ON THE HYPERBOLIC DIRICHLET TO NEUMANN FUNCTIONAL IN ABELIAN LIE GROUPS ∗ CLAUDIO CUEVAS

We prove the injectivity of the linearization of the hyperbolic Dirichlet to Neumann functional in a “small” compact neighborhood of the identity element e of an abelian Lie group G, under some suitable transversality condition. AMS subject classification : 35L20, 58G20. ∗This work was partially supported by DIUFRO, IN-16/98.


Introduction and Statement of the Result
Let Ω be a compact manifold of dimension n with smooth boundary ∂Ω and let Met(Ω) denote the set of all Riemannian metrics g on Ω.
We consider the ansitropic wave equation (1.1) There is a unique solution to (1.1); hence we may define the hyperbolic Dirichlet to Neumann map as the linear operator Λ g : where u is the unique solution to (1.1) and v g is the g-interior unit normal to ∂Ω.The hyperbolic Dirichlet to Neumann Functional: where O p (Γ) denotes the space of all linear operators from C ∞ 0 (Γ) into C ∞ (Γ), is known to be invariantly defined on the orbit obtained by the action over Met(Ω) of the group D of all diffeomorphism ψ of Ω, each of which restricts to the identity on ∂Ω.A natural conjecture is that this is the only obstruction to the uniqueness of Λ. From the point of view of applications, an even more important problem is to give a method to reconstruct g from Λ g .The elliptic Dirichlet to Neumann map was treated by several authors and is closely related to the physical problem referred as Electrical Impedance Tomography, of determining the conductivity of a body from measurement of voltage potential and corresponding current fluxes at the boundary (see [4]).For fixed g, we consider the following map: It is easy to see that the tangent space T I D of D at the identity mapping I is the vector space Γ 0 (T Ω) of all smooth vector fields on Ω which vanish on ∂Ω.On the other hand, the tangent space T g (Met(Ω)) of Met(Ω) at g is the vector space Γ(S 2 Ω) of all smooth sections of symmetric 2-tensors on Ω.We introduce on the spaces Γ 0 (T Ω) and Γ(S 2 Ω) the inner products where υ g (resp., tr), denote the volume element (resp., the trace) associated to g and m is the unique linear map defined by Considerer as in [3], the formal linearizations of A g at I and of Λ at g, respectively : (1.8) and Let (A g ) * denote the formal adjoint of A g with respect to the inner products (1.5) and (1.6) and diam g (Ω) the diameter of Ω in the metric g.In [3], the authors stated the following : Conjecture 1.1 Let Ω 0 ⊂ Ω be a submanifold, m ∈ Γ(S 2 Ω) have support in Ω 0 i.e., m ∈ Γ 0 (S 2 Ω) and assume that a) Λ g (m) = 0, b) (A g ) * (m) = 0 and c) diam g (Ω 0 ) < T is sufficiently small that the exponential map for g is a global diffeomorphism.Then m is identically zero.
As in [3] we refer to condition b) as the Transversality Condition.We remind that condition c) is necessary to avoid the appearance of caustics.The Tranversality Condition replace the harmonic hypothese used in [4].
The main results of this paper is:

holds if G be an abelian Lie group and
Ω is a compact neighborhood of G at the identity element e, and g is an invariant metric on G and m ∈ Γ 0 (S 2 Ω).
We are able to obtain an generalization Cardozo-Mendoza Theorem 1.1 [3] as corollary of Theorem 1.1.
Corollary They also proved the conjecture when n=2 and g is near the Euclidean metric in the C 3 -topology.In [1] they prove that Conjecture 1.1 holds if Ω is a bounded domain of the hyperbolic space (resp.,n-sphere)and g is the canonical metric in this spaces.In [2], the authors proved the uniqueness conjecture for the case when the manifold is a sufficiently small bounded domain of IR 3 , under suitable geometric conditions and the metric g is C 3 -close to Euclidean metric.We shall make use of the invariant formulas for A g , (A g ) * and Λ g proved in Section 2 [3].

Proof of Theorem 1.1
Let G be an abelian Lie group, and Lie(G) the algebra formed by the set of all left invariant vector fields on G, and Ω is a compact neighborhood of G at the identity element e, and Γ 0 (S 2 Ω) denote the vector space of all smooth sections of symmetric 2-tensors on G which are supported on Ω.
Proof of Theorem 1.1 Let x ∈ Ω, v ∈ Lie(G).The geodesic with initial tangent vector (x, v) ∈ T G ∼ = G × Lie(G) is given by γ(t) = x exp(tv).Since x ∈ Ω there exists an element A ∈ Lie(G) such that x = exp(A).
Let {E µ } µ=1,...,n a orthonormal frame field in Γ(T G) then we can define a orthonormal system { Ẽµ , V µ } µ=1,...,n in Γ(T T G) as follows.We consider the diagram and the operator We obtain 1.1 Conjecture 1.1 holds if Ω is a bounded domain of IR n , and g is a metric on IR n and m ∈ Γ 0 (S 2 Ω).Conjecture 1.1 holds if Ω is a bounded domain of the Torus T n , and g is an invariant metric on T n and m ∈ Γ 0 (S 2 Ω).
[3]ollary 1.3 Conjecture 1.1 holds if Ω is a bounded domain of the product IR n × T m and g is an invariant metric on IR n × T m and m ∈ Γ 0 (S 2 Ω).Remark 1.1 In[3], the authors proved that Conjecture 1.1 holds if Ω is a bounded domain of IR n , n ≥ 2 and g is the Euclidean metric.