dist(x, )= 1 2 (λ+1) 1 } (e j (x)) 2. Proof of Proposition 3.1. We shall see that the estimate in this Proposition is an immediate consequence of the similarly results as Theorem in Hörmander [6] for Neumann or Ψ 1 -Robin boundary problem, if we replace the Dirichlet wave kernel by the Neumann wave kernel or Ψ 1 -Robin wave kernel in Chapter 17 in [6], we can modify straightforward the Hörmander s argument for Dirichlet Laplacian in his book line by line to work for the Neumann Laplacian as well (with different constants), which in turn is based on earlier work of Seeley [10] and Pham The Lai [8]. One can also see detail proof of this argument in [4]. To state this result, we let e(x, λ) = (2π) n {ξ R n ξ λ} 15 (1 e i2dist(x, )ξn) )dξ,

16 If we assume that the local coordinates have been chosen so that the Riemannian volume form is dx 1 dx n, then the result just quoted says that there is a uniform constant C so that for λ 1, (e j (x)) 2 e(x, λ) Cλ(λ + dist(x, ) 1 ) n 2. λ j λ Since λ(λ + dist(x, ) 1 ) n 2 = O(λ n 1 ) for all x satisfying dist(x, ) ɛλ 1. This yields Proposition 3.1 since e(x, λ + 1) e(x, λ) C ɛ λ n 1, x with dist(x, ) ɛλ 1. Hence we have our assertion hold. Q.E.D. Proof of Proposition 3.2. Here we use the geodesic coordinates with respect to the boundary. we use the same function H(x) as in Proof of Proposition 2.4 in [13], where the estimate for Dirichlet Laplacian is discussed, to apply the maximum principle to get bound at boundary of the λ 1 strip, and furthermore, we show the outward normal derivatives of H(x) on the boundary must be strictly positive as pointed in [2] for single eigenfunctions of Neumann Laplacian. In what follows we shall assume that λ is large enough so that λ C/c, where C is some universe constant and c is the uniform constant with respect to the local coordinates. Assume further that spec( g ) [λ, λ + 1], and consider the function where H(x) = 1 (w(x)) 2 (e j (x)) 2, w(x) = 1 (λ + 1) 2 x 2 n. Support that in the strip {x : 0 x n 1(λ + 2 1) 1 } the function H(x) has a maximum at an interior point x = x 0. Then v(x) = 1 w(x) e j (x 0 w(x 0 ) e j(x) must have a positive maximum at x = x 0. For because of our assumptions on the spectrum we then have v(x 0 ) = 2 > 0, while at other points in the strip e j (x 0 w(x 0 ) 16

17 v(x) 1 w(x) ( = (H(x)) 1/2 (H(x 0 )) 1/2 H(x 0 ) = v(x 0 ) e j (x) 2 ) 1/2 1 w(x 0 ) ( e j (x 0 ) 2 ) 1/2 Note that in the strip {x : 0 x n 1 2 (λ + 1) 1 } we have ( + λ 2 j)w = 2(λ + 1) 2 2b n (x)x n (λ + 1) 2 + λ j (1 (λ + 1) 2 x 2 n) (λ + 1)2, for λ j λ + 1, 2 assuming that λ is large enough so that 2b n (x)x n 1/2 in the strip. Also, in this strip we have that 1 w(x) 1. 2 Let us set v j (x) = e j(x) e j (x 0 ) w(x) w(x 0 ), so that v(x) = λ j [λ, λ + 1)v j (x). We also set u j (x) = e j(x 0 ) w(x 0 ) e j(x), and note that + λ 2 j)u j (x) = 0. A computation (one may see p.72, [9]) shows that for a given j we have 0 = = 1 w(x) ( + λ2 j)u 2 j(x) n n g kl (x) k l v j + k,l=1 k=1( 2 w n l=1 g kl l w + b k ) k v j + v j w ( + λ2 j)w. Therefore, if we sum over λ j [λ, λ + 1), we get n k,l=1 n g kl (x) k l v + k=1( 2 w n l=1 In particular, at point x = x 0, we have g kl v j l w + b k ) k v = sum λj [λ,λ+1) w ( + λ2 j)w. 17

