Some elliptic PDEs on Riemannian manifolds with boundary
aa r X i v : . [ m a t h . A P ] J u l SOME ELLIPTIC PDESON RIEMANNIAN MANIFOLDS WITH BOUNDARY
YANNICK SIRE AND ENRICO VALDINOCI
Abstract.
The goal of this paper is to investigate some rigidityproperties of stable solutions of elliptic equations set on manifoldswith boundary.We provide several types of results, according to the dimensionof the manifold and the sign of its Ricci curvature.
Contents
1. Introduction 11.1. Results for product manifolds 41.2. Results for the hyperbolic space 52. The case of product manifolds and a weighted Poincar´einequality for stable solutions of (1.3) 63. Proof of theorems 1.1–1.4 93.1. Proof of theorem 1.1 93.2. Proof of theorem 1.2 103.3. Proof of theorem 1.3 113.4. Proof of theorem 1.4 134. The case of the hyperbolic space 134.1. Motivations and scattering theory 134.2. Proof of theorem 1.5 17References 171.
Introduction
Let ( M , ¯ g ) be a complete, connected, smooth, n + 1-dimensionalmanifold with boundary ∂ M , endowed with a smooth Riemannianmetric ¯ g = { ¯ g ij } i,j =1 ,...,n . YS : Universit´e Aix-Marseille 3, Paul C´ezanne – LATP – Marseille, France andLaboratoire Poncelet, UMI 2615– Moscow, Russia– [email protected] EV : Universit`a di Roma Tor Vergata – Dipartimento di Matematica – Rome, Italy– [email protected] . The volume element writes in local coordinates as(1.1) dV ¯ g = p | ¯ g | dx ∧ · · · ∧ dx n , where { dx , . . . , dx n } is the basis of 1-forms dual to the vector basis { ∂ i , . . . , ∂ n } and we use the standard notation | ¯ g | = det(¯ g ij ) > ¯ g X the divergence of a smooth vector field X on M , that is, in local coordinates,div ¯ g X = 1 p | ¯ g | ∂ i (cid:16)p | ¯ g | X i (cid:17) , with the Einstein summation convention.We also denote by ∇ ¯ g the Riemannian gradient and by ∆ ¯ g theLaplace-Beltrami operator, that is, in local coordinates,(1.2) ( ∇ ¯ g φ ) i = ¯ g ij ∂ j φ and ∆ ¯ g φ = div ¯ g ( ∇ ¯ g φ ) = 1 p | ¯ g | ∂ i (cid:16)p | ¯ g | ¯ g ij ∂ j φ (cid:17) , for any smooth function φ : M → R .We set h· , ·i to be the scalar product induced by ¯ g .Given a vector field X , we also denote | X | = p h X, X i . Also (see, for instance Definition 3.3.5 in [Jos98]), it is customary todefine the Hessian of a smooth function φ as the symmetric 2-tensorgiven in a local patch by( H ¯ g φ ) ij = ∂ ij φ − Γ kij ∂ k φ, where Γ kij are the Christoffel symbols, namelyΓ kij = 12 ¯ g hk ( ∂ i ¯ g hj + ∂ j ¯ g ih − ∂ h ¯ g ij ) . Given a tensor A , we define its norm by | A | = √ AA ∗ , where A ∗ is theadjoint.The present paper is devoted to the study of special solutions ofelliptic equations on manifolds with boundary and is, in some sense, afollow up to the paper by the authors and Farina (see [FSV08b]) wherethe case without boundary was investigated. In an Euclidean context,i.e. M = R n +1+ with the flat metric, the rigidity features of the stablesolutions has been investigated in [SV09, CS09].Boundary problems are related (via a theorem of Caffarelli and Sil-vestre [CS07]) to non local equations involving fractional powers of theLaplacian. An analogue of the results of [CS07] has been obtained in a LLIPTIC PDES ON MANIFOLDS WITH BOUNDARY 3 geometric context, by means of scattering theory (see [FG02, GJMS92,GZ03]).In this paper, we will focus on the following two specific models: • product manifolds of