arXiv subjects

Amir Moradifam

Publications and source records attributed to Amir Moradifam.

At least 19 recordsLinked to original sources

arXiv2026

Symmetry and Rigidity Results for the Mean Field Equation and Hawking Mass on $\mathbb{S}^2$

In this paper, we establish symmetry results for solutions of the mean field equation \[ \fracα{2} Δu + e^u - 1 = 0 \] on $ \mathbb{S}^2 $ for $\frac{1}{3}\leq α\leq 1$, under a geometric condition $(\mathcal{H})$ introduced below. The proofs utilize the Sphere Covering Inequality and incorporate topological arguments on $ \mathbb{S}^2 $. These results are further applied to demonstrate a rigidity property of the Hawking mass for stable constant mean curvature (CMC) spheres, addressing a question posed by Robert Bartnik in 2002. Our results unify and extend previous rigidity results obtained under symmetry or hemispherical balance assumptions, and apply beyond the nearly spherical setting.

math.AP↗

arXiv2026

What Remains Human in Mathematics in the Age of AI

Mathematics is often valued for its role as the language of science, but it is equally important as a human practice that teaches us how to think and cultivates creativity, problem-solving skills, attention, and judgment. The rapid rise of modern AI systems, which can solve many standard problems and produce fluent mathematical explanations, forces a reconsideration of the nature of mathematical understanding and how it is developed. This article reflects on the impact of AI across three intertwined domains: mathematics education, mathematical research, and the broader role of mathematics as a way of learning to think. In education, AI changes how students interact with mathematical material, making high-quality assistance widely available while weakening the traditional link between assigned work and demonstrated understanding. The challenge concerns not only academic integrity but also the risks of cognitive offloading and the illusion of explanatory depth. In research, AI can accelerate exploration in various ways, including generating conjectures and examples, performing symbolic manipulation, suggesting proofs, and assisting in the communication of ideas. It has also begun to contribute to discoveries on problems that had resisted human efforts. Yet its limitations remain serious, especially in long arguments requiring global structure, careful verification, and the synthesis of multiple ideas. The central claim is that in an era when routine tasks become easier to automate, the most human aspects of mathematics, including conceptual insight, problem framing, genuine understanding, and creative judgment, become more visible and more valuable.

math.HO↗

arXiv2026

Simultaneous Recovery of the Initial Source and Sound Speed for the Wave Equation under a Constitutive Constraint

We study the simultaneous recovery of the initial source and sound speed for the scalar wave equation from a single boundary measurement. Although the recovery of either parameter separately is well understood under suitable geometric hypotheses, simultaneous recovery remains open in general, and stable recovery is further obstructed by the inherent instability of the linearized problem. We show that both uniqueness and stability can be obtained when the two unknowns are coupled through a prescribed constitutive relation arising from a common underlying material. Under a quantitative nondegeneracy condition, motivated by calibrated material regimes, the coupled inverse problem reduces to a single inverse source problem. Applying the microlocal and Carleman framework of Stefanov and Uhlmann to this reduced problem, we establish global uniqueness and Lipschitz stability under geometric conditions involving strictly convex foliations and geodesic visibility. We also obtain partial-data results, proving local uniqueness and Lipschitz stability in compact visible regions where the constitutive nondegeneracy condition holds.

math.AP↗

arXiv2026

Quantitative Stability of Generalized $p$-Area Minimizing Surfaces

We study the stability of $p$-area minimizing surfaces in the Heisenberg group under perturbations of the weight function and the drift vector field in generalized least gradient problems of the form \[ \inf_{w\in BV_0(Ω)} \int_Ω\left(a(x)|Dw+F(x)|+H(x)w\right)\,dx. \] Owing to the lack of strict convexity, establishing stability of minimizers is challenging. We derive quantitative stability estimates for minimizers. In particular, under suitable nondegeneracy and geometric assumptions, we obtain $L^1$ and $W^{1,1}$ stability estimates with respect to perturbations of the weight function $a$ and the drift vector field $F$. We further establish unified quantitative stability estimates under simultaneous perturbations of all principal parameters, namely $a$, $F$, and $H$. Numerical simulations illustrating the stability theory are also presented.