18 n n g kl (x 0 ) k l v(x 0 ) + k,l=1 k=1( 2 n g kl (x 0 ) l w(x 0 ) + b k (x 0 )) k v(x 0 ) w l=1 = 1 ( e j(x 0 ) w(x 0 ) λ j w(x [λ,λ+1) 0 ) )2 ( + λ 2 j)w(x 0 ) (λ + 1)2 2w(x 0 ) ( e j(x 0 ) w(x 0 ) )2 > 0. (7) But this is impossible since v have a positive maximum at x 0, which implies that k v(x 0 ) = 0 for every k, and n k,l=1 g kl (x 0 ) k l v(x 0 ) 0. Thus, we conclude that the function H(x) cannot have a maximum at an interior point of the strip {x : 0 x n 1(λ + 2 1) 1 }. Now we shall prove the function H(x) cannot have a maximum value of the strip on. Suppose that there is a maximum at the boundary point x = (x, 0) on. Then by the same argument as above, we have v(x) must have a positive maximum at x = (x, 0) and v((x, 0)) > 0. It implies v has a positive maximum at (x, 0) on, in our local coordinates,which means that k v((x, 0)) = 0 for every k n 1, and n 1 k,l=1 gkl ((x, 0)) k l v((x, 0)) 0. And we also have n v(x) = ( n 1 w(x) ) = 2(λ + 1)2 x n w(x) 2 e j ((x, 0)) w((x, 0)) e j(x) + 1 w(x) e j ((x, 0)) w((x, 0)) e j(x) + 1 w(x) Since the Ψ 1 -Robin boundary condition, we have n v((x, 0)) = K((x, 0)) e 2 j((x, 0)). e j ((x, 0)) w((x, 0)) ne j (x) e j ((x, 0)) w((x, 0)) ne j (x). For our local coordinates, we have g nn = 1, and g nk = g kn = 0 for k n. Hence from (7), we have nv((x 2 (λ + 1)2, 0)) = [ 2 K((x, 0))] e 2 j((x, 0)) > 0, for large λ. But it cannot be true since in local coordinates, we have that v(x 0, x n ) gets its maximum at (x, 0) = (x 0, 0) from our assumption and n v(x, 0) = 0, which implies 2 nv(x, 0) 0. It is a contradiction. Because of this and our lower bound for w, we get that 18

19 max {x : dist(x, ) 1 2 (λ+1) 1 } (e j (x)) 2 4 max {x : dist(x, )= 1 2 (λ+1) 1 } (e j (x)) 2 As desired, which completes the proof of Proposition 3.2. Q.E.D. Combine Proposition 3.1 and Proposition 3.2, we proved the L estimates on χ λ for Neumann or Ψ 1 -Robin Laplacian. Notice that since the gradient estimates for boundary points is trivial because of n χ λ (x) = K(x)χ λ (x) for x and the L estimates on χ λ, we need only do the gradient estimates for interior points and for points on the λ 1 strip near the boundary, as did in Section 2, to get the L gradient estimates for Neumann or Ψ 1 -Robin Laplacian. Notice in the proofs of Lemma 2.1 and Lemma 2.4, we did not use the assumption χ λ f = 0 for Dirichlet Laplacian, hence we can apply Lemma 2.1 and Lemma 2.4 to χ λ f for Neumann or Ψ 1 -Robin Laplacian directly. Then we prove Theorem Upper bounds for normal derivatives In this section, we study the upper bounds for normal derivatives of the unit spectral projection operators of Dirichlet Laplacian on L 2 (), where (, g) is a compact Riemannian manifold of dimension n 2 with C 2 boundary. Let 0 < λ 2 1 λ 2 2 λ 2 3 denote the eigenvalues for the Dirichlet Laplacian, and {e j (x)} be an associated real orthogonal normalized basis in L 2 (), and define e j (f)(x) and the unit band spectral projection operators, χ λ f, for function f L 2 (), and we also denote (χ (1) λ f) = gχ λ f = λ 2 je j (f). Following the idea using in the proof of the Rellich-type estimates in [3], we have the following Rellich-type Lemma for χ λ f. Lemma 4.1 Let χ λ f be the operators on f L 2 () defined as above, then for any differential operator A, + (χ λ f) A((χ λ f))dσ = ((χ λ f), [ g, A]((χ λ f)))dx ν ((χ λ f), A((χ (1) λ f)))dx ((χ (1) λ f), A((χ λf)))dx, (8) where g is the Laplace-Beltrami operator associated to the Riemannian metric g, ν is the outward normal derivative on the boundary, dσ is the area element on the boundary, and dx is the volume element on. 19

