eigenvalues and eigenfunctions of metric measure manifolds

EIGENVALUES AND EIGENFUNCTIONS OF METRIC MEASURE MANIFOLDS YUXIN GE and ZHONGMIN SHEN [Received 18 June 1999; revised 23 May 2000]

1. Introduction A metric measure space is a metric space equipped with a j-®nite Borel measure. These spaces appear in many mathematical problems. See [14] for the fundamental theory on metric measure spaces. In this paper we are concerned with regular metric measure spaces, for which we can employ calculus to study geometric properties. The most general regular metric measure spaces are Finsler manifolds …M; F † equipped with a volume form dm. We will call them metric measure manifolds. In this paper, we will study the regularity of the eigenfunctions of a metric measure manifold and the relationship between the ®rst eigenvalue and the isoperimetric constants. Refer to [16] for related discussions. Let …M; F; dm† be a compact metric measure manifold (possibly with p the Sobolev space of functions u 2 H 1; p with boundary). Denote by H 1; o R = , and with uj ¶M ˆ 0 if ¶M 6ˆ 0= . Let H o1 :ˆ H o1; 2 . Following M u dm ˆ 0 if ¶M ˆ 0 [13], we de®ne the canonical energy functional E on H 1o ÿ f0g by R F  …du†2 dm ; E…u† :ˆ M R 2 M u dm where F  : T  M ! ‰0; 1† denotes the Finsler co-metric de®ned by Fx …y† :ˆ sup Fx … y† ˆ 1 y… y† for y 2 Tx M. Note that E is C 1 on H 1o ÿ f0g. By de®nition, l is an eigenvalue of …M; F; dm† if there is a function u 2 H 1o ÿ f0g such that d u E ˆ 0 with l ˆ E…u†. In this case, u is called an eigenfunction corresponding to l. Denote by El the union of the zero function and the set of all eigenfunctions corresponding to l. In general, El is a cone, not a subspace in H 1o . Therefore, we will callpE l the eigencone corresponding to l. If F is Riemannian, that is,  F… y† ˆ g… y; y†, where g is a Riemannian metric, then all eigenfunctions are C 1 and the eigencone El is a subspace. For general Finsler metrics, we can prove the following theorem. Theorem 1.1. Let …M; F; dm† be a compact metric measure manifold without or with boundary. Any eigenfunction f is C 1; a for some a with 0 < a < 1. Moreover, it is C 1 on the open subset fd f 6ˆ 0g. Examples show that the eigenfunctions (of the Dirichlet problem) are at most C 1; 1 (see § 6). Thus the best regularity of eigenfunctions one can expect is C 1; 1 . 2000 Mathematics Subject Classi®cation: 47J05, 47J10, 53C60, 58E05, 58C40. Proc. London Math. Soc. (3) 82 (2001) 725±746. q London Mathematical Society 2001.

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In fact, there are several notions of eigenvalues of an energy functional on H 1o . Compare [21] and [13]. In [13], M. Gromov introduced the notion of eigenvalues of an energy functional E on H 1o , using a dimension-like function on the subsets of H 1o . Among many dimension-like functions is the so-called essential dimensionlike function Dim. Gromov studied the eigenvalues fl i g with respect to Dim. In particular, he gave some relations between l i and the other geometric quantities such as isoperimetric constants, etc. Set l 1 :ˆ inf 1 E…u†: u 2 Ho

It is easy to verify that l 1 is the smallest critical value of E, which is called the ®rst eigenvalue of …M; F; dm† (see Proposition 2.1). The eigenfunctions corresponding to l 1 have some special properties when the Finsler metric F is reversible, that is, F…ÿy† ˆ F… y†. Theorem 1.2. Let …M; F; dm† be a compact reversible metric measure n-manifold with a smooth boundary ¶M. Suppose that u 2 H 1o is an eigenfunction corresponding to l 1 . Then either u > 0 or u < 0 in M n¶M. Moreover, if u 2 H o2; n , then du 6ˆ 0 on ¶M; in this case, u is smooth in a neighborhood of ¶M. Furthermore, the eigencone El 1 corresponding to l 1 is a one-dimensional space. The ®rst eigenvalue is closely related to the isoperimetric pro®le. Let …M; F; dm† be a compact metric measure manifold without boundary. We de®ne the isoperimetric pro®le h M : ‰0; 1Š ! R ‡ by h M …s† :ˆ inf

