Injectivity of the Dirichlet-to-Neumann Functional and the Schwarzian Derivative

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. Silva^I; Pedro A.G. Venegas^II; Ramón O.M. Ahumada^I

^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}

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* (e^2φ 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.

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* (e^2φ g₀), no anel circular A_R, e a Derivada Schwarziana da função argumento f, da restrição do difeomorfismo F à fronteira de A_R.

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,

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 Green’s 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 Green’s 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 Green’s 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 Green’s function of is given by

where , , , , , .

LEMA . The Schwartz kernel of Λ_g0, being of the form, is

The equality above is in the distributions sense.

PROPOSITION 2.7 . Let p ,q Î ¶ A_r then,

where denotes the geodesic distance between and with respect to the Euclidean metric in ∂ A_r.

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