Inverse problem by Cauchy data on arbitrary subboundary for system of elliptic equations

We consider an inverse problem of determining coefficient matrices in an $N$-system of second-order elliptic equations in a bounded two dimensional domain by a set of Cauchy data on arbitrary subboundary. The main result of the article is as follows: If two systems of elliptic operators generate the same set of partial Cauchy data on an arbitrary subboundary, then the coefficient matrices of the first-order and zero-order terms satisfy the prescribed system of first-order partial differential equations. The main result implies the uniqueness of any two coefficient matrices provided that the one remaining matrix among the three coefficient matrices is known

Inverse parabolic problems by Carleman estimates with data taken initial or final time moment of observation

We consider a parabolic equation in a bounded domain \OOO over a time interval $(0,T)$ with the homogeneous Neumann boundary condition. We arbitrarily choose a subboundary \Gamma \subset \ppp\OOO. Then, we discuss an inverse problem of determining a zeroth-order spatially varying coefficient by extra data of solution $u$: $u\vert_{\Gamma \times (0,T)}$ and $u(\cdot,t_0)$ in \OOO with $t_0=0$ or $t=T$. First we establish a conditional Lipschitz stability estimate as well as the uniqueness for the case $t_0=T.$ Second, under additional condition for $\Gamma$, we prove the uniqueness for the case $t_0=0$. The second result adjusts the uniqueness by M.V. Klibanov (Inverse Problems {\bf 8} (1992) 575-596) to the inverse problem in a bounded domain \OOO. We modify his method which reduces the inverse parabolic problem to an inverse hyperbolic problem, and so even for the inverse parabolic problem, we have to assume conditions for the uniqueness for the corresponding inverse hyperbolic problem. Moreover we prove the uniqueness for some inverse source problem for a parabolic equation for $t_0=0$ without boundary condition on the whole \ppp\OOO