the type (cid:16) M = M × R + , ¯ g = g + | dx | (cid:17) where ( M, g ) is a complete, smooth Riemannian manifold with-out boundary, and • the hyperbolic halfspace , i.e. (cid:16) M = H n +1 , ¯ g = | dy | + | dx | x (cid:17) where x > y ∈ R n .Notice that the above models comprise both the positive and the neg-ative curvature cases.We denote by ν the exterior derivative at points of ∂ M .We will investigate the two following problems(1.3) (cid:26) ∆ ¯ g u = 0 in M = M × R + ,∂ ν u = f ( u ) on M × { } . and(1.4) (cid:26) − ∆ ¯ g u − s ( n − s ) u = 0 in M = H n +1 ,∂ ν u = f ( u ) on ∂ H n +1 . where f is a C ( M ) nonlinearity (in fact, up to minor modifications,the proofs we present also work for locally Lipschitz nonlinearities).The real parameter s in (1.4) is chosen to be s = n γ, where γ ∈ (0 , u is weak solution of (1.3) if, for every ξ ∈ C ∞ ( M × R ), we have(1.5) Z M h∇ ¯ g u, ∇ ¯ g ξ i = Z ∂ M f ( u ) ξ. Analogously, we say that u is weak solution of (1.4) if, for every ξ ∈ C ∞ ( M ), we have that(1.6) Z M h∇ g u, ∇ g ξ i − s ( n − s ) Z M uξ = Z ∂ M f ( u ) ξ. YANNICK SIRE AND ENRICO VALDINOCI
We focus on an important class of solutions of (1.3) and (1.4), namelythe so called stable solutions.These solutions play an important role in the calculus of variationsand are characterized by the fact that the second variation of the energyfunctional is non negative definite. This condition may be explicitlywritten in our case by saying that a solution u of either (1.3) and or(1.4) is stable if(1.7) Z M |∇ ¯ g ξ | dV ¯ g − s ( n − s ) ε Z M ξ − Z ∂ M f ′ ( u ) ξ dV ¯ g > ξ ∈ C ∞ ( M × R ) with ε = 0 in case of (1.3), and for every ξ ∈ C ∞ ( M ) and with ε = 1 in case of (1.4).1.1. Results for product manifolds.
Now we present our results inthe case of product manifolds M = M × R + . Theorem 1.1.
Assume that the metric on M = M × R + is given by ¯ g = g + | dx | .Assume furthermore that M is compact and satisfies Ric g > with Ric g not vanishing identically.Then every bounded stable weak solution u of (1.3) is constant. We remark that the assumption on the boundedness of u is neededas the following example shows: the function u ( x, y ) = x is a stablesolution of (cid:26) ∆ ¯ g u = 0 in M × R + ,∂ ν u = − M × { } . From theorem 1.1, one also obtains the following Liouville-type theo-rem for the half-Laplacian on compact manifolds (for the definition andbasic functional properties of fractional operators see, e.g., [Kat95]):
Theorem 1.2.
Let ( M, g ) be a compact manifold and u : M → R be asmooth bounded solution of (1.8) ( − ∆ g ) / u = f ( u ) , with (1.9) Z M ( |∇ g ξ | + |∇ x ξ | ) − Z ∂ M f ′ ( u ) ξ > , for every ξ ∈ C ∞ ( M ) .Assume furthermore that Ric g > LLIPTIC PDES ON MANIFOLDS WITH BOUNDARY 5 and
Ric g does not vanish identically.Then u is constant. Results for ( − ∆ g ) α with α ∈ (0 ,
1) may be obtained with similartechniques as well.
Theorem 1.3.
Assume that the metric on M = M × R + is given by ¯ g = g + | dx | , that M is complete, and Ric g > , with Ric g not vanishing identically.Assume also that, for any R > , the volume of the geodesic ball B R in M (measured with respect to the volume element dV g ) is boundedby C ( R + 1) , for some C > .Then every bounded stable weak solution u of (1.3) is constant. Next theorem is a flatness result when the Ricci tensor of M vanishesidentically: Theorem 1.4.
