Abstracts
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 A R, and the Schwarzian Derivative of the argument function f of the restriction of the diffeomorphism F to the boundary of A R.
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 A R, e a Derivada Schwarziana da função argumento f, da restrição do difeomorfismo F à fronteira de A R.
anel; Funcional Dirichlet-Neumann; Derivada Schwarziana
MATHEMATICAL SCIENCES
Injectivity of the Dirichlet-to-Neumann Functional and the Schwarzian Derivative
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
Correspondence to Correspondence to: Ramón Orestes Mendoza Ahumada E-mail: ramon@dmat.ufpe.br
ABSTRACT
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.
RESUMO
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.
INTRODUCTION
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,
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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
that is,
which implies:
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
which implies:
or, more precisely:
Let us λ= λ (R) denote, the expression
Then, it, follows that
where
From equations (2.14) and (2.15) we have that
which implies:
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,
that is,
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.
ACKNOWLEDGMENTS
We thank Gustavo Tamm and Henrique Araújo for helping us to solve and to understand the non-linear ordinary differential equation (2.18).
Manuscript received on October 1, 2008; accepted for publication on August 11, 2010
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Publication Dates
-
Publication in this collection
28 Feb 2011 -
Date of issue
Dec 2010