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

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(eg0) 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 .


INTRODUCTION
Let M( ) denote the space of all Riemannian metrics on a compact manifold , with C ∞ boundary ∂ , and denote by O p (∂ ) the space of continuous linear operators acting on C ∞ (∂ ).
The Dirichlet-to-Neumann functional is a mapping from M( ) into O p (∂ ) such that, for each g ∈ M( ), g takes Dirichlet boundary values to Neumann boundary values.More precisely, if u ∈ C ∞ ( ) is the unique solution of the Dirichlet problem g u = 0 in , u| ∂ = ϕ ∈ C ∞ (∂ ), then g (ϕ) = du(ν g ) ∈ C ∞ (∂ ), where g (resp.ν g ) is the Laplace-Beltrami operator (resp.unit interior normal vector field) associated to the metric g.The study of this functional goes back to the seminal paper of (Calderón 1980).
It is known (Lee and Uhlmann 1989) that g is in fact an elliptic self-adjoint pseudo-differential operator of order one, whose principal symbol is |ξ | h 0 , ξ ∈ T * ∂ , and h 0 := g | ∂ .
Let D( ) be the group of diffeomorphism of .The semi-direct product D( ) C ∞ ( ) (Polyakov 1987) of the groups D( ) and C ∞ ( ) defined by 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 realvalued 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 D 0 ( ) With respect to the determination of the metric g 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 h = F * e 2ϕ g 0 , for g 0 coming from the pull-back of the euclidean metric in the annulus of radius 1 and R 2 , R > 1.We prove, in this special case, that the equality of the Dirichlet-to-Neumann Operators associated to both metrics h and g 0 gives us a relation involving the Schwarzian derivative of f ( f the lifting to R of the restriction to the boundary of the diffeomorphism F).
Furthermore, we also show that the conformal factor restricted to the boundary of the annulus is determined by f .
More precisely, we shall prove in Section 2 that, if is the annulus ) is conformal to the euclidean metric, h = F * (e 2ϕ g 0 ), where F ∈ D( A R ) and ϕ ∈ C ∞ ( A R ); the equality of the Schwartz kernels of g 0 and of h implies that the argument function f , of the restriction of F to ∂ A R , satisfies the differential equation and S( f ) denotes the Schwarzian Derivative of f .It follows that, if λ(R) ≥ 0, then f = 1, f (θ ) = θ + c and ϕ equals zero on the boundary.

GEOMETRIC FORMULATION
Here on we will denote by N g the Schwartz kernel of g .We start with two lemmas.
where F denotes the real, valued function on changing variables F(y) = z we get: where F denotes the unique real, valued function defined on ∂ such that and E is the tangent unitary vector field on ∂ such that h(E, E) = 1 and v h (E) = +1.The above equation means at every point p ∈ ∂ the following: belong to the same onedimensional tangent space T F( p) (∂ ); consequently, the first one is a real multiple of the second.In fact, this multiple is unique and it is equal to F ( p).
For the second equality, finishing the proof.
where χ is any smooth function on such that χ = ϕ + O(x 2 ).Now, using Green's formula and On the other hand, since then, taking the limit when y → x in (2.3), the demonstration follows.
REMARK 2.5.From Lemma 2.2 and Corollary 2.4 we have the following equation, (2.4) The set of solutions of equation (2.4) is a group with multiplication law given by composition of functions, that is, if F and G are solutions of the equation (2.4), then, G • F is solution of (2.4).In fact, In what follows, we use an explicit formula for the Green's Function of g 0 on the annulus A R (Bârza and Guisa 1998).There, g 0 is given in polar coordinates by: and it is conformal to the euclidean metric, with conformal factor f (ρ, θ) = 1 2 1 + 1 ρ 2 .Then, the normal derivative of u ∈ C ∞ ( A R ), with respect to g 0 on | z |= R, is: Analogously, the normal derivative of u, with respect to g 0 on | z |= 1 R , is: LEMA 2.6.The Schwartz kernel of g 0 , g 0 ∈ M( A R ) being of the form (2.5), is (2.6) (2.7) The equality above is in the distributions sense.
PROPOSITION 2.7.Let p, q ∈ ∂ A R , then, and lim p→q d 2 g eucl ( p, q)N g 0 ( p, q) = −4 where d g eucl denotes the geodesic distance between p and q with respect to the Euclidean metric in ∂ A R .
PROOF.In order to prove equation (2.8), we write Then, the sequence b n = n(a n − 1) has the following property: On the other hand, using the Fourier series of the function which implies: the equality being in the distributions sense.Then, multiplying (2.6) by d 2 g eucl ( p, q) and taking the limit as q → p, we get the following: Analogously, we get (2.9).
REMARK 2.8.It follows from the proof of the Proposition (2.6) that the Schwartz kernel of g 0 can be written as: where H is a C ∞ function given by (2.10) TEOREMA 2.9.Let g 0 be a metric as in (2.5), h = F * (e 2ϕ g 0 ) where where S( f ) denotes the Schwarzian Derivative of f (see (2.18) and the line right after it).
PROOF.Using the equality of the Dirichlet-to-Neumann operators, it follows from Lemma 2.2 that Writing x = Re iθ , y = Re iα and using (2.8), we have that .
On the other hand, we have from Corollary 2.4 and (2.8) that e −ϕ•F = f on the boundary.Hence, .
We obtain, then, . (2.12) Since the left hand side of the equation (2.12) is the C ∞ component of the Schwartz kernel, then if we take α → θ , we get In what concerns the right hand side of the equation (2.12), we use Taylor expansion of order 4 of the expression in brackets, for α near θ ; we get, with δ = α − θ , , which can be written as follows, (2.13) Since the limit exists, when δ → 0, we obtain from (2.12) and (2.13) that or, more precisely: (2.15) An Acad Bras Cienc (2010) 82 (4) Then, it, follows that 2λ From equations (2.14) and (2.15) we have that (2.17) which implies: (2.18) The right-hand side of (2.18) is called the Schwarzian Derivative S( f ) of f (Navas 2007).which implies that y = 0.Because there is 0 ≤ θ 0 ≤ 2π such that f (θ 0 ) = 1, we get y = 0. Therefore, f = 1.
It follows that F restricted to the exterior boundary is a rotation and ϕ equals zero there.The same conclusion holds for the restriction of F 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.

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FERNANDO A.F.C. SILVA, PEDRO A.G. VENEGAS and RAMÓN O.M. AHUMADA provides a natural right action on M( ), given by g • (F, ϕ) = F * e 2ϕ g, where F * denotes the pull-back of F.
PROOF.Let x 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∂ n = ∇ g x of x.The flow e t∂ n of ∇ g x induces a diffeomorphism φ : [0, ) t × ∂ → φ([0, ) t × ∂ )defined by φ(t, y) := e t∂ n (y), and we have x(φ(t, y)) = t.This induces natural coordinates z = (x, y) near the boundary, these are normal geodesic coordinates.The function u is the unique solution of the Dirichlet problem g u = 0 in , and u| ∂ An Acad Bras Cienc (2010) 82 (4) 816 FERNANDO A.F.C. SILVA, PEDRO A.G. VENEGAS and RAMÓN O.M. AHUMADA The next Lemma establish, a relation between N g and the Green function G(z, z ) of the Laplacian g with Dirichlet condition on ∂ (Guillarmou and Sá Barreto 2009).LEMMA 2.3.The Schwartz kernel N g (y, y ) of g is given for y, y ∈ ∂ , y = y , byN g y, y = ∂ n ∂ n G z, z z=y,z =ywhere ∂ n , ∂ n are, respectively, the inward pointing vector fields to the boundary in variable z and z .