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Print version ISSN 0001-3765
An. Acad. Bras. Ciênc. vol.82 no.4 Rio de Janeiro Dec. 2010
Fernando A.F.C. SilvaI; Pedro A.G. VenegasII; Ramón O.M. AhumadaI
IUniversidade Federal de Pernambuco, CCEN Departamento de Matemática, Av. Prof. Luiz Freire s/n Cidade Universitária, 50740-540 Recife, PE, Brasil
IIUniversidade Federal da Paraíba, CCEN Departamento de Matemática, Cidade Universitária 58059-970 João Pessoa, PB, Brasil
In this article, we show the relation between the Schwartz kernels of the Dirichlet-to-Neumann operators associated to the metrics g0 and h = F* (e2φ g0) on the circular annulus AR, and the Schwarzian Derivative of the argument function f of the restriction of the diffeomorphism F to the boundary of AR.
Key words: annulus, Dirichlet-to-Neumann Functional, Schwarzian Derivative.
Neste artigo mostramos a relação entre os núcleos de Schwartz dos operadores Dirichlet-to-Neumann associados à métrica g0 e h = F* (e2φ g0), no anel circular AR, e a Derivada Schwarziana da função argumento f, da restrição do difeomorfismo F à fronteira de AR.
Palavras-chave: anel, Funcional Dirichlet-Neumann, Derivada Schwarziana.
Let denote the space of all Riemannian metrics on a compact manifold, with boundary, and denote by the space of continuous linear operators acting on C∞(∂Ω).
The Dirichlet-to-Neumann functional is a mapping from into such that, for each, , takes Dirichlet boundary values to Neumann boundary values. More precisely, if is the unique solution of the Dirichlet problem in Ω, , then, where (resp. ) is the Laplace-Beltrami operator (resp. unit interior normal vector field) associated to the metric. The study of this functional goes back to the seminal paper of (Calderón 1980).
It is known (Lee and Uhlmann 1989) that is in fact an elliptic self-adjoint pseudo-differential operator of order one, whose principal symbol is , , and .
Let be the group of diffeomorphism of . The semi-direct product (Polyakov 1987) of the groups and defined by
provides a natural right action on , given by
where denotes the pull-back of F.
The main obstruction to injectivity, in the two-dimensional case, is the semidirect product of the groups of diffeomorphisms that restricts to the identity on the boundary, and the Abelian group of real-valued functions that equals zero on it. In fact, as formula (2.1) shows, the Dirichlet-to-Neumann Functional is constant on the orbits determined by ; this is a normal subgroup of .
With respect to the determination of the metric from the Dirichlet-to-Neumann Operator, we recommend the papers (Lee and Uhlmann 1989), (Lassas and Uhlmann 2001) and (Lassas et al. 2003). In these papers, they solve, in a more general setting, the problem of recovering the manifold and the metric.
In the case of a fixed annulus, all metrics can be written as for coming from the pull-back of the euclidean metric in the annulus of radius and . We prove, in this special case, that the equality of the Dirichlet-to-Neumann Operators associated to both metrics and gives us a relation involving the Schwarzian derivative of ( the lifting to of the restriction to the boundary of the diffeomorphism ).
Furthermore, we also show that the conformal factor restricted to the boundary of the annulus is determined by .
More precisely, we shall prove in Section 2 that, if is the annulus
is conformal to the euclidean metric, , where and ; the equality of the Schwartz kernels of and of implies that the argument function , of the restriction of to , satisfies the differential equation
and denotes the Schwarzian Derivative of . It follows that, if , then, and equals zero on the boundary.
2 GEOMETRIC FORMULATION
Here on we will denote by the Schwartz kernel of . We start with two lemmas.
LEMMA . Given a two-dimensional compact manifold with boundary and,, and, we have
Proof. See (Gómez and Mendoza 2006).
LEMMA . Let be a two-dimensional compact manifold with boundary, where , , and the unitary vector field to, with respect to the metric. Then,
where denotes the real, valued function on such that .
Proof. Let ,
changing variables we get:
where denotes the unique real, valued function defined on such that
and is the tangent unitary vector field on such that and . The above equation means at every point the following: and belong to the same one-dimensional tangent space ; consequently, the first one is a real multiple of the second. In fact, this multiple is unique and it is equal to .
For the second equality,
finishing the proof.
The next Lemma establish, a relation between and the Green function of the Laplacian with Dirichlet condition on (Guillarmou and Sá Barreto 2009).
