This article shows that knowledge of the Dirichlet-Neumann (DN) map on certain subsets of the boundary for functions supported roughly on the rest of the boundary uniquely determines a magnetic Schrödinger operator. With some geometric conditions on the domain, either the subset on which the DN map is measured or the subset on which the input functions have support may be made arbitrarily small. This is a response to a question posed in [DKSU]. The method involves modifying the Carleman estimate in that paper by conjugation with certain pseudodifferential-like operators.2000 Mathematics Subject Classification. Primary 35R30.

Abstract. We propose a method to reconstruct the density of an optical source in a highly scattering medium from ultrasound-modulated optical measurements. Our approach is based on the solution to a hybrid inverse source problem for the radiative transport equation (RTE). A controllability result for the RTE plays an essential role in the analysis.

We consider the inverse problem of recovering the optical properties of a highly-scattering medium from acousto-optic measurements. Using such measurements, we show that the scattering and absorption coefficients of the radiative transport equation can be reconstructed with Lipschitz stability by means of algebraic inversion formulas.

Abstract. We show that measurements of the Neumann-to-Dirichlet map on a certain part of the boundary of a domain in R N , N ≥ 3, for inputs with support restricted to the other part, determine an electric potential on that domain. Given a convexity condition on the domain, either the set on which measurements are taken, or the set on which input functions are supported, can be made to be arbitrarily small. The result is analogous to the result by Kenig, Sjöstrand, and Uhlmann for the Dirichlet-to-Neumann map. The main new ingredient in the proof is a Carleman estimate for the Schrödinger operator with appropriate boundary conditions. IntroductionConsider the Euclidean space R n+1 , n ≥ 2, and suppose Ω is a smooth bounded domain in R n+1 . Now suppose that q ∈ L ∞ (Ω) is such that the problemThe basic inverse problem here is whether N q determines q. This question is related to the corresponding question for the Dirichlet-to-Neumann (DN) map, which has been studied in several papers. Notably, Sylvester and Uhlmann proved uniqueness for the DN problem in [19], and Nachman gave a reconstruction method in [17]. For the Neumannto-Dirichlet map, the fact that N q determines q is a consequence of the argument in [19].A more difficult question is whether partial knowledge of N q determines q. Some recent papers on this subject have been written for the two-dimensional case. In [11], Imanuvilov, Uhlmann, and Yamamoto in [11] proved that measuring N q on arbitrary open domains determines q for q ∈ W 1,p , p > 2. A slightly different problem, where assumptions are made on the potential in the neighbourhood of the boundary, was addressed by Hyvönen, Piiroinen, and Seiskari, in [8]. Earlier work by Imanuvilov, Uhlmann, and Yamamoto was also done for the DN map in two dimensions in [10]. The results of [10] were then generalized to Riemannian surfaces by Guillarmou and Tzou in [6].For three or higher dimensions, the only work known to this author on the partial data ND map problem comes from Isakov, who proved in [9] that subsets of the boundary which coincide with a hyperplane or hypersphere may be ignored in the measurements, for both the ND and DN problems. For more general subsets of the boundary, in three or more dimensions, on the DN problem, there are several previous results. Bukhgeim and Uhlmann in [2] and Kenig, Sjöstrand, and Uhlmann in [13] show for the DN problem, 2000 Mathematics Subject Classification. Primary 35R30.

Abstract. We prove uniqueness results for a Calderón type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometrical optics solutions which reduce the Calderón type problem to a tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar results due to Kenig-Sjöstrand-Uhlmann.