arXiv subjects

Ahmad Sabra

Publications and source records attributed to Ahmad Sabra.

13 recordsLinked to original sources

arXiv2026

Functional Ordinary Differential Equations and the Design of Dichromatic Lenses

This paper establishes an existence and uniqueness theorem for a class of functional ordinary differential equations under assumptions weaker than those found in the literature. We then consider the problem of designing a lens that refracts rays of two distinct frequencies emitted from a point source into a common collimated beam. We derive a system of functional ordinary differential equations whose solvability characterizes the existence of such lenses. Using the existence theorem, we prove that such a lens exists in two dimensions. We then extend the corresponding result to three dimensions.

math.AP↗

arXiv2025

Fractional Infinity Laplacian with Obstacle

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, \Omega\\ u=g &\qquad on\, \partial \Omega\end{cases},$$ with $$L[u](x)=\sup_{y\in \Omega,\,y\neq x}\dfrac{u(y)-u(x)}{|y-x|^{\alpha}}+\inf_{y\in \Omega,\,y\neq x} \dfrac{u(y)-u(x)}{|y-x|^\alpha},\qquad 0<\alpha<1.$$ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,\beta}(\partial\Omega)$ for some $0<\beta<\alpha$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $\beta-$H\"olderian on $\overline{\Omega}$. Our proof is based on an approximation of $f$ by an appropriate sequence of functions $f_\varepsilon$ where we prove using Perron's method the existence of solutions $u_\varepsilon$, for every $\varepsilon>0$. Then, we show some uniform H\"older estimates on $u_\varepsilon$ that guarantee that $u_\varepsilon \rightarrow u$ where this limit function $u$ turns out to be a solution to our obstacle problem.

math.AP↗

arXiv2024

The non-convex planar Least Gradient Problem

We study the least gradient problem in bounded regions with Lipschitz boundary in the plane. We provide a set of conditions for the existence of solutions in non-convex simply connected regions. We assume the boundary data is continuous and in the space of functions of bounded variation, and we are interested in solutions that satisfy the boundary conditions in the trace sense. Our method relies on the equivalence of the least gradient problem and the Beckman problem which allows us to use the tools of the optimal transportation theory.

math.AP↗

arXiv2023

Generalized Snell's law and Maxwell equations

This paper examines the Maxwell system of electrodynamics within the framework of distributions. A primary objective is to establish boundary conditions for fields at interfaces when the charge and current densities are measures localized on the interface. From this analysis, the paper presents a derivation of the generalized Snell's law, along with formulas for the amplitudes of the reflected and transmitted waves in terms of the incident amplitude.

math.AP↗

arXiv2020

The planar Least Gradient problem in convex domains: the discontinuous case

We study the two dimensional least gradient problem in convex polygonal sets in the plane, $Ω$. We show the existence of solutions when the boundary data $f$ are attained in the trace sense. The main difficulty here is a possible discontinuity of $f$. Moreover, due to the lack of strict convexity of $Ω$, the classical results are not applicable. We state the admissibility conditions on the boundary datum $f$, that are sufficient for establishing an existence result. One of them is that $f\in BV(\partialΩ)$. The solutions are constructed by a limiting process, which uses solutions to known problems

math.AP↗

arXiv2019

The planar Least Gradient problem in convex domains, the case of continuous datum

We study the two dimensional least gradient problem in a convex polygonal set in the plane. We show existence of solutions when the boundary data are attained in the trace sense. Due to the lack of strict convexity, the classical results are not applicable. We state the admissibility conditions on the continuous boundary datum $f$ that are sufficient for establishing an existence and uniqueness result. The solutions are constructed by a limiting process, which uses the well-known geometry of superlevel sets of least gradient functions.

math.AP↗

arXiv2019

Chromatic Aberration in Metalenses

This paper provides a mathematical approach to study chromatic aberration in metalenses. It is shown that radiation of a given wavelength is refracted according to a generalized Snell's law which together with the notion of envelope yields the existence of phase discontinuities. This is then used to establish a quantitative measure of dispersion in metalenses concluding that in the visible spectrum it has the same order of magnitude as for standard lenses.

physics.class-ph↗

arXiv2018

Refractor surfaces determined by near-field data

In this paper we study the near-field refractor problem with point source at the origin and prescribed target on the given receiver surface $Σ$. This nonvariational problem can be studied in the framework of prescribed Jacobian equations. We construct the corresponding generating function and show that the Aleksandrov and the Brenier type solutions are equivalent. Our main result establishes local smoothness of Aleksandrov's solutions when the data is smooth and when the medium containing the source has smaller refractive index than the medium containing the target. This is done by deriving the Monge-Ampere type equation that smooth solutions satisfy and establishing the validity of the MTW condition for a large class of receiver surfaces, which in turn implies the local $C^2 $ regularity of the refactor.

math.AP↗

arXiv2018

On the existence of dichromatic single element lenses

Due to dispersion, light with different wavelengths, or colors, is refracted at different angles. Our purpose is to determine when is it possible to design a lens made of a single homogeneous material so that it refracts light superposition of two colors into a desired fixed final direction. Two problems are considered: one is when light emanates in a parallel beam and the other is when light emanates from a point source. For the first problem, and when the direction of the parallel beam is different from the final desired direction, we show that such a lens does not exist; otherwise we prove the solution is trivial, i.e., the lens is confined between two parallel planes. For the second problem we prove that is impossible to design such a lens when the desired final direction is not among the set of incident directions. Otherwise, solving an appropriate system of functional equations we show that a local solution exists.

math.AP↗

arXiv2017

The planar Least Gradient problem in convex domains

We study the two dimensional least gradient problem in a convex, but not necessary strictly convex region. We look for solutions in the space of $BV$ functions satisfying the boundary data $f$ in trace sense. We assume that $f$ is in $BV$ too. We state admissibility conditions on the trace and on the domain that are sufficient for existence of solutions.

math.AP↗

arXiv2016

Special cases of the planar least gradient problem

We study two special cases of the planar least gradient problem. In the first one, the boundary conditions are imposed on a part of the strictly convex domain. In the second case, we impose the Dirichlet data on the boundary of a rectangle, an example of convex but not strictly convex domain. We show the existence of solutions and study their properties for particular cases of data.

math.AP↗

arXiv2016

Aspherical lens design and imaging

We design freeform lenses refracting an arbitrarily given incident field into a given fixed direction. In the near field case, we study the existence of lenses refracting a given bright object into a predefined image. We also analyze the existence of systems of mirrors that solve the near field and the far field problems for reflection.

math.AP↗

arXiv2013

The reflector problem and the inverse square law

We introduce a model to design reflectors that take into account the inverse square law for radiation. We prove existence of solutions, both in the near and far field cases, when the input and output energies are prescribed.

math.AP↗