We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain Ω ⊂ R n which vanish on a closed proper subset of the boundary is dense in L 1 (Ω). We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.

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We study inverse boundary problems for magnetic Schrödinger operators on a compact Riemannian manifold with boundary of dimension ≥ 3. In the first part of the paper we are concerned with the case of admissible geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Cauchy data on the boundary of the manifold for the magnetic Schrödinger operator with L ∞ magnetic and electric potentials, determines the magnetic field and electric potential uniquely.In the second part of the paper we address the case of more general conformally transversally anisotropic geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a compact manifold, which need not be simple. Here, under the assumption that the geodesic ray transform on the transversal manifold is injective, we prove that the knowledge of the Cauchy data on the boundary of the manifold for a magnetic Schrödinger operator with continuous potentials, determines the magnetic field uniquely. Assuming that the electric potential is known, we show that the Cauchy data determines the magnetic potential up to a gauge equivalence.

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We study inverse boundary problems for the advection diffusion equation on an admissible manifold, i.e. a compact Riemannian manifold with boundary of dimension ≥ 3, which is conformally embedded in a product of the Euclidean real line and a simple manifold. We prove the unique identifiability of the advection term of class H 1 ∩ L ∞ and of class H 2/3 ∩ C 0,1/3 from the knowledge of the associated Dirichlet-to-Neumann map on the boundary of the manifold. This seems to be the first global identifiability result for possibly discontinuous advection terms.Here and in what follows H s (M 0 ), s ∈ R, is the standard Sobolev space on M 0 , see [28, Chapter 4], and M 0 = M \ ∂M stands for the interior of M. We also let ν be the unit outer normal to the boundary of M. We shall define the trace of the normal derivative ∂ ν u ∈ H −1/2 (∂M) as follows. Let ϕ ∈ H 1/2 (∂M). Then letting v ∈ H 1 (M 0 ) be a continuous extension of ϕ, we set ∂ ν u, ϕ H −1/2 (∂M )×H 1/2 (∂M ) = M ∇u, ∇v dV + M X(u)vdV, (1.2) 1