Assume that the metric on M = M × R + is given by ¯ g = g + | dx | and Ric g vanishes identically.Assume also that, for any R > , the volume of the geodesic ball B R in M (measured with respect to the volume element dV g ) is boundedby C ( R + 1) , for some C > .Then for every x > and c ∈ R , every connected component of thesubmanifold S x = { y ∈ M, u ( x, y ) = c } is a geodesic. Results for the hyperbolic space.
The next theorem providesa flatness result when the manifold M is H . Theorem 1.5.
Let n = 2 .Let u be a smooth weak solution of (1.4) and let s = n + γ where γ ∈ (0 , .Also, suppose that either (1.10) ∂ y u > or (1.11) f ′ on ∂ H n +1 . Then, for every x > and c ∈ R , each of the submanifold S x = (cid:8) y ∈ R n , | u ( x, y ) = cx n − s (cid:9) is a Euclidean straight line. YANNICK SIRE AND ENRICO VALDINOCI
As discussed in details in section 4.2, the proof of theorem 1.5 con-tains two main ingredients:(1) We first notice that the metric on H n +1 is conformal to the flatmetric on R n +1+ .(2) We then use some results by the authors in [SV09] (see also[CS09] for related problems) to get the desired result.The rest of this paper is structured as follows. In section 2 we provea geometric inequality for stable solutions in product manifolds, fromwhich we obtain the proofs of theorems 1.1–1.4, contained in section 3.Then, in section 4, we consider the hyperbolic case and we prove the-orem 1.1.2. The case of product manifolds and a weightedPoincar´e inequality for stable solutions of (1.3)Now we deal with the case of product manifolds M × R + .In order to simplify notations, we write ∇ instead of ∇ ¯ g for the gra-dient on M × R + but we will keep the notation ∇ g for the Riemanniangradient on M .Recalling (1.2), we have that(2.12) ∇ = ( ∇ g , ∂ x ) . In the subsequent theorem 2.1, we obtain a formula involving the ge-ometry, in a quite implicit way, of the level sets of stable solutions of(1.3).Such a formula may be considered a geometric version of the Poincar´einequality, since the L -norm of the gradient of any test function boundsthe L -norm of the test function itself. Remarkably, these L -normsare weighted and the weights have a neat geometric interpretation.These type of geometric Poincar´e inequalities were first obtainedby [SZ98a, SZ98b] in the Euclidean setting, and similar estimates havebeen recently widely used for rigidity results in PDEs (see, for in-stance, [FSV08a, SV09, FV09]). Theorem 2.1.
Let u be a stable solution of (1.3) such that ∇ g u isbounded.Then, for every ϕ ∈ C ∞ ( M × R ) , the following inequality holds: Z M × R + n Ric g ( ∇ g u, ∇ g u ) + | H g u | − |∇ g |∇ g u || o ϕ Z M × R + |∇ g u | |∇ ϕ | . (2.13)Notice that only the geometry of M comes into play in formula (2.13). LLIPTIC PDES ON MANIFOLDS WITH BOUNDARY 7
Proof.
