Abstract. In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed condition on the principal part.

We consider the transport equation ∂ t u(x, t)+(H(x)·∇u(x, t))+p(x)u(x, t) = 0 in Ω × (0, T ) where Ω ⊂ R n is a bounded domain, and discuss two inverse problems which consist of determining a vector-valued function H(x) or a real-valued function p(x) by initial values and data on a subboundary of Ω. Our results are conditional stability of Hölder type in a subdomain D provided that the outward normal component of H(x) is positive on ∂D ∩ ∂Ω. The proofs are based on a Carleman estimate where the weight function depends on H.

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In this paper, we discuss an inverse problem for the Vlasov-Poisson system. We prove local uniqueness and stability theorems by using the method in Anikonov and Amirov [5] under the specular re ection boundary condition and with a prescribed outward electrical eld at the boundary.

In this paper, we discuss the uniqueness in an integral geometry problem in a strongly convex domain. Our problem is related to the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation on a Riemannian manifold and then the uniqueness theorem is proved in semi-geodesic coordinates by using the tools of Fourier analysis.Throughout this paper, we assume that a ij = a ji , 2 ≤ i, j ≤ n.which are known for each pair of the points (x, y) ∈ ∂D × ∂D.In this paper, we investigate the uniqueness of solution of Problem 1. Our method for proving the main result, which is stated in Section 2, relies on the reduction of Problem 1 to some kinetic equation (see (3.5) below) on a Riemannian manifold where the metric is considered in a semi-geodesic system of coordinates.Here we note that when we replace i, j = 2 in the summation by i, j = 1 in (1.1), we do not know the uniqueness in determining a ij , 1 ≤ i, j ≤ n. The choice of the indices "i, j = 1" depends on the semi-geodesic coordinates which we use in this paper. For detailed explanations, see also Remark 1 in Section 4. Problem 1 is related to an inverse problem of determining the Riemannian metric by the distances between boundary points, see Chapter 1 of [30]; also [2], [3]. Such an inverse problem is the mathematical model of several important medical imaging techniques and geophysical problems, and has called wide attention. As for the uniqueness theorem and stability estimates, see Muhometov [19] in two dimensions, and a recent work Pestov and Uhlmann [25]. For higher dimensions, we refer to Bernstein and Gerver [6], Beylkin [7], Muhometov and Romanov [20]. Also see e.g., [21], [22]. As for other kinds of inverse problems from the integral geometry, we refer to [5], [8], [14], [23], [24], [30] -[35]. Here we do not intend any complete lists of references. The connections between these problems and the inverse problems for parabolic, hyperbolic and kinetic equations are described in the works [1], [4], [16], [17], [18], [29].