n…¶Q† m…M†

…1†

where the in®mum is taken over all regular domains Q Ì M such that m…Q† ˆ s ´ m…M†. Here n denotes the induced smooth measure on ¶Q (see § 2.2 below). Note that h M …s† ˆ h M …1 ÿ s† > 0, for all s 2 …0; 1†, and h M …0† ˆ 0 ˆ h M …1†. A continuous function h: ‰0; 1Š ! R ‡ is called an isoperimetric function if it satis®es h…s† ˆ h…1 ÿ s† > 0 for all s 2 …0; 1†, and h…0† ˆ 0 ˆ h…1†. For an isoperimetric function h, we de®ne the ®rst eigenvalue of h by R1 0 2 0 jf …s†h…s†j ds ; …2† l 1 …h† ˆ inf R1 2 f inf a 2 R 0 jf…s† ÿ aj ds where the in®mum is taken over all bounded piecewise C 1 -functions f on ‰0; 1Š. By symmetrization based on the one-dimensional model, we can prove the following. Theorem 1.3. Let …M; F; dm† be a compact metric measure n-manifold without boundary. Suppose that h M > h for some isoperimetric function h. Then l 1 …M† > l 1 …h†: For an isoperimetric function h, if h M …s† > 0; 0 < s < 1 h…s†

I h …M† :ˆ inf

…3†

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then h M …s† > e h…s† :ˆ I h …M† ´ h…s†. It follows from (2) and Theorem 1.3 that h† ˆ I h …M†2 l 1 …h†: l 1 …M† > l 1 … e See more discussion in § 8. 2. Preliminaries 2.1. Finsler metrics Let V n be the canonical n-dimensional real vector space without any metric structure. A Minkowski functional on V n is a function Fo : V n ! ‰0; 1† which has the following properties: (i) Fo …tv† ˆ tFo …v† for all t > 0 and all v 2 V n ; (ii) Fo is C 1 on V n nf0g; and (iii) Fo is strongly convex, that is, for any non-zero w 2 V n , the associated quadratic form gw in V n is positive de®nite, where gw …u; v† :ˆ

1 ¶2 …‰F 2 Š…w ‡ su ‡ tv††j s ˆ t ˆ 0 : 2 ¶s¶t o

Let V n ˆ …V n ; Fo †. We call V n a Minkowski space. The function Fo is said to be reversible if Fo …ÿv† ˆ Fo …v† for all v 2 V n . Denote the standard n-dimensional Euclidean space by R n and the Euclidean norm by j ? j. We denote by B 1 the unit ball in R n . Let M be an n-dimensional manifold. A Finsler metric on M is a non-negative function F on TM such that F is C 1 on TM ÿ f0g and the restriction Fx :ˆ F j T x M is a Minkowski functional on Tx M for all x 2 M. A non-negative function on T  M with analogous properties is called a Finsler co-metric. For any Finsler metric F, its dual F  …y† :ˆ supF x … y† ˆ 1 y… y† is a Finsler co-metric. Let w :ˆ …x i †: U Ì M ! R n be a local coordinate system in M. It induces a standard local coordinate system …x i ; hi † in T  M by mapping v ˆ hi dx i j x ! …x i ; hi †, where …x i † are the coordinates of x. For the sake of simplicity, we simply write x ˆ …x i †, h ˆ …hi † and F  …x; h† :ˆ F  …hi dx i j x †, if no confusion is caused. Set 1 ¶‰F  2 Š Ai …x; h† :ˆ …x; h†; 2 ¶hi and ¶Ai 1 ¶ 2 ‰F  2 Š …x; h† ˆ …x; h†: g i j …x; h† ˆ 2 ¶hi ¶h j ¶h j By choosing a smaller coordinate neighborhood U Ì M if necessary, one may assume that there exists a positive constant C > 1, such that for x; x 0 2 w…U † and non-zero y; h 2 R n , jA i …x; y†j < C jyj; …4† i ¶A ¶x s …x; y† < C jyj;

…5†

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yuxin ge and zhongmin shen C ÿ1 jz j 2
jy ÿ z j 2 q 2 d x: C B1 B1

731

…17†

…18†

Taking J…x† ˆ j…x†ÿ1 u k; « …x†q 2 …x† in (17), we obtain Z ‰Ai …x; y† ÿ Ai …x; z†Š…y i ÿ z i †q 2 d x B1