20 Proof. Since χ λ f vanishes at the boundary, we apply Green s formula to get = = = (χ λ f) A((χ λ f))dσ ν ((χ λ f), g (Aχ λ f)))dx ( g ((χ λ f)), A((χ λ f)))dx ((χ λ f), g (A(χ λ f)))dx (χ λ f, A((χ λ f)))dx + ((χ λ f), [ g, A]((χ λ f)))dx ((χ λ f), A((χ (1) λ f)))dx ((χ (1) λ f), A((χ λf)))dx Here we use the fact that e j (f) satisfying the following Dirichlet boundary problem ( g + λ 2 j)u(x) = 0, x ; u(x) = 0, x. Q.E.D. To prove an upper bound for the L 2 ( ) norm of ν (χ λ f) on the boundary, we shall choose an operator A so that the left hand side of (8) is a positive form in ν (χ λ f). As before, we use the geodesic coordinates with respect to the boundary. We can find a small constant c > 0 so that the map (x, x n ) [0, c), sending (x, x n ) to the endpoint, x, of the geodesic of length x n which starts a x and is perpendicular to is a local diffeomorphism. In this local coordinates x = (x, x n ) = (x 1,, x n 1, x n ), the metric has the form n g ij (x)dx i dx j = (dx n ) 2 + and the Laplacian can be written as n 1 g ij (x )dx i dx j, g = = n n 2 g ij (x) + x i x j g ij (x), x i x j n b i (x) x i Where (g ij (x)) 1 i,j n is the inverse matrix of (g ij (x)) 1 i,j n, and g nn = 1, and g nk = g kn = 0 for k n. Also the b i (x) are C and real valued. We use the above local coordinates x = (x, x n ), then we choose the differential operator A = η(x n ) xn on this local coordinates for each χ λ f, where η C c (R) is identically 1 for x n close to 0, and vanishes for x n δ, satisfying η (x n ) C/δ and η (x n ) C/δ 2. 20

21 Here 0 < δ < c, where c is the constant above associated with the local coordinates, since the manifold is compact, the constant c has uniformly lower bound respect to the local coordinates. First we show the following result: Lemma 4.2 For above differential operator A, and for each N 0, we have (χ λf) 2 L ν 2 ( ) Cδ 1 λ 2 χ λ f 2 L 2 () + Cλ 2 χ λ f 2 L (), 2 holds for some constant C independent of f L 2, N and λ, as λ. Proof. From Lemma 4.1, we have the identity (8). Now plug the above differential operator A in (8). The left hand side of (8) is precisely the square of the L 2 norm of ν (χ λ f). Here we estimate the right hand side of (8) in two cases: Case I: Estimate on the first term of the right hand side of (8). In the local coordinates, we have the representation of [ g, A] as n 1 [ g, A] = ( n 2 n + g ij (x) i j + b i (x) i )(η(x n ) n ) n 1 +η(x n ) n ( n 2 n + g ij (x) i j + b i (x) i ) n = 2η (x n ) n 2 η (x n ) n b n (x)η (x n ) n + ( n b i (x)) i n 1 = 2η (x n ) n 2 (η (x n ) + b n (x)η (x n ) n b n (x)) n + ( n b i (x)) i Denote C = max { b n, n b i }. Plus the above representation of [ g, A] in the first term of the right hand side of (8), and integral by parts for first term, we have {2η ( n (χ λ f)) 2 (η + b n η n b n )(χ λ f) n (χ λ f) n 1 + ( n b i )(χ λ f) i (χ λ f)}dx C { η ( n (χ λ f)) 2 + ( η + η + 1) (χ λ f) n (χ λ f) n 1 + (χ λ f) i (χ λ f) }dx C {δ 1 N ( n (χ λ f)) 2 + (δ 2 + δ 1 + 1) (χ λ f) n (χ λ f) 21

