Searcharxiv⌕ Search

arXiv subjects

Amir Moradifam

Publications and source records attributed to Amir Moradifam.

At least 37 records · Page 2Linked to original sources
arXiv2017

Least Gradient Problems with Neumann Boundary Condition

We study existence of minimizers of the least gradient problem \[\inf_{v \in BV_g} \int_Ωφ(x, Dv),\] where $BV_g=\{v \in BV(Ω): \int_{\partial Ω}gv=1\}$, $φ(x,p): Ω\times \R^n \rightarrow \R$ is a convex, continuous, and homogeneous function of degree $1$ with respect to the $p$ variable, and $g$ satisfies the comparability condition $\int_{\partial Ω} g dS=0$. We prove that for every $0\not \equiv g \in L^{\infty}(\partial Ω)$ there are infinitely many minimizers in $BV(Ω)$. Moreover there exists a divergence free vector field $T\in (L^{\infty}(Ω))^n$ that determines the structure of level sets of all minimizers, i.e. $T$ determines $\frac{Du}{|Du|}$, $|Du|-$ a.e. in $Ω$, for every minimizer $u$. We also prove some existence results for general 1-Laplacian type equations with Neumann boundary condition. A numerical algorithm is presented that simultaneously finds $T$ and a minimizer of the above least gradient problem. Applications of the results in conductivity imaging are discussed.

math.AP↗
arXiv2016

Existence and structure of minimizers of least gradient problems

We study existence of minimizers of the general least gradient problem \[\inf_{u \in BV_f} \int_Ωφ(x,Du),\] where $BV_f=\{u \in BV(Ω): \ \ u|_{\partial Ω}=f\}$, $f\in L^{1}(\partial Ω)$, and $φ(x,ξ)$ is convex, continuous, and homogeneous function of degree $1$ with respect to the $ξ$ variable. It is proven that there exists a divergence free vector field $T\in (L^{\infty}(Ω))^n$ that determines the structure of level sets of all (possible) minimizers, i.e. $T$ determines $\frac{Du}{|Du|}$, $|Du|-$ a.e. in $Ω$, for all minimizers $u$. We also prove that every minimizer of the above least gradient problem is also a minimizer of \[\inf_{u\in \mathcal{A}_f} \int_{\R^n}φ(x,Du),\] where $\mathcal{A}_f=\{v\in BV(\R^n): \ \ v=f \ \ \hbox{on}\ \ Ω^c\}$ and $f\in W^{1,1}(\R^n)$ is a compactly supported extension of $f\in L^1(\partial Ω)$, and show that $T$ also determines the structure of level sets of all minimizers of the latter problem. This relationship between minimizers of the above two least gradient problems could be exploited to obtain information about existence and structure of minimizers of the former problem from that of the latter, which always exist.

math.AP↗
arXiv2016

The Sphere Covering Inequality and Its Applications

In this paper, we show that the total area of two distinct surfaces with Gaussian curvature equal to 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We refer to this lower bound of total area as the Sphere Covering Inequality. The inequality and its generalizations are applied to a number of open problems related to Moser- Trudinger type inequalities, mean field equations and Onsager vortices, etc, and yield optimal results.

math.AP↗
arXiv2016

Photo-acoustic tomography in a rotating measurement setting

Photo-acoustic tomography (PAT) aims to leverage the photo-acoustic coupling between optical absorption of light sources and ultrasound (US) emission to obtain high contrast reconstructions of optical parameters with the high resolution of sonic waves. Quantitative PAT often involves a two-step procedure: first the map of sonic emission is reconstructed from US boundary measurements; and second optical properties of biological tissues are evaluated. We consider here a practical measurement setting in which such a separation does not apply. We assume that the optical source and an array of ultrasonic transducers are mounted on a rotating frame (in two or three dimensions) so that the light source rotates at the same time as the US measurements are acquired. As a consequence, we no longer have the option to reconstruct a map of sonic emission corresponding to a given optical illumination. We propose here a framework where the two steps are combined into one and an absorption map is directly reconstructed from the available ultrasound measurements.

