NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 85
THEOREM 1.1
Let (M0,g) be a smooth compact Riemann surface with boundary, let Ɣ ⊂ ∂M0 be
an open subset of the boundary, and let g be the nonnegative Laplacian on M0.For
α ∈ (0, 1),letV1,V2 ∈ C1,α (M0) be two potentials and, for i = 1, 2,let
C Ɣ
i =: (u|Ɣ,∂νu|Ɣ); u ∈ H 1 (M0), (g + Vi)u = 0, u = 0on∂M0 \ Ɣ
(1)
be the respective Cauchy partial data spaces. If C Ɣ 1 = C Ɣ 2 ,thenV1 = V2.
Here the space C1,α (M0) is the usual Hölder space for α ∈ (0, 1). Notice that when
g + Vi do not have L2 eigenvalues for the Dirichlet condition, the statement above
can be given in terms of Dirichlet-to-Neumann operators. Since ˆg = e−2ϕg when
ˆg = e2ϕg for some function ϕ, it is clear that in the statement in Theorem 1.1, we
need only to fix the conformal class of g instead of the metric g (or equivalently,
we need only to fix the complex structure on M0). In particular, the smoothness
assumption of the Riemann surface with boundary is not really essential since we
can change it conformally to make it smooth; for the Cauchy data space, this just
has the effect of changing the potential conformally (all we need is that this new
potential be C1,α ). Observe also that Theorem 1.1 implies that, for a fixed Riemann
surface with boundary (M0,g), the Dirichlet-to-Neumann map on Ɣ for the operator
u →−divg(γ ∇gu) determines the isotropic conductivity γ if γ ∈ C3,α (M0) in the
sense that two conductivities giving rise to the same Dirichlet-to-Neumann map are
equal. This is a standard observation by transforming the conductivity problem to a
potential problem with potential V := (gγ 1/2 )/γ 1/2 . So our result also extends that
of Henkin and Michel [17] in the case of isotropic conductivities. Notice also that
reconstruction methods for isotropic conductivities are obtained in a recent work of
Henkin and Novikov [18] for full boundary data measurements on Riemann surfaces.
The method to identify the potential follows [6] and[19] and is based on the
construction of a large set of special complex geometric optic solutions of (g +
V )u = 0, also called Faddeev-type solutions and introduced first by Faddeev in [10].
More precisely, if Ɣ0 = ∂M0 \ Ɣ is the set where we do not know the Dirichletto-Neumann
operator, then we construct solutions of the form u = Re e/h (a +
r(h)) + eRe()/hs(h) with u|Ɣ0 = 0, where h>0 is a small parameter, and a
are holomorphic functions on (M0,g) independent of h,and||r(h)||L2 = O(h) while
||s(h)||L2 = O(h3/2 | log h|) as h → 0. The idea of [6] to reconstruct V (p) for p ∈ M0
is to take with a nondegenerate critical point at p and then use stationary phase as
h → 0. In our setting, the function needs to be purely real on Ɣ0 and Morse with a
prescribed critical point at p. One of our main contributions is a geometric construction
of the holomorphic Carleman weights satisfying such conditions. We should point
out that we use a method quite different from that in [19] to construct this weight,
and we believe that our method simplifies their construction even in their case. A