First of all, we recall the classical Bochner-Weitzenb¨ock formulafor a smooth function φ : M → R (see, for instance, [BGM71, Wan05]and references therein):(2.14) 12 ∆ ¯ g |∇ ¯ g φ | = | H ¯ g φ | + h∇ ¯ g ∆ ¯ g φ, ∇ ¯ g φ i + Ric ¯ g ( ∇ ¯ g φ, ∇ ¯ g φ ) . The proof of theorem 2.1 consists in plugging the test function ξ = |∇ g u | ϕ in the stability condition (1.7): after a simple computation, thisgives Z M ϕ |∇|∇ g u || + 12 h∇|∇ g u | , ∇ ϕ i + |∇ g u | |∇ ϕ | − Z M f ′ ( u ) |∇ g u | ϕ > . (2.15)Also, by recalling (2.12), we have(2.16) h∇|∇ g u | , ∇ ϕ i = h∇ g |∇ g u | , ∇ g ϕ i + ∂ x |∇ g u | ∂ x ϕ . Moreover, since M is boundaryless, we can use on M the Green formula(see, for example, page 184 of [GHL90]) and obtain that Z M h∇ g |∇ g u | , ∇ g ϕ i = Z R + Z M h∇ g |∇ g u | , ∇ g ϕ i = − Z R + Z M ∆ g |∇ g u | ϕ = − Z M ∆ g |∇ g u | ϕ . (2.17)Hence, using (2.14), (2.16) and (2.17), we conclude that12 Z M h∇|∇ g u | , ∇ ϕ i = 12 Z M ∂ x |∇ g u | ∂ x ϕ − Z M ϕ n | H g u | + h∇ g ∆ g u, ∇ g u i + Ric g ( ∇ g u, ∇ g u ) o . (2.18)Using the equation in (1.3), we obtain that∆ g u = − ∂ xx u, so (2.18) becomes12 Z M h∇|∇ g u | , ∇ ϕ i = 12 Z M ∂ x |∇ g u | ∂ x ϕ − Z M ϕ | H g u | + Z M ϕ h∇ g ∂ xx u, ∇ g u i − Z M ϕ Ric g ( ∇ g u, ∇ g u ) . (2.19) YANNICK SIRE AND ENRICO VALDINOCI
Furthermore, Integrating by parts, we see that Z M ∂ x |∇ g u | ∂ x ϕ = Z M Z + ∞ ∂ x |∇ g u | ∂ x ϕ = − Z M (cid:16) ∂ x |∇ g u | ϕ (cid:17) | x =0 − Z M Z + ∞ ∂ xx |∇ g u | ϕ = − Z M (cid:16) ∂ x |∇ g u | ϕ (cid:17) | x =0 − Z M ∂ xx |∇ g u | ϕ . Consequently, (2.19) becomes12 Z M h∇|∇ g u | , ∇ ϕ i = − Z M ϕ n ∂ xx |∇ g u | + | H g u | + Ric g ( ∇ g u, ∇ g u ) o (2.20) + Z M ϕ h∇ g ∂ xx u, ∇ g u i − (cid:16) ∂ x |∇ g u | ϕ (cid:17) | x =0 . Now, we use the boundary condition in (1.3) to obtain that, on M , f ′ ( u ) ∇ g u = ∇ g ( f ( u )) = ∇ g ∂ ν u = −∇ g ∂ x u. Therefore, − Z M (cid:16) ∂ x |∇ g u | ϕ (cid:17) | x =0 − Z M h∇ g u x , ∇ g u i ϕ = Z M f ′ ( u ) |∇ g u | ϕ . (2.21)All in all, by collecting the results in (2.15), (2.20), and (2.21), weobtain that Z M ϕ |∇|∇ g u || − Z M ϕ n ∂ xx |∇ g u | + | H g u | + Ric g ( ∇ g u, ∇ g u ) o + Z M ϕ h∇ g ∂ xx u, ∇ g u i + Z M |∇ g u | |∇ ϕ | > . (2.22)Also, we observe that | ∂ x |∇ g u || + h∇ g ∂ xx u, ∇ g u i − ∂ xx |∇ g u | = | ∂ x |∇ g u || − | ∂ x ∇ g u | |∇|∇ g u || = |∇ g |∇ g u || + | ∂ x |∇ g u || ∂ xx |∇ g u | − h∇ g ∂ xx u, ∇ g u i . LLIPTIC PDES ON MANIFOLDS WITH BOUNDARY 9
This and (2.22) give (2.13). (cid:3) Proof of theorems 1.1–1.4
With (2.13) at hand, one can prove theorems 1.1–1.4.For this scope, first, we recall the following lemma, whose proof canbe found in section 2 of [FSV08b].
Lemma 3.1.
For any smooth φ : M → R , we have that (3.23) | H ¯ g φ | > (cid:12)(cid:12) ∇ ¯ g |∇ ¯ g φ | (cid:12)(cid:12) almost everywhere. Moreover, we have the following result:
Lemma 3.2.
