PANEITZ-TYPE OPERATORS AND APPLICATIONS PDF

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PANEITZ-TYPE OPERATORS AND APPLICATIONS

The Paneitz operator is conformally invariant in the sense that if ˜g = e2?g is a we deal with definition (0.3) for Paneitz-type operators.

PANEITZ-TYPE OPERATORS AND APPLICATIONS

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PANEITZ-TYPE OPERATORS AND APPLICATIONS

ZINDINE DJADLI, EMMANUEL HEBEY,andMICHEL LEDOUX

To the memory of André Lichnerowicz

Given(M,g)a smooth 4-dimensional Riemannian manifold, letS g be the scalar curvature ofg, and letRc gbe the Ricci curvature ofg. The Paneitz operator, discov- ered in [21], is the fourth-order operator defined by P 4g u=? 2g u-div g ?2 7S g g-2Rcg du, where? g u=-div g ?uis the Laplacian ofuwith respect tog. When(M,g)is the

4-dimensional standard unit sphere(S

4 ,h), we get that P4h u=? 2h u+2? h u. The Paneitz operator is conformally invariant in the sense that ifg=e 2? gis a conformal metric tog, then for allu?

C∞

(M), P

4g

u=e -4? P 4g (u).

The 2-dimensional analogue of this relation is

g u=e -2? ?g u. When the dimension is 2, it is well known that the scalar curvatures ofgandgare related by the equation g ?+1 WS g =1 WS g e2?

When the dimension is 4, we get that

P 4g ?+Q 4g =Q

4g

e 4? where Q 4g =1=?? g S g +S 2g -3|Rc g 2

Received 10 March 1999.

2000Mathematics Subject Classification. Primary 58E35; Secondary 35A15.

129

130DJADLI, HEBEY, AND LEDOUX

With respect to the Euler-Poincaré characteristic, in dimension 2, we have that

χ(M)=1

M S g dv g while in dimension 4,

χ(M)=1

2 M ?1 .|W g 2 +Q 4g dv g whereW g 4g havebeendeveloped recently by Beckner [2], Branson-Chang-Yang [4], Chang-Yang [7], Chang-Gursky- Yang [6], and Gursky [12]. Also see the survey Chang [5]. The Paneitz operator was generalized to higher dimensions by Branson [3]. Given(M,g)a smooth compact

Riemanniann-manifold,n≥5, letP

ng be the operator defined by P ng u=? 2g u-div g ?a n S g g+b n Rc g ?du+n-4 WQ ng u, where Q ng =1

W(n-1)?

g S g +n 3 -4n 2 +16n-16 @(n-1) 2 (n-2) 2 S 2g -2 (n-2) 2 |Rc g 2 and??? ?a n =(n-2) 2 +4

W(n-1)(n-2),

b n =-4 n-2.

Ifg=?

4/(n-4)

gis a conformal metric tog, then for allu?C (M), P ng (u?)=? (n+4)/(n-4) P ng u.

In particular,

P ng ?=n-4 WQ ng (n+4)/(n-4) These two relations have well-known analogues when dealing with the conformal

LaplacianL

ng , the second order operator whose expression is given by L ng u=? g u+n-2 .(n-1)S g u.

Ifg=?

4/(n-2)

gis a conformal metric tog, for allu?C (M), we get that L ng (u?)=? (n+2)/(n-2) L ng (u).

PANEITZ-TYPE OPERATORS AND APPLICATIONS131

In particular,

L ng ?=n-2 .(n-1)S g (n+2)/(n-2)

On the standard unit sphere(S

n ,h),n≥5, the expression ofP nh is P nh u=? 2h u+c n h u+d n u,(0.1) where ?c n =1 W(n 2 -2n-4), d n =n-4 e=n(n 2 -4),(0.2) that is,c n =n(n-1)a n +(n-1)b n ,d n =(n-4)Q nh /2. Note thatP nh =(L nh 2 -2L nh Given(M,g)a smoothn-dimensional compact Riemannian manifold,n≥5, and

α>0 real, we letP

g be the fourth-order operator defined by P g u=? 2g u+α? g u.(0.3)

Keeping in mind the expression ofP

4g on the standard sphere, we refer toP g as a Paneitz-type operator. Results and remarks often shift between operators likeP g , and

Paneitz-Branson-type operators like

P g u=? 2g u+α? g u+βu, whereαandβare real numbers. Here we should regardP g as the essential part of P g , like the Laplacian is the essential part of the conformal Laplacian, and note that the Paneitz-Branson operatorP ng reduces toP g whengis Einstein. A natural space when studyingP g is the Sobolev spaceH 22
(M)defined as the completion ofC (M) with respect to the norm ?u? 2 2 u? 22
+??u? 22
+?u? 22

Following standard notations,?·?

p in the above expression stands for theL p -norm (with respect to the Riemannian measuredv g ). As is well known and easy to see, for allu?C (M), g u) 2 2 u| 2 M 2 u| 2 dv g M g u| 2 dv g M Rc g (?u,?u)dv g M g u| 2 dv g +k? M |?u| 2 dv g

132DJADLI, HEBEY, AND LEDOUX

wherekis such thatRc g ≥-k. Hence,?·? H

22defined by

quotesdbs_dbs19.pdfusesText_25

[PDF] PANEITZ-TYPE OPERATORS AND APPLICATIONS

PANEITZ-TYPE OPERATORS AND APPLICATIONS

ZINDINE DJADLI, EMMANUEL HEBEY,andMICHEL LEDOUX

To the memory of André Lichnerowicz

4-dimensional standard unit sphere(S

C∞

4g

The 2-dimensional analogue of this relation is

When the dimension is 4, we get that

4g

Received 10 March 1999.

2000Mathematics Subject Classification. Primary 58E35; Secondary 35A15.

130DJADLI, HEBEY, AND LEDOUX

χ(M)=1

χ(M)=1

Riemanniann-manifold,n≥5, letP

W(n-1)?

W(n-1)(n-2),

Ifg=?

4/(n-4)

In particular,

LaplacianL

Ifg=?

4/(n-2)

PANEITZ-TYPE OPERATORS AND APPLICATIONS131

In particular,

On the standard unit sphere(S

α>0 real, we letP

Keeping in mind the expression ofP

Paneitz-Branson-type operators like

Following standard notations,?·?

132DJADLI, HEBEY, AND LEDOUX

22defined by

4g

4g

Ifg=?

Ifg=?