math.AP↗

arXiv2026

Stability of p-area minimizing surfaces in the Heisenberg group

We study the stability of minimizers of weighted $p$-area functionals associated with prescribed $p$-mean curvature surfaces in the Heisenberg group. While existence and uniqueness results are well established, quantitative stability with respect to perturbations of the mean curvature $H$ remains largely unexplored in the nonzero-$H$ regime. Using a Rockafellar--Fenchel duality framework, we identify a unique underlying vector field associated with each minimizer and prove its stability under perturbations of $H$. This yields quantitative control of the direction field of the horizontal gradient. Building on this structure, we establish $L^1$ stability of admissible minimizers under natural geometric assumptions on level sets. In dimensions two and three, we also derive $W^{1,1}$ stability estimates under additional regularity and structural hypotheses, with explicit rates in terms of $\|H-\tilde H\|_{L^\infty}$. Our results provide the first quantitative stability theory for $p$-area minimizing graphs with prescribed nonzero $p$-mean curvature, even in the unweighted case. Numerical simulations are included to illustrate the robustness of the theoretical results.

math.AP↗

arXiv2025

A Generalization of the Sphere Covering Inequality

The Sphere Covering Inequality was introduced in \cite{GM} (\emph{Invent. Math.}, 2018) as a sharp geometric inequality that provides a lower bound for the total area of two distinct surfaces of Gaussian curvature 1. These surfaces are assumed to be conformal to the Euclidean unit disk and share the same conformal factor along the boundary. In this paper, we establish a quantitative generalization that relaxes the boundary matching condition by allowing the conformal factors to differ by a constant $ c \ge 0 $ on the boundary. This refinement reveals a new stability-type structure underlying the inequality. Our results show that the Sphere Covering Inequality is stable with respect to perturbations in the boundary data and provide a precise quantitative description of how the total-area bound varies under such perturbations. The generalized inequality provides new analytic and geometric tools for the study of elliptic equations with exponential nonlinearities, conformal geometry, and related problems in mathematical physics.

math.AP↗

arXiv2024

Existence and structure of solutions for general $P$-area minimizing surface

We study existence and structure of solutions to the Dirichlet and Neumann boundary problems associated with minimizers of the functional $I(u)=\int_Ω (ϕ(x, D u + F)+Hu) \, dx$, where $ϕ(x, ξ)$, among other properties, is convex and homogeneous of degree $1$ with respect to $ξ$. We show that there exists an underlying vector field $N$ that characterizes the existence and structure of all minimizers. We also investigate existence of solutions under the barrier condition on $\partial Ω$. The results in this paper generalize and unify many results in the literature about existence of minimizers of least gradient problems and $P-$area minimizing surfaces.

math.AP↗

arXiv2021

Existence and structure of P-area minimizing surfaces in the Heisenberg group

We study existence and structure of $P-$area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field $N$ that characterized existence and structure of $P$-area minimizing surfaces. This vector field exists even if there is no $P$-area minimizing surface satisfying the prescribed boundary conditions. We prove that if $\partial Ω$ satisfies a so called Barrier condition, it is sufficient to guarantee existence of such surfaces. Our approach is completely different from previous methods in the literature and makes major progress in understanding existence of $P$-area minimizing surfaces.

math.DG↗

arXiv2020

Stability of Current Density Impedance Imaging

We study stability of reconstruction in current density impedance imaging (CDII), that is, the inverse problem of recovering the conductivity of a body from the measurement of the magnitude of the current density vector field in the interior of the object. Our results show that CDII is stable with respect to errors in interior measurements of the current density vector field, and confirm the stability of reconstruction which was previously observed in numerical simulations, and was long believed to be the case.

math.AP↗

arXiv2019

Determining both the source of a wave and its speed in a medium from boundary measurements