Let u be a bounded solution of (1.3) . Assume that Ric g > and that Ric g does not vanish identically on M .Suppose that (3.24) Ric g ( ∇ g u, ∇ g u ) vanishes identically on M .Then, u is constant on M .Proof. By assumption, we have that
Ric g is strictly positive definite ina suitable non empty open set U ⊆ M .Then, (3.24) gives that ∇ g u vanishes identically in U × R + .This means that, for any fixed x ∈ R + , the map U ∋ y u ( x, y )does not depend on y . Accordingly, there exists a function ˜ u : R + → R such that u ( x, y ) = ˜ u ( x ), for any y ∈ U .Thus, from (1.3), 0 = ∆ ¯ g u = ˜ u xx in U × R + and so there exist a , b ∈ R for which u ( x, y ) = ˜ u ( x ) = a + bx for any x ∈ R + and any y ∈ U .Since u is bounded, we have that b = 0, so u is constant in U × R + .By the unique continuation principle (see Theorem 1.8 of [Kaz88]),we have that u is constant on M × R + . (cid:3) Proof of theorem 1.1.
Points in M will be denoted here as ( x, y ),with x ∈ R + and y ∈ M .Take ϕ in (2.13) to be the function ϕ ( x, y ) = φ ( xR )where R > φ is a smooth cut-off, that is φ = 0 on | x | > φ = 1 on | x | We remark that this is an admissible test function, since M is as-sumed to be compact in theorem 1.1. Moreover, we remark that(3.25) |∇ ϕ ( x, y ) | k φ k C ( R ) χ (0 , R ) ( x ) R .
Also, since u is bounded, elliptic regularity gives that ∇ u is boundedin M × R + .Therefore, using (2.13), lemma 3.1 and (3.25), we obtain(3.26) Z M × R + n Ric g ( ∇ g u, ∇ g u ) o ϕ CR Z M × (0 , R ) dV ¯ g CR for some constant C > R → + ∞ and using the fact that Ric g >
0, we concludethat
Ric g ( ∇ g u, ∇ g u ) vanishes identically.Thus, by lemma 3.2, we deduce that u is constant.3.2. Proof of theorem 1.2.
We put coordinates x ∈ R + and y ∈ M for points in M = M × R + .Given a smooth and bounded u o : M → R , we can define the har-monic extension E u o : M × R + → R as the unique bounded functionsolving(3.27) (cid:26) ∆ ¯ g ( E u o ) = 0 in M × R + , E u o = u o on M × { } . See Section 2.4 of [CSM05] for furter details.Then, we define(3.28) L u o := ∂ ν ( E u o ) (cid:12)(cid:12) x =0 . We claim that, for any point in M → R ,(3.29) − ∂ x ( E u o ) = E ( L u o ) . Indeed, by differentiating the PDE in (3.27),∆ ¯ g ∂ x ( E u o ) = 0 . On the other hand, − ∂ x ( E u o )(0 , y ) = ∂ ν ( E u o )(0 , y ) = L u o , thanks to (3.28).Moreover, ∂ x ( E u o ) is bounded by elliptic estimates, since so is u o .Consequently, − ∂ x ( E u o ) is a bounded solution of (3.27) with u o re-placed by L u o .Thus, by the uniqueness of bounded solutions of (3.27), we ob-tain (3.29). LLIPTIC PDES ON MANIFOLDS WITH BOUNDARY 11
By exploiting (3.28) and (3.29), we see that L u o = ∂ ν (cid:0) E ( L u o ) (cid:1)(cid:12)(cid:12) x =0 = − ∂ x (cid:0) E ( L u o ) (cid:1)(cid:12)(cid:12) x =0 = − ∂ x (cid:0) − ∂ x ( E u o ) (cid:1)(cid:12)(cid:12) x =0 = ∂ xx ( E u o ) (cid:12)(cid:12) x =0 . (3.30)On the other hand, using the PDE in (3.27),0 = ∆ ¯ g ( E u o ) = ∆ g ( E u o ) + ∂ xx ( E u o ) , so (3.30) becomes L u o ( y ) = ∂ xx ( E u o )(0 , y ) = − ∆ g ( E u o )(0 , y ) = − ∆ g u o ( y ) , for any y ∈ M , that is(3.31) L = ( − ∆ g ) / . With these observations in hand, we now take u as in the statement oftheorem 1.2 and we define v := E u .From (3.28) and (3.31), ∂ ν v (cid:12)(cid:12) x =0 = ∂ ν ( E u ) (cid:12)(cid:12) x =0 = L u = ( − ∆ g ) / u. Consequently, recalling (1.8), we obtain that v is a bounded solutionof (1.3).Furthermore, the function v is stable, thanks to (1.9).Hence v is constant by theorem 1.1, and so we obtain the desiredresult for u = v | x =0 .3.3. Proof of theorem 1.3.