math.AP↗
arXiv2016

Symmetry of solutions of a mean field equation on flat tori

We study symmetry of solutions of the mean field equation \[ Δu +ρ(\frac{Ke^u}{\int_{T_ε} Ke^u} -\frac{1}{|T_ε|} )=0\] on the flat torus $T_ε=[-\frac{1}{2ε}, \frac{1}{2ε}] \times [-\frac{1}{2}, \frac{1}{2}]$ with $0<ε\leq 1$, where $K\in C^2({T}_ε)$ is a positive function with $-Δ\ln K \leq \fracρ{|T_ε|}$ and $ρ\leq 8π$. We prove that if $(x_0,y_0)$ is a critical point of the function $u+ln(K)$, then $u$ is evenly symmetric about the lines $x=x_0$ and $y=y_0$, provided $K$ is evenly symmetric about these lines. In particular we show that all solutions are one-dimensional if $K\equiv 1$ and $ρ\leq 8π$. The results are sharp and answer a conjecture of Lin and Lucia affirmatively. We also prove some symmetry results for mean field equations on annulus.

math.AP↗
arXiv2014

Uniqueness of minimizers of weighted least gradient problems arising in conductivity imaging

We prove uniqueness for minimizers of the weighted least gradient problem \[\inf \left\lbrace \int_Ω a|Du|: \ \ u\in BV(Ω), \ \ u|_{\partial Ω}=f \right\rbrace.\] The weight function $a$ is assumed to be continuous and it is allowed to vanish in certain subsets of $Ω$. Existence is assumed a priori. Our approach is motivated by the hybrid inverse problem of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current density vector field.

math.AP↗
arXiv2013

Existence and uniqueness of minimizers of general least gradient problems

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems - under certain sharp conditions - for minimizers of the general least gradient problem \[\inf_{u\in BV_f(Ω)} \int_Ωφ(x,Du),\] where $f:\partial Ω\to \R$ is continuous, \[ BV_f(Ω):=\{v\in BV(Ω): \ \ \forall x\in \partial Ω, \ \ \lim_{r\to 0} \ \esssup_{y\in Ω, |x-y|<r} |f(x) - v(y)| = 0 \ \} %BV_f(Ω)=\{u\in BV(Ω): {0.1cm} u|_{\partial Ω}=f {0.1cm} \hbox{and} {0.1cm} {0.1cm} u {0.1cm} \hbox{is continuous at} {0.1cm} \partial Ω\}. \] and $φ(x,ξ)$ is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the $ξ$ variable. In particular we prove that if $a\in C^{1,1}(Ω)$ is bounded away from zero, then minimizers of the weighted least gradient problem $\inf_{u \in BV_f}\int_Ω a|Du|$ are unique in $BV_f(Ω)$. We construct counterexamples to show that the regularity assumption $a\in C^{1,1}$ is sharp, in the sense that it can not be replaced by $a\in C^{1,α}(Ω)$ with any $α<1$.

math.FA↗
arXiv2013

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

We present a procedure for recovering the conformal factor of an anisotropic conductivity matrix in a known conformal class in a domain in Euclidean space of dimension greater than or equal to 2. The method requires one internal measurement, together with a priori knowledge of the conformal class (local orientation) of the conductivity matrix. This problem arises in the coupled-physics medical imaging modality of Current Density Impedance Imaging (CDII) and the assumptions on the data are suitable for measurements determinable from cross-property based couplings of the two imaging modalities CDII and Diffusion Tensor Imaging (DTI). We show that the corresponding electric potential is the unique solution of a constrained minimization problem with respect to a weighted total variation functional defined in terms of the physical data. Further, we show that the associated equipotential surfaces are area minimizing with respect to a Riemannian metric obtained from the data. The results are also extended to allow the presence of perfectly conducting and/or insulating inclusions.

math.AP↗
arXiv2012

Functional Inequalities: New Perspectives and New Applications

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.

math.AP↗
arXiv2011

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 or insulating inclusions from the interior measurement of the magnitude of one current density field $|J|$. We prove that the conductivity outside the inclusions, and the shape and position of the perfectly conducting and insulating inclusions are uniquely determined (except in an exceptional case) by the magnitude of the current generated by imposing a given boundary voltage. We have found an extension of the notion of admissibility to the case of possible presence of perfectly conducting and insulating inclusions. This also makes it possible to extend the results on uniqueness of the minimizers of the least gradient problem $F(u)=\int_Ωa|\nabla u|$ with $u|_{\partial Ω}=f$ to cases where $u$ has flat regions (is constant on open sets).

math.AP↗
arXiv2011

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

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $Ω$ from interior Current Density Imaging data obtainable using MRI measurements. We only require knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partialΩ$. As previously shown, the corresponding voltage potential u in $Ω$ is a minimizer of the weighted least gradient problem \[u=\hbox{argmin} \{\int_Ωa(x)|\nabla u|: u \in H^{1}(Ω), \ \ u|_{\partial Ω}=f\},\] with $a(x)= |J(x)|$. In this paper we present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(Ω)$ non-negative and $f\in H^{1/2}(\partial Ω)$. We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm.

