In this review, concerning parabolic equations, we give self-contained descriptions on (1) derivations of Carleman estimates; (2) methods for applications of the Carleman estimates to estimates of solutions and to inverse problems. Moreover limited to parabolic equations, we survey the previous and recent results in view of the applicability of the Carleman estimate. We do not intend to pursue any general treatments of the Carleman estimate itself but by showing it in a direct manner, we mainly aim to demonstrate the applicability of the Carleman estimate to the estimation of solutions and inverse problems. Contents

We consider a system y t (t, x) = −Ay(t, x) + g(t, x)x ∈ with a suitable boundary condition, where ⊂ R n is a bounded domain, −A is a uniformly elliptic operator of the second order whose coefficients are suitably regular for (t, x), θ ∈]0, T [ is fixed, and a function g(t, x) satisfiesOur inverse problems are determinations of g using overdetermining data y |]0,T [×ω or {y |]0,T [× 0 , ∇y |]0,T [× 0 }, where ω ⊂ and 0 ⊂ ∂ . Our main result is the Lipschitz stability in these inverse problems. We also consider the determination of f = f (x), x ∈ in the case of g(t, x) = f (x)R(t, x) with given R satisfying R(θ, •) > 0 on . Finally, we discuss an upper estimation of our overdetermining data by means of f .

We consider a one-dimensional fractional diffusion equation: ∂ α t u(x, t) = ∂ ∂x p(x) ∂u ∂x (x, t) , 0 < x < , where 0 < α < 1 and ∂ α t denotes the Caputo derivative in time of order α. We attach the homogeneous Neumann boundary condition at x = 0, and the initial value given by the Dirac delta function. We prove that α and p(x), 0 < x < , are uniquely determined by data u(0, t), 0 < t < T. The uniqueness result is a theoretical background in experimentally determining the order α of many anomalous diffusion phenomena which are important for example in the environmental engineering. The proof is based on the eigenfunction expansion of the weak solution to the initial value/boundary value problem and the Gel'fand-Levitan theory. §1. Introduction. Recently there are many anomalous diffusion phenomena observed which show different aspects from the classical diffusion. For example, Adams and Gelhar [1] pointed that field data in the saturated zone of a highly heterogeneous aquifer are not well simulated by the classical advection-diffusion equation which is based on

The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for the continuously differentiable functions. Accordingly, in the publications devoted to the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of smooth functions that appear to be too narrow for several important applications. In this paper, we propose a definition of the Caputo derivative on a finite interval in the fractional Sobolev spaces and investigate it from the operator theoretic viewpoint. In particular, some important equivalences of the norms related to the fractional integration and differentiation operators in the fractional Sobolev spaces are given. These results are then applied for proving the maximal regularity of the solutions to some initial-boundary-value problems for the time-fractional diffusion equation with the Caputo derivative in the fractional Sobolev spaces. MSC 2010 : Primary 26A33; Secondary 35C05, 35E05, 35L05, 45K05, 60E99 Key Words and Phrases: Riemann-Liouville integral, Caputo fractional derivative, fractional Sobolev spaces, norm equivalences, fractional diffusion equation in Sobolev spaces, norm estimates of the solutions, initialboundary-value problems, weak solution, existence and uniqueness results c 2015 Diogenes Co., Sofia pp. 799-820 ,

We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can uniquely determine the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.