Given p = ( m, x ) ∈ M × R + , we de-fine d g ( m ) to be the geodesic distance of m in M (with respect to afixed point) and d ( p ) := q d g ( m ) + x . Let also ˆ B R := { p ∈ M × R + s.t. d ( p ) < R } , for any R >
0. Noticethat |∇ g u | ∈ L ∞ ( M × R + ), by elliptic estimates, and that ˆ B R ⊆ B R × [0 , R ], where B R is the corresponding geodesic ball in M .As a consequence, by our assumption on the volume of B R , we obtain Z ˆ B R |∇ g u | dV ¯ g k∇ g u k L ∞ ( M × R + ) Z B R × [0 ,R ] dV ¯ g = R k∇ g u k L ∞ ( M × R + ) Z B R dV g CR ( R + 1) k∇ g u k L ∞ ( M × R + ) . That is, by changing name of C ,(3.32) Z ˆ B R |∇ g u | dV ¯ g CR for any R > Also, since d g is a distance function on M (see pages 34 and 123of [Pet98]), we have that(3.33) |∇ d ( p ) | = (cid:12)(cid:12)(cid:0) d g ( m ) ∇ g d g ( m ) , x (cid:1)(cid:12)(cid:12) d ( p ) . Also, given R >
1, we define φ R ( p ) := d ( p ) √ R ,(log √ R ) − (cid:0) log R − log( d ( p )) (cid:1) if d ( p ) ∈ ( √ R, R ),0 if d ( p ) > R .Notice that (up to a set of zero V ¯ g -measure) |∇ φ R ( p ) | χ ˆ B R \ ˆ B √ R ( p )log √ R d ( p ) , due to (3.33).As a consequence,(log √ R ) Z M × R + |∇ g u | |∇ φ R | dV ¯ g Z ˆ B R \ ˆ B √ R |∇ g u ( p ) | d ( p ) dV ¯ g ( p )= Z ˆ B R \ ˆ B √ R |∇ g u ( p ) | (cid:16) R + Z Rd ( p ) dtt (cid:17) dV ¯ g ( p ) R Z ˆ B R |∇ g u ( p ) | dV ¯ g ( p ) + Z R √ R Z ˆ B t |∇ g u ( p ) | t dV ¯ g ( p ) dt. Therefore, by (3.32),(log √ R ) Z M × R + |∇ g u | |∇ φ R | dV ¯ g C (cid:18) Z R √ R dtt (cid:19) C log R. Consequently, from (2.13), Z M × R + n Ric g ( ∇ g u, ∇ g u ) + | H g u | − |∇ g |∇ g u || o φ R C log R . (3.34)From this and (3.23), we conclude that Z M × R + Ric g ( ∇ g u, ∇ g u ) φ R C log R .
By sending R → + ∞ , we obtain that Ric g ( ∇ g u, ∇ g u ) vanishes identi-cally.Hence, u is constant, thanks to lemma 3.2, proving theorem 1.3. LLIPTIC PDES ON MANIFOLDS WITH BOUNDARY 13
Proof of theorem 1.4.
The proof of theorem 1.3 can be carriedout in this case too, up to formula (3.34).Then, (3.34) in this case gives that Z M × R + n | H g u | − |∇ g |∇ g u || o φ R C log R .