Z ˆl

B1

‰u k; « Š 2 q 2 d x

Z ÿ Z ÿ

B1

B1

‰Ai …x; y† ÿ Ai …x; z†Šu k; « j

¶ …j ÿ1 q 2 † d x ¶x i

‰Ai …x ‡ «e k ; y† ÿ Ai …x; y†Šj

¶ …j ÿ1 u k; « q 2 † d x: ¶x i

Using (7), (8), (16) and the inequality 2ab < da 2 ‡ d ÿ1 b 2

for all d > 0;

we can easily prove that there exists a number « o such that for any « < « o , Z ‰Ai …x; y† ÿ Ai …x; z†Š…y i ÿ z i †q 2 d x B1


0, and f i 2 L p …Q† 1 …Q† be a local solution of (20). Then, for some p > 2 …i ˆ 1; . . . ; n†. Let u 2 H loc there exist constants K and r0 , independent of r, such that for any r < r0 we have the following estimates: (i) if n > p > 2,   n X 1 k f i k L p …B r …x o †† ; …23† kuk L p  …B r = 2 …x o †† < K n…1 ÿ 2 = p  † = 2 kuk L 2 …B r …x o †† ‡ r iˆ1 (ii) where 1=p  ˆ 1=p ÿ 1=n; (ii) if p > n,   n X 1 …1 ÿ n = p† i k f k L p …B r …x o †† : kuk L 1 …B r = 2 …x o †† < K n = 2 kuk L 2 …B r …x o †† ‡ r r iˆ1

…24†

If u 2 H o1 …Q†, the above estimates also hold if B r …x o † is replaced by B r …x o † Ç Q. Lemma 4.2 (De Giorgi). Assume that b i 2 L r …Q†, c 2 L r = 2 …Q† with r > n and 1 …Q† be a local solution of (20). The following statements hold. let u 2 H loc p i 0; a …Q† with p > n, then u is in C loc …Q†. More precisely, for every (i) If f 2 L loc x o 2 Q and R < dist…x o ; ¶Q†, there are constants a, with 0 < a < 1, and K such

metric measure manifolds that qx …r† :ˆ sup B r = 2 …x† u… y† ÿ inf B r = 2 …x† u… y† satis®es   a X …1 ÿ n = p† i 1 r R k f k L p …Q Ç B R …x†† ; q x …r† < K n = 2 kuk L 2 …Q Ç B R …x†† ‡ R R i

733

…25†

for 0 < r < 12 R and x 2 B R = 2 …x o †. (ii) If f i 2 L p …Q Ç B R …x o †† with p > n, and for x o 2 ¶Q, if uj ¶Q Ç B R …x o † ˆ 0, then u is in C 0; a …B R = 2 …x o † Ç Q†. More precisely, (25) also holds for x 2 B R = 2 …x o † Ç Q and r < 12 R. Thus, if u 2 H o1 …Q† and f i 2 L p …Q† for some p > n, then u is in C 0; a …Q†. Proof of Theorem 1.1. Let u be an eigenfunction corresponding to an eigenvalue l. In a coordinate neighborhood B 1 ˆ w…U †, u satis®es ÿ

¶ ‰j…x†Ai …x; =x u†Š ˆ lj…x†u…x†: ¶ xi

…26†

We shall ®rst prove the theorem for a compact metric measure manifold M without boundary. By Lemma 3.1, we have u 2 H o2; 2 . Differentiating (26) with respect to x s , for s ˆ 1; . . . ; n , one obtains an equation for w ˆ ¶u=¶ x s :   ¶ ¶w ¶f i Lw ˆ ÿ i a i j j ˆ i ¶x ¶x ¶x with a i j …x† :ˆ j…x†g  i j …x; =x u†; f i …x† :ˆ j…x†

¶Ai ¶j …x; =x u† ‡ s …x†Ai …x; =x u† ‡ d is lj…x†u…x†: ¶x s ¶x

By (6), one can see that the a i j satisfy (22). Inequalities (4) and (5) imply that j f i j < C…juj ‡ j=x uj†: Thus f i 2 L 2 …Q†. Applying Lemma 4.1, we deduce that

¶u

< C…k=x uk L 2 …B r † ‡ kuk L 2 …B r † † for all s ˆ 1; 2; . . . ; n:

¶x s 2  L …B r = 2 † Iterating the above procedure and applying the Sobolev embedding theorem, we obtain =x u 2 L qloc for some q > n: It follows from Lemma 4.2 that Thus u 2

1; a . C loc

0; a =x u 2 C loc

for some 0 < a < 1:

We now study the regularity of u in a neighborhood of ¶M if ¶M 6ˆ 0= . Take a coordinate system w: U ! R n at the boundary such that w…U † ˆ B ‡ 1 and w…U Ç ¶M† ˆ G1 :ˆ B 1 Ç ¶R n‡ . Fix a point x o 2 B 1 Ç ¶R n‡ and r < Dist…x o ; ¶B 1 †. n n Let B ‡ r …x o † ˆ B r …x o † Ç R ‡ and Gr ˆ B r …x o † Ç ¶R ‡ .

