Monthly Notices is one of the three largest general primary astronomical research publications. It is an international journal, published by the Royal Astronomical Society. This article 1 describes its publication policy and practice.

Monthly Notices of the Royal Astronomical Society

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

t. This paper surveys some of the work our group has done in electrical impedance tomography.

Electrical Impedance Tomography

The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Observation and Control for Operator Semigroups

We show that the coefficient -y(x) of the elliptic equation Vie (QyVu) = 0 in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and give a reconstruction pro

https://www.jstor.org/stable/pdfplus/2118653.pdf

Global uniqueness for a two-dimensional inverse boundary value problem

We consider the problem of determining the scalar coefficient γ in the elliptic equation div(γ grad u) = 0 in ω when, for every Dirichlet datum u = ∅ on ∂ω , the Neumann datum γ(∂/ ∂ n)u = ∧.γ∅ is known. We prove a continuous dependence result

Stable determination of conductivity by boundary measurements

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.

The stability for the Cauchy problem for elliptic equations

Singular solutions of elliptic equations and the determination of conductivity by boundary measurements

Lipschitz stability for the inverse conductivity problem

We study mappings from ℝ2 into ℝ2 whose components are weak solutions to the elliptic equation in divergence form, div (σ∇u)= 0, which we call σ-harmonic mappings. We prove sufficient conditions for the univalence, i.e., injectivity, of such mappings. Moreover we prove local bounds in BMO on the logarithm of the Jacobian determinant of such univalent mappings, thus obtaining the a.e. nonvanishing of their Jacobian. In particular, our results apply to σ-harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization.

Giovanni Alessandrini

Papers

Stable determination of conductivity by boundary measurements

The stability for the Cauchy problem for elliptic equations

Singular solutions of elliptic equations and the determination of conductivity by boundary measurements

Lipschitz stability for the inverse conductivity problem

Univalent σ-Harmonic Mappings