By sending R → + ∞ , and by recalling (3.23), we conclude that | H g u | is identically equal to |∇ g |∇ g u || on (cid:0) M × { x } (cid:1) ∩ {∇ g u = 0 } , for anyfixed x > k =1 , . . . , n there exist κ k : M → R such that ∇ g (cid:0) ∇ g u (cid:1) k ( p ) = κ k ( p ) ∇ g u ( p ) for any p ∈ (cid:0) M × { x } (cid:1) ∩ {∇ g u = 0 } .From this and [FSV08b] (see the computation starting there on for-mula (23)), one concludes that every connected component of { y ∈ M, u ( x, y ) = c } is a geodesic.4. The case of the hyperbolic space
We now come to problem (1.4). Notice that up to now we assumedfor the manifold M to be positively curved. We deal here with spe-cial equations on negatively curved manifolds. As a consequence, thegeometric formula (2.13) is not useful since the Ricci tensor does nothave the good sign and so we need a different strategy to deal with thehyperbolic case.For this, we will make use here of the fact that the manifold H n +1 with the metric ¯ g | dy | + | dx | x is conformal to R n +1+ with the flat metric,and, in fact, ( H n +1 , ¯ g ) is the main example of conformally compactEinstein manifold, as we discuss in section 4.1 here below.4.1. Motivations and scattering theory.
In order to justify thestudy of problem (1.4), we describe the link between problem (1.4) andfractional order conformally covariant operators.Let M be a compact manifold of dimension n . Given a metric h on M , the conformal class [ h ] of h is defined as the set of metrics ˆ h thatcan be written as ˆ h = f h for a positive conformal factor f .Let M be a smooth manifold of dimension n + 1 with boundary ∂ M = M .A function ρ is a defining function of ∂ M in M if ρ > M , ρ = 0 on ∂ M , dρ = 0 on ∂ M We say that g is a conformally compact metric on X with conformalinfinity ( M, [ h ]) if there exists a defining function ρ such that the man-ifold ( ¯ M , ¯ g ) is compact for ¯ g = ρ g , and ¯ g | M ∈ [ h ].If, in addition ( M n +1 , g ) is a conformally compact manifold and Ric g = − ng , then we call ( M n +1 , g ) a conformally compact Einstein manifold.Given a conformally compact, asymptotically hyperbolic manifold( M n +1 , g ) and a representative ˆ g in [ˆ g ] on the conformal infinity M ,there is a uniquely defining function ρ such that, on M × (0 , ǫ ) in M , g has the normal form g = ρ − ( dρ + g ρ ) where g ρ is a one parameterfamily of metrics on M (see [GZ03] for precise statements and furtherdetails).In this setting, the scattering matrix of M is defined as follows.Consider the following eigenvalue problem in ( M , g ), with Dirichletboundary condition,(4.35) (cid:26) − ∆ g u s − s ( n − s ) u s = 0 in M u s = f on M for s ∈ C and f defined on M .Problem (4.35) is solvable unless s ( n − s ) belongs to the spectrumof − ∆ g .However, σ ( − ∆ g ) = (cid:2) ( n/ , ∞ (cid:1) ∪ σ pp (∆ g )where the pure point spectrum σ pp (∆ g ) (i.e., the set of L eigenvalues),is finite and it is contained in (0 , ( n/ ).Moreover, given any f on M , Graham-Zworski [GZ03] obtained ameromorphic family of solutions u s = P ( s ) f such that, if s n/ N ,then P ( s ) f = F ρ n − s + Hρ s . And if s = n/ γ , γ ∈ N , P ( s ) f = F ρ n/ − γ + Hρ n/ γ log ρ where F, H ∈ C ∞ ( X ), F | M = f , and F, H mod O ( ρ n ) are even in ρ .It is worth mentioning that in the second case H | M is locally deter-mined by f and ˆ g . However, in the first case, H | M is globally deter-mined by f and g . We are interested in