Amir Moradifam

On the best possible remaining term in the Hardy inequality

We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain Ω of ℝn that makes it an admissible candidate for an improved Hardy inequality of the following type. For every ∈ H10(Ω) A characterization of the best possible constant c(V) is also given. This result yields easily the improved Hardys inequalities of Brezis-Vázquez [Brezis H, Vázquez JL (1997) Blow up solutions of some nonlinear elliptic problems. Revista Mat Univ Complutense Madrid 10:443–469], Adimurthi et al. [Adimurthi, Chaudhuri N, Ramaswamy N (2002) An improved Hardy Sobolev inequality and its applications. Proc Am Math Soc 130:489–505], and Filippas-Tertikas [Filippas S, Tertikas A (2002) Optimizing improved Hardy inequalities. J Funct Anal 192:186–233] as well as the corresponding best constants. Our approach clarifies the issue behind the lack of an optimal improvement while yielding the following sharpening of known integrability criteria: If a positive radial function V satisfies lim infr→oln(r)∫ro,sV(s)ds>−∞,then there exists ρ:=ρ(Ω) > 0 such that the above inequality holds for the scaled potential vρ(x)=v(|x|ρ).On the other hand, if lim r→0 ln(r)∫ro,sV(s)ds=−∞, then there is no ρ > 0 for which the inequality holds for Vρ.

Conductivity Imaging from One Interior Measurement in the Presence of Perfectly Conducting and Insulating Inclusions

We consider the problem of recovering an isotropic conductivity outside some perfectly conducting inclusions or insulating inclusions from the interior measurement of the magnitude of one current density field

Current Density Impedance Imaging of an Anisotropic Conductivity in a Known Conformal Class

|J|

The singular extremal solutions of the bi-Laplacian with exponential nonlinearity

. We show that the conductivity outside the inclusions and the shape and position of the inclusions are uniquely determined (except in an exceptional case) by the magnitude of the current generated by imposing a given boundary voltage. Our results show that even when the minimizer of the least gradient problem

The sphere covering inequality and its applications

\min \int_{\Omega} a |\nabla u|

A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data

with

On the existence of bounded positive solutions of Schrödinger equations in two-dimensional exterior domains

u|_{\partial \Omega}=f

Photo-acoustic tomography in a rotating measurement setting

exhibits flat regions (i.e., regions with

Symmetry and uniqueness of solutions to some Liouville-type equations and systems

\nabla u=0

A note on simultaneous preconditioning and symmetrization of non-symmetric linear systems

) it can be identified as the voltage potential of a conductivity problem with perfectly conducting inclusions.