22 n 1 + C C (χ λ f) i (χ λ f) }dx {(δ 1 + δ 2 ɛ)( n (χ λ f)) 2 + n 1 i (χ λ f) 2 + (ɛ 1 + 1) (χ λ f) 2 }dx {(δ 1 + δ 2 ɛ + 1)( (χ λ f)) 2 + (ɛ 1 + 1) (χ λ f) 2 }dx Here we use the geometric mean inequality and ɛ > 0 is an arbitrary constant. On the other hand, we have the following estimates χ λ f 2 L 2 () = = ((χ λ f), g ((χ λ f)))dx (((χ λ f)), e j (f))dx λ 2 j (λ + 1) 2 = (λ + 1) 2 χ λ f 2 L 2 () Hence we know that the first term is bounded by e j (f) 2 L 2 () C{(δ 1 + δ 2 ɛ)(λ + 1) 2 + ɛ 1 } f 2 L 2 () Cδ 1 (λ + 1) 2 f 2 L 2 (), Here we let ɛ = δ. Case II: Estimate on the other two terms of the right hand side of (8). For our differential operator A = η(x n ) n, by integration by parts, we have = = (χ λ f, A(χ (1) f))dx (χ (1) λ λ (χ (1) λ f, n(η N χ λ f))dx η (χ (1) λ f, χ λf)dx 2 f, A(χ λf))dx (χ (1) λ f, η n(χ λ f))dx ((χ (1) λ f), η n(χ λ f))dx Notice that χ (1) λ f L 2 (λ + 1)2 f L 2 and (χ (1) λ f)) L 2 (λ + 1)3 f L 2, then there are the following estimates η (χ (1) λ f, χ λf)dx Cδ 1 χ (1) λ f χ λf dx Cδ 1 (χ (1) λ f) L 2 (χ λf) L 2 Cδ 1 (λ + 1) 2 χ λ f 2 L 2; 22

23 (χ (1) λ f, η n(χ λ f))dx η n (χ λ f) χ λ f dx χ (1) λ f L 2 η n(χ λ f) L 2 χ (1) λ f L 2 (χ λf) L 2 (λ + 1) 3 χ λ f 2 L 2. Combine above two Cases, we have the estimates as (χ λf) 2 L ν 2 ( ) Cδ 1 (λ + 1) 2 f 2 L + 2 χ(1) λ f L 2 n(χ λ f) L 2 Cδ 1 (λ + 1) 2 χ λ f 2 L + (λ + 2 1)3 χ λ f 2 L2. (9) Now let δ 0 = λ 1 in Lemma 4.2, we prove Theorem 1.3. Q.E.D. References [1] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 2001 [2] D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Comm. P.D.E., 27 (7-8), (2002). [3] A. Hassell and T. Tao, Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions. (arxiv: math. AP/ ) Preprint. [4] A. Hassell and S. Zelditch, Quantum ergodicity of boundary values of eigenfunctions. preprint. [5] L. Hörmander, The spectral function of an elliptic operator, Acta ath. 88 (1968), [6] L. Hörmander, The analysis of linear partial differential operators III-IV, Springer- Verlag, [7] S. Ozawa, Asymptotic property of eigenfunction of the Laplacian at the boundary. Osaka J. ath. 30 (1993), [8] Pham The Lai eilleures estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatifs au laplacien. ath. Scand. 48 (1981), no. 1, [9].H. Protter and H.F. Weinberger, aximum principles in differential equations, Springer Verlag

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