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Let v be the weak solution to the following Dirichlet problem: 8 < ¶ ‰Ai …x ; = v†Š ˆ 0 in B ‡ …x †; r o x o ¶x i : ‡ v ˆ u on ¶B r …x o †:

…27†

We shall use v to estimate u. By the same arguments as in Lemma 3.1, we conclude that v 2 H 2; 2 …B r‡ …x o †† for any r < r. For each s ˆ 1; 2; . . . ; n ÿ 1, differentiating (27) with respect to x s yields that the function w :ˆ ¶v=¶x s is a solution of the following equation:   8 < ¶ g  i j …x ; = v† ¶w ˆ 0 in B ‡ …x †; r o x o …28† ¶x i ¶x j : w ˆ 0 on G1 : By Lemma 4.2, w 2 C 0; d for some d 2 …0; 1† (see [7]). Moreover, for any r < 14 r,  n ‡ 2 d Z Z r 2 jwj d x < C jwj 2 d x: …29† ‡ r B r …x o † B r‡= 2 …x o † Using the Caccioppoli inequality and the Poincare inequality, we obtain, from (29), Z Z j=x wj 2 d x < Crÿ2 jwj 2 d x B‡ 2 r …x o †

B‡ r …x o †

 n ÿ 2 ‡ 2 d Z r jwj 2 d x ‡ r B r = 2 …x o †  n ÿ 2 ‡ 2 d Z r j=x wj 2 d x: 0 and a number k > 1 such that for every 1 xo 2 B‡ 3 = 4 and for every r < 4 , we have Z …1 ‡ j=x uj 2 † d x < Cr …k ÿ 1†v ; …44† B‡ r …x o †

with kv < n ‡ 2d and 0 < v < 1. Then Z 2 kv j=x u ÿ f=x ug‡ r j d x < Cr ; B‡ r …x o †

…45†

for the same x o and r. As in Chapter 3 of [7], we denote Campanato spaces by L p; q and Morrey spaces by L p; q. Take v such that, for some integer h , …h ÿ 1†v < n < hv < n ‡ 2d: 2; hv

…B ‡ We assert that =x u 2 L 3 = 4 †. We argue by induction. Clearly, (44) holds for k ˆ 1. Suppose that it holds for some k < h. Using Lemma 4.3, we ®nd that =x u belongs to the Campanato space L 2; kv …B ‡ 3 = 4 † which is isomorphic to the Morrey space L 2; kv …B ‡ † since kv < n (see [7, Proposition 1.2, Chapter 3]). Thus 3=4 Z j=x uj 2 d x < Cr kv : B‡ r …x o †

Eventually, we obtain the following identity: Z j=x uj 2 d x < Cr …h ÿ 1†v : B‡ r …x o †

Thus, it follows from Lemma 4.3 again that u 2 L 2; hv …B ‡ 3 = 4 †. Therefore 1 † with a ˆ …hv ÿ n†. u 2 C 1; a …B ‡ 3=4 2

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Remarks. (1) As in [9], if we can prove that the solution v is in C 1; a , then u is also in C 1; a . (2) We can prove HoÈlder continuity for eigenfunctions corresponding to l 1 under much weaker conditions (see [8]). This is because of the fact that every such eigenfunction is a minimum point of the following energy functional: Z Z 2 F …x; =x u† dm ÿ l 1 u 2 dm: E 1 …u† ˆ M

M

(3) By a result in [17, p. 336, Theorem 6.1], any eigenfunction u is regular on the regular set fd u 6ˆ 0g since u is continuous and F  is regular on T  M nf0g. 5. Eigenfunctions in the Dirichlet problem In this section, we are going to study the eigenfunctions corresponding to the ®rst eigenvalue in the Dirichlet problem. First, we give two classical Maximum Principles without proof (see [20, 10, 4]). Consider a linear operator   ¶ ¶u ¶u ij …46† Lu ˆ ÿ j a …x† i …x† ‡ b i …x† i …x† ‡ c…x†u…x†; ¶x ¶x ¶x where the functions a i j are such that C ÿ1 jyj 2 < a i j …x†y i y j < C jyj 2

for all x 2 Q; y 2 R n :

…47†

Lemma 5.1. Let L be the differential operator in (46). Assume that b i 2 L r …Q†, p …Q† with p > n …i ˆ 1; . . . ; n†. Let u be a c 2 L r = 2 …Q† with r > n and f i 2 L loc non-negative solution of the following equation: Lu ˆ Then u > 0 in Q.