LEMMA . The Schwartz kernel of is given for y, y' ∈ ∂Ω, y ≠ y', by
where, are, respectively, the inward pointing vector fields to the boundary in variable and .
Proof. Let be the distance function to the boundary in ; it is smooth in a neighborhood of and the normal vector field to the boundary is the gradient of . The flow of induces a diffeomorphism defined by, and we have . This induces natural coordinates near the boundary, these are normal geodesic coordinates. The function is the unique solution of the Dirichlet problem in, and can be obtained by taking
where is any smooth function on such that . Now, using Greens formula and , where is the Dirac mass on the diagonal, we obtain for
We have Taylor expansion near the boundary. Let and take supported near. Thus, pairing with gives
Now taking with support disjoint to the support of , thus , and differentiating (2.2) in x, we see, in view of the fact that Greens function is smooth outside the diagonal, that
which proves the claim.
Let be a Riemannian manifold, and let us denote by the geodesic distance between , and we denote . If
does not depend on x, we have the following result:
COROLLARY . If then for .
PROOF. Using the equalities of the Dirichlet-to- Neumann operators and Lemma 2.2 we have
On the other hand, since
then, taking the limit when in (2.3), the demonstration follows.
REMARK 2.5. From LEMMA 2.2 and COROLLARY 2.4 we have the following equation,
The set of solutions of equation (2.4) is a group with multiplication law given by composition of functions, that is, if and are solutions of the equation (2.4), then, is solution of (2.4). In fact,
In what follows, we use an explicit formula for the Greens Function of on the annulus (Bârza and Guisa 1998). There, is given in polar coordinates by:
and it is conformal to the euclidean metric, with conformal factor.
Then, the normal derivative of with respect to on, is , is:
Analogously, the normal derivative of, with respect to on, is :
The Greens function of is given by
where , , , , , .
LEMA . The Schwartz kernel of Λg0, being of the form, is
The equality above is in the distributions sense.
PROPOSITION 2.7 . Let p ,q Î ¶ Ar then,
where denotes the geodesic distance between and with respect to the Euclidean metric in ∂ Ar.
Proof. In order to prove equation (2.8), we write
Then, the sequence has the following property: for all where is a constant that depends only on k and. In fact,. Hence, the series represents a function. On the other hand, using the Fourier series of the function, with, we have that
the equality being in the distributions sense.
Then, multiplying (2.6) by and taking the limit as , we get the following:
Analogously, we get (2.9).
REMARK . It follows from the proof of the Proposition (2.6) that the Schwartz kernel of can be written as:
where is a function given by
TEOREMA . Let be a metric as in, where, , and. If, then,
where denotes the Schwarzian Derivative of (see and the line right after it).
Proof. Using the equality of the Dirichlet-to-Neumann operators, it follows from Lemma 2.2 that
Writing, and using (2.8), we have that
On the other hand, we have from Corollary 2.4 and (2.8) that on the boundary. Hence,
We obtain, then,
Since the left hand side of the equation (2.12) is the component of the Schwartz kernel, then if we take , we get
In what concerns the right hand side of the equation (2.12), we use Taylor expansion of order of the expression in brackets, for near ; we get, with ,
which can be written as follows,
Since the limit exists, when d ® 0, we obtain from (2.12) and (2.13) that
or, more precisely:
Let us λ= λ (R) denote, the expression
Then, it, follows that
From equations (2.14) and (2.15) we have that
The right-hand side of (2.18) is called the Schwarzian Derivative of (Navas 2007).
REMARK . The numerical study of defined in (2.16) is done in Mendoza et al. 2009.
COROLLARY . The solution of the equation for is .
Proof. Making the change of variables: , the equation (2.18) becomes
Since,f(θ + 2π) =f (θ + 2π) we have that and are periodic of period 2π. Then, integrating (2.20) between and 2π we obtain
On the other hand,
which implies that . Because there is such that , we get . Therefore,.
It follows that restricted to the exterior boundary is a rotation and equals zero there. The same conclusion holds for the restriction of to the interior boundary.
The general solution of the equations (2.18) can be obtained using the formulas of Chuaqui et al. 2003, page 1.
We thank Gustavo Tamm and Henrique Araújo for helping us to solve and to understand the non-linear ordinary differential equation (2.18).
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Ramón Orestes Mendoza Ahumada
Manuscript received on October 1, 2008; accepted for publication on August 11, 2010