math.AP↗
arXiv2011

Convergence of the alternating split Bregman algorithm in infinite-dimensional Hilbert spaces

We prove results on weak convergence for the alternating split Bregman algorithm in infinite dimensional Hilbert spaces. We also show convergence of an approximate split Bregman algorithm, where errors are allowed at each step of the computation. To be able to treat the infinite dimensional case, our proofs focus mostly on the dual problem. We rely on Svaiter's theorem on weak convergence of the Douglas-Rachford splitting algorithm and on the relation between the alternating split Bregman and Douglas-Rachford splitting algorithms discovered by Setzer. Our motivation for this study is to provide a convergent algorithm for weighted least gradient problems arising in the hybrid method of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current.

math.FA↗
arXiv2010

Isolated Singularities of Polyharmonic Inequalities

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-Δ^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these solutions $u$ satisfy a pointwise a priori bound as $x\to 0$.

math.AP↗
arXiv2009

Optimal weighted Hardy-Rellich inequalities on $H^2\cap H^{1}_{0}$

We give necessary and sufficient conditions on a pair of positive radial functions $V$ and $W$ on a ball $B$ of radius $R$ in $R^{n}$, $n \geq 1$, so that the following inequalities hold \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x) u^{2}dx+b\int_{\partial B}u^2 ds$ for all u $\in H^1(B)$,} \end{equation*} and \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|Δu |^{2}dx \geq \int_{B} W(x)|\nabla u|^{2}dx+b\int_{\partial B}|\nabla u|^2 ds$ for all u $\in H^2(B)$.} \end{equation*} Then we present various classes of optimal weighted Hardy-Rellich inequalities on $H^{2}\cap H^{1}_{0}$. The proofs are based on decomposition into spherical harmonics. These types inequalities are important in the study of fourth order elliptic equations with Navier boundary condition and systems of second order elliptic equations.

math.AP↗
arXiv2009

The singular extremal solutions of the bilaplacian with exponential nonlinearity

Consider the problem {ll} Δ^2 u= λe^{u} &\text{in} B u=\frac{\partial u}{\partial n}=0 &\text{on}\partial B, where $B$ is the unit ball in $\R^N$ and $λ$ is a parameter. Unlike the Gelfand problem the natural candidate $u=-4\ln(|x|)$, for the extremal solution, does not satisfy the boundary conditions and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. Dávila et al. in \cite{DDGM} used a computer assisted proof to show that the extremal solution is singular in dimensions $13\leq N\leq 31$. Here by an improved Hardy-Rellich inequality which follows from the recent result of Ghoussoub-Moradifam \cite{GM} we overcome this difficulty and give a simple mathematical proof to show the extremal solution is singular in dimensions $N\geq13$.

math.AP↗
arXiv2009

On the critical dimension of a fourth order elliptic problem with negative exponent

We study the regularity of the extremal solution of the semilinear biharmonic equation $βΔ^2 u-τΔu=\fracλ{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=Δu=0$ on $\partial B$, where $λ>0$ is a parameter, while $τ>0$, $β>0$ are fixed constants. It is known that there exists a $λ^{*}$ such that for $λ>λ^{*}$ there is no solution while for $λ<λ^{*}$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{*}$ is regular ($\sup_{B}u^{*}<1$) for $N\leq 8$ and $β, τ>0$ and it is singular ($\sup_{B}u^{*}=1$) for $N\geq 9$, $β>0$, and $τ>0$ with $\fracτβ$ small. Our proof for the singularity of extremal solutions in dimensions $N\geq 9$ is based on certain improved Hardy-Rellich inequalities.

math.AP↗
arXiv2009

The critical dimension for a 4th order problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\fλ{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u=\partial_νu=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0 \la^*$. Our main result asserts that the extremal solution $u_{λ^*}$ is regular $(\sup_B u_{λ^*} <1)$ provided $ N \le 8$ while $u_{λ^*} $ is singular ($\sup_B u_{λ^*} =1$) for $N \ge 9$, in which case $1-C_0|x|^{4/3}\leq u_{λ^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $ C_0:= (\frac{λ^*}{\overlineλ})^{1/3}$ and $ \barλ:= {8/9} (N-{2/3}) (N- {8/3})$.

math.AP↗