¶f i ¶x i

in Q:

…48†

Consider another elliptic operator, Lu :ˆ ÿa i j where the a i j satisfy (47).

¶2u i ¶u ; i j‡b ¶x ¶x ¶x i

Lemma 5.2 (Alexandroff, Bakelman,ÿ P Pucci). Let
d be a positive n constant n i 2 1=2 jb j < b and f 2 L …Q†. Let such that Diam…Q† < d. Assume that iˆ1 2; n u 2 H loc …Q† Ç C…Q† be a subsolution of the following equation: Then

Lu < f : sup u < sup u ‡ ‡ K k f ‡ k L n …Q† ; Q

¶Q

where K is a constant depending only on n, b, d and C. Proof of Theorem 1.2. Let u be an eigenfunction corresponding to l 1 . Namely, u satis®es 1 ¶ ‰j…x†Ai …x; =x u†Š ‡ l 1 u ˆ 0: …49† j…x† ¶ x i

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By the assumption that F is reversible, we have E…juj† ˆ E…u†. This implies that E…juj† ˆ l 1 . Thus juj is a minimum point of E. Hence, juj is an eigenfunction corresponding to l 1 , satisfying 1 ¶ ‰j…x†Ai …x; =x juj†Š ‡ l 1 juj ˆ 0: j…x† ¶ x i It follows from Lemma 5.1 that juj > 0 in M n¶M: Since u 2 C 1; a , u is either positive or negative in M n¶M. Suppose that u 2 H 2; n . Now we write the Euler equation (49) in a nondivergence form: ¶2u ¶u ˆ l1 u Lu :ˆ ÿa i j i j ‡ b j ¶x ¶x ¶x j where a i j :ˆ g  i j …x; =x u† and ¶ ln j  i j ¶g  i j g …x; = u† ÿ …x; =x u†: b j :ˆ ÿ x ¶ xi ¶x i Clearly, b i …x† 2 L 1 and a i j satisfy (47). Suppose that u < 0 in M n¶M and ®x x o 2 ¶M. Without loss of generality, we assume that B r … p† Ì M such that ¶B r … p† is tangent to ¶M at x o . Let « > 0 and a > 0. De®ne v…x† ˆ «…eÿ a r ÿ eÿ aj x ÿ pj †: Our goal is to choose suitable « and a such that u…x† < v…x†

for all x 2 B r … p†nB r = 2 … p†

…50†

and u…x o † ˆ v…x o † ˆ 0. Then we have ¶u ¶v …x † < …x † ˆ ÿ«a < 0; ¶n o ¶n o where n denotes the inward-pointing normal vector. This implies that =x u 6ˆ 0 for x 2 ¶M. Pn i 2 2 Assume that i ˆ 1 jb j < b . Observe that for x 2 B r … p†nB r = 2 … p†, Lv ˆ ÿa i j …x†

¶2v ¶v …x† ‡ b i …x† i …x† ¶x i ¶x j ¶x

   …x i ÿ p i †…x j ÿ p j † a i j …x† …x i ÿ p i †…x j ÿ p j † ij d ÿ ÿ ˆ «a aa i j …x† jx ÿ pj jx ÿ pj 2 jx ÿ pj 2  …x i ÿ p i † ÿ a j x ÿ pj i e ‡ b …x† jx ÿ pj   2nC C ‡ ÿ b e ÿ a j x ÿ pj : > «a aC ÿ r r

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First choose a large a > 0 such that Lv > 0. Let f ˆ u ÿ v. We have L… f † ˆ Lu ÿ Lv ˆ l 1 u ÿ Lv < 0: Note that for x 2 ¶B r … p†, f …x† ˆ u…x† < 0 ˆ f …x o †. Choose a small « > 0 such that for any x 2 ¶B r = 2 … p†, f …x† ˆ u…x† ÿ v…x† < ÿ

inf

x 2 B r = 2 … p†

juj ‡ «…e ÿ a r = 2 ÿ e ÿ a r † < 0:

By Lemma 5.2, we see that for any x 2 B r … p†, f …x†