We study the inverse problem of determining both the source of a wave and its speed inside a medium from measurements of the solution of the wave equation on the boundary. This problem arises in photoacoustic and thermoacoustic tomography, and has important applications in medical imaging. We prove that if the solutions of the wave equation with the source and sound speed $(f_1,c_1)$ and $(f_2,c_2)$ agree on the boundary of a bounded region $Ω$, then \[ \int_Ω(c_2^{-2}-c_1^{-2})φdy=0,\] for every harmonic function $φ\in C(\barΩ)$, which holds without any knowledge of the source. We also show that if the wave speed $c$ is known and only assumed to be bounded then, under a natural admissibility assumption, the source of the wave can be uniquely determined from boundary measurements.

math.AP↗

arXiv2019

Remarks on a mean field equation on $\mathbb{S}^{2}$

In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -Δ_{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.

math.AP↗

arXiv2019

General Least Gradient Problems with Obstacle

We study existence, structure, uniqueness and regularity of solutions of the obstacle problem \begin{equation*} \inf_{u\in BV_f(Ω)}\int_{\mathbb{R}^n}ϕ(x,Du), \end{equation*} where $BV_f(Ω)=\{u\in BV(Ω): u\geq ψ\text{ in }Ω\text{ and } u|_{\partial Ω}=f|_{\partial Ω}\}$, $f \in W^{1,1}_0(\mathbb{R}^n)$, $ψ$ is the obstacle, and $ϕ(x,ξ)$ is a convex, continuous and homogeneous function of degree one with respect to the $ξ$ variable. We show that every minimizer of this problem is also a minimizer of the least gradient problem \[\inf_{u\in \mathcal{A}_f(Ω)}\int_{\mathbb{R}^n}ϕ(x,Du),\] where $\mathcal{A}_f(Ω)=\{u\in BV(Ω): u\geq ψ, \text{ and } u=f \text{ in }Ω^c\}$. Moreover, there exists a vector field $T$ with $\nabla \cdot T \leq 0$ in $Ω$ which determines the structure of all minimizers of these two problems, and $T$ is divergence free on $\{x\in Ω: u(x)>ψ(x)\}$ for any minimizer $u$. We also present uniqueness and regularity results that are based on maximum principles for minimal surfaces. Since minimizers of the least gradient problems with obstacle do not hit small enough obstacles, the results presented in this paper extend several results in the literature about least gradient problems without obstacle.

math.AP↗

arXiv2018

The sphere covering inequality and its dual

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension two. The approach in this paper extends, improves, and unifies several inequalities about solutions of elliptic equations with exponential nonlinearities.

math.AP↗

arXiv2018

Electrical Networks with Prescribed Current and Applications to Random Walks on Graphs

We study the inverse problem of determining the conductivity matrix of an electrical network from the prescribed knowledge of the magnitude of the induced current along the edges coupled with the imposed voltage or injected current on the boundary nodes. This problem leads to a weighted $l^1$ minimization problem for the corresponding voltage potential. We also investigate the problem of determining the transition probabilities of random walks on graphs from the prescribed net number of times the walker passes along the edges of the graph. We also show that a mass preserving flow $J=(J_{i.j})$ on a network can be uniquely recovered from the knowledge of $|J|=(|J_{i,j}|)$ and the flux of the flow on the boundary nodes, where $J_{i,j}$ is the flow from node $i$ to node $j$ and $J_{i,j}=-J_{j,i}$. Convergent numerical algorithms for solving such problems are also presented.

math.AP↗

arXiv2018

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

We derive a singular version of the Sphere Covering Inequality which was recently introduced in [42], suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.

math.AP↗

arXiv2017

Uniqueness of solutions of mean field equations in $\R^2$

In this paper, we prove uniqueness of solutions of mean field equations with general boundary conditions for the critical and subcritical total mass regime, extending the earlier results for null Dirichlet boundary condition. The proof is based on new Bol's inequalities for weak radial solutions obtained from rearrangement of the solutions.

math.AP↗

arXiv2017

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

We prove symmetry and uniqueness results for three classes of Liouville-type problems arising in geometry and mathematical physics: asymmetric Sinh-Gordon equation, cosmic string equation and Toda system, under certain assumptions on the mass associated to these problems. The argument is in the spirit of the Sphere Covering Inequality which for the first time is used in treating different exponential nonlinearities and systems.

math.AP↗

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↗