the study of these nonlocaloperators.We define the scattering operator as S ( s ) f = H | M , which is a mero-morphic family of pseudo-differential operators in Re ( s ) > n/ s = n/ N of finite rank residues. The relation between f LLIPTIC PDES ON MANIFOLDS WITH BOUNDARY 15 and S ( s ) f is like that of the Dirichlet to Neumann operator in standardharmonic analysis. Note that the principal symbol is σ ( S ( s )) = 2 n − s Γ( n/ − s )Γ( s − n/ σ (cid:0) ( − ∆ g ) s − n/ (cid:1) The operators obtained when s = n/ γ , γ ∈ N have been wellstudied. Indeed, at those values of s the scattering matrix S ( s ) has asimple pole of finite rank and its residue can be computed explicitly,namely Res s = n/ γ S ( s ) = c γ P γ , c γ = ( − γ [2 γ γ !( γ − − and P γ are the conformally invariant powers of the Laplacian con-structed by [FG02, GJMS92].In particular, when γ = 1 we have the conformal Laplacian, P = − ∆ + n − n − R and when γ = 2, the Paneitz operator P = ∆ + δ ( a n Rg + b n Ric ) d + n − Q n We can similarly define the following fractional order operators on M of order γ ∈ (0 ,
1) as P γ f := d γ S ( n/ γ ) f, d γ = 2 γ Γ( γ )Γ( − γ ) . It is important to mention that these operators are conformally covari-ant. Indeed, for a change of metric g u = u n − γ g , we have P g u γ f = u − n +2 γn − γ P g γ ( uf ) . The following result, which can be found in [CG08], establishes a linkbetween scattering theory on M and a local problem in the half-space.We provide the proof for sake of completeness. Lemma 4.1.
Fix < γ < and let s = n + γ . Assume that u is asmooth solution of (4.36) (cid:26) − ∆ ¯ g u − s ( n − s ) u = 0 in H n +1 ,∂ ν u = v on ∂ H n +1 . for some smooth function v defined on ∂ H n +1 .Then the function U = x s − n u solves (4.37) div ( x − γ ∇ U ) = 0 for y ∈ R n , x ∈ (0 , + ∞ ) U (0 , . ) = u | x =0 , in R n − lim x → x − γ ∂ x U = Cv for some constant C .Proof. By the results in [GZ03], one has the following representationof u in H n +1 u = x n − s u | x =0 + x s ∂ ν u. From this we deduce that U = u | x =0 + x s − n ∂ ν u. Since s = n + γ , we have 2 s − n = 2 γ > U | x =0 = u | x =0 and − lim x → x − γ ∂ x U = C∂ ν u = Cv.
We now prove that U satisfies the desired equation. This only comesfrom the conformality of the metric on H n +1 to the flat one in thehalf-space. Indeed, the conformal Laplacian is given by L ¯ g = − ∆ ¯ g + n − n R ¯ g where R ¯ g is the scalar curvature of H n +1 , which is equal to − n ( n + 1) . On the other hand, if we have h = e w ¯ g for some function w (i.e. themetrics h and g are conformal) then the conformal law of L ¯ g is givenby L h ψ = e − n +32 w L ¯ g ( e n − w ψ )for any smooth ψ .In our case, we have h = | dx | + | dy | the flat metric on R n +1+ and e w = x . Thus, using the conformal law, we have − ∆ ¯ g ψ = − x ∆ ψ + ( n − ∂ x ψ for any ψ smoothly on R n +1+ . Plugging ψ = u and using equation (4.36) leads s ( n − s ) u = − x ∆ u + ( n − x∂ x u. Finally, plugging U = x s − n u leads to the equation∆ U + 1 − γx ∂ x U = 0 , which is equivalent to div ( x − γ ∇ U ) = 0. (cid:3) LLIPTIC PDES ON MANIFOLDS WITH BOUNDARY 17
Proof of theorem 1.5.
Let u be a solution as requested in The-orem 1.5. By Lemma 4.1, the function U satisfies in a weak sense(4.38) div ( x − γ ∇ U ) = 0 for y ∈ R , x ∈ (0 , + ∞ ) U (0 , . ) = u | x =0 , − lim x → x − γ ∂ x U = f ( U ) . Notice that either ∂ y U > f ′
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