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

Electrical Impedance Tomography

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Mental health concerns of people impacted by the coronavirus pandemic have not been adequately addressed. The objective of this study was to develop and evaluate the properties of the Coronavirus Anxiety Scale (CAS), which is a brief mental health screener to identify probable cases of dysfunctional anxiety associated with the COVID-19 crisis. This 5-item scale, which was based on 775 adults with anxiety over the coronavirus, demonstrated solid reliability and validity. Elevated CAS scores were found to be associated with coronavirus diagnosis, impairment, alcohol/drug coping, negative religious coping, extreme hopelessness, suicidal ideation, as well as attitudes toward President Trump and Chinese products. The CAS discriminates well between persons with and without dysfunctional anxiety using an optimized cut score of ≥ 9 (90% sensitivity and 85% specificity). These results support the CAS as an efficient and valid tool for clinical research and practice.

Coronavirus Anxiety Scale: A brief mental health screener for COVID-19 related anxiety.

We review theoretical and numerical studies of the inverse problem of electrical impedance tomography which seeks the electrical conductivity and permittivity inside a body, given simultaneous measurements of electrical currents and potentials at the boundary.

/pdf/electrical-impedance-tomography-3po2ct3t0v.pdf

Electrical impedance tomography

Let Ω be a domain in R n whose boundary is C 1 if n≥3 or C 1,β if n=2. We consider a magnetic Schrodinger operator L W , q in Ω and show how to recover the boundary values of the tangential component of the vector potential W from the Dirichlet to Neumann map for L W , q . We also consider a steady state heat equation with convection term Δ+2W·∇ and recover the boundary values of the convection term W from the Dirichlet to Neumann map. Our method is constructive and gives a stability result at the boundary.

/pdf/identifiability-at-the-boundary-for-first-order-terms-2tpxg3fnz8.pdf

Identifiability at the boundary for first-order terms

(1997). Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Communications in Partial Differential Equations: Vol. 22, No. 5-6, pp. 1009-1027.

Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions

We establish uniqueness in the inverse conductivity problem for conductivities which have 3/2 derivatives in L
 p
 , p > 2n. Our results are in dimensions three and higher.

http://www.ms.uky.edu/~rbrown/publications/sle5.pdf

Uniqueness in the Inverse Conductivity Problem for Conductivities with 3/2 Derivatives in L^p, p > 2n

We study the Stokes operator A in a three- dimensional Lipschitz domain Ω. Our main result asserts that the domain of A is contained in W 1,p 0 (Ω) ∩ W 3/2,2 (Ω) for some p> 3. Certain L ∞ -estimates are also established. Our results may be used to improve the regularity of strong solutions of Navier-Stokes equations in nonsmooth domains. In the ap- pendix we provide a simple proof of area integral estimates for solutions of Stokes equations.

/pdf/estimates-for-the-stokes-operator-in-lipschitz-domains-2pcai3s6s5.pdf

Estimates for the Stokes Operator in Lipschitz Domains

If $L_\gamma = {\operatorname{div}}\gamma \nabla $ is an elliptic operator with scalar coefficient $\gamma $, we show that we can recover the coefficient $\gamma $ from the Dirichlet-to-Neumann map under the assumption that $\gamma $ has only ${3 / 2} + \epsilon $ derivatives. Previously, the best result required $\gamma $ to have two derivatives.

Russell M. Brown

Papers

Identifiability at the boundary for first-order terms

Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions

Uniqueness in the Inverse Conductivity Problem for Conductivities with 3/2 Derivatives in L^p, p > 2n

Estimates for the Stokes Operator in Lipschitz Domains

Global Uniqueness in the Impedance-Imaging Problem for Less Regular Conductivities