Abstract:
We prove uniqueness for Calder\'on's problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three and four dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for $C^1$ conductivities and Lipschitz conductivities sufficiently close to the identity.

Abstract:
We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb R^n$ is Lipschitz, $3\leq n \leq 6$, and $s$ and $p$ are such that $ W^{s,p}(\Omega)\not \subset W^{1,\infty}(\Omega)$. In particular, we obtain uniqueness for conductivities in $W^{1,n}(\Omega)$ ($n=3,4$). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.

Abstract:
In this paper we study the inverse conductivity problem with partial data. Moreover, we show that, in dimension $n\geq 3$ the uniqueness of the Calder\'{o}n problem holds for the $C^{1}\bigcap H^{3/2, 2}$ conductivities.

Abstract:
We consider the stability issue of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet-to-Neumann map. We extend here the stability result obtained by Alessandrini and Vessella in Advances in Applied Mathematics 35:207-241, where the authors considered the piecewise constant isotropic case.

Abstract:
We consider the electrostatic inverse boundary value problem also known as electrical impedance tomography (EIT) for the case where the conductivity is a piecewise linear function on a domain $\Omega\subset\mathbb{R}^n$ and we show that a Lipschitz stability estimate for the conductivity in terms of the local Dirichlet-to-Neumann map holds true.

Abstract:
We consider the inverse problem of determining the Lam\'{e} parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse problem when the Lam\'{e} parameters and the density are assumed to be piecewise constant on a given domain partition.

Abstract:
In these notes we prove log-type stability for the Calder\'on problem with conductivities in $ C^{1,\varepsilon}(\bar{\Omega}) $. We follow the lines of a recent work by Haberman and Tataru in which they prove uniqueness for $ C^1(\bar{\Omega}) $.

Abstract:
In this paper we consider the problem of determining an unknown pair $\lambda$, $\mu$ of piecewise constant Lam\'{e} parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $\lambda$ and $\mu$ from the Dirichlet to Neumann map.

Abstract:
We extend a global uniqueness result for the Calder\'on problem with partial data, due to Kenig-Sj\"ostrand-Uhlmann, to the case of less regular conductivities. Specifically, we show that in dimensions $n\ge 3$, the knowledge of the Diricihlet-to-Neumann map, measured on possibly very small subsets of the boundary, determines uniquely a conductivity having essentially $3/2$ derivatives in an $L^2$ sense.

Abstract:
We study the Dirichlet problem for the equation in the exterior of nonclosed Lipschitz surfaces in . The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable. Weak solvability of elliptic boundary value problems with Dirichlet, Neumann, and mixed Dirichlet-Neumann boundary conditions in Lipschitz domains has been studied in [1–6]. It is pointed out in the book [1, page 91] that domains with cracks (cuts) are not Lipschitz domains. So, solvability of elliptic boundary value problems in domains with cracks does not follow from general results on solvability of elliptic boundary value problems in Lipschitz domains. In the present paper, the weak solvability of the Dirichlet problem for the equation in the exterior of nonclosed Lipschitz surfaces (cracks) in is studied. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution are proved, integral representation for a solution in the form of single-layer potential is obtained, the problem is reduced to the uniquely solvable operator equation. The weak solvability of the Neumann problem for the Laplace equation in the exterior of several smooth nonclosed surfaces in has been studied in [7]. Boundary value problems for the Helmholtz equation in the exterior of smooth nonclosed screens have been studied in [8, 9]. In Cartesian coordinates in consider bounded Lipschitz domain with the boundary , that is, is closed Lipschitz surface. Let be nonempty subset of the boundary and . Assume that is a nonclosed Lipschitz surface with Lipschitz boundary in the space , and assume that includes its limiting points, or, alternatively, assume that is a union of finite number of such nonclosed surfaces, which do not have common points, in particular, they do not have common boundary points. In the latter case, is not a connected set. Notice that is a closed set. Let us introduce Sobolev spaces on as follows: Spaces and are dual spaces in the sense of scalar product in [1, pages 91-92]. Furthermore, one can set for (see [1, page 79]), and (see [1, pages 77, 99]). Spaces and on closed Lipschitz surface and their norms are defined, for example, in [1, page 98]. Let be Laplacian in , then for the equation consider the single-layer