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Cristian E. Gutiérrez

Publications and source records attributed to Cristian E. Gutiérrez.

14 recordsLinked to original sources
arXiv2026

Refraction laws in spatio-temporal media

We study the time-dependent Maxwell system, formulated in the sense of distributions, for electromagnetic waves propagating through media with temporal and spatial material interfaces. Under explicit trace and regularity assumptions on the permittivity and permeability, we derive the jump conditions produced by temporal discontinuities and by subsequent spatial interfaces. These conditions are used to obtain generalized Snell laws for the wave vectors generated by temporal splitting, reflection, and transmission. We also derive the associated amplitude equations and organize them as a finite-dimensional linear system for a space-time slab geometry. Finally, we provide an explicit oblique-incidence example. Our approach does not impose a smooth-field ansatz and allows material parameters that need not be constant away from the interfaces.

math.AP↗
arXiv2026

Double metasurfaces and Optimal transport

This paper constructs metalenses that separate homogeneous media with different refractive indices, refracting one domain into another while conserving a prescribed energy distribution. Using optimal transport theory, we design singlet and doublet metalenses satisfying energy conservation by refraction, and obtain partial regularity of the optimal maps involved.

math.AP↗
arXiv2026

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↗
arXiv2025

Refraction laws in temporal media

We consider the time dependent Maxwell system in the sense of distributions in the context of temporal interfaces. Just as with spatial interfaces, electromagnetic waves at temporal interfaces scatter and create a transmitted and reflected wave. We provide a rigorous derivation of boundary conditions for the electric and magnetic fields at temporal interfaces with precise assumptions on the material parameters. In turn, we use this to obtain a general Snell's Law at such interfaces. From this, we obtain explicit formulas for the reflection and transmission coefficients. Unlike previous works, we do not make any simplifying ansatz on the solution to the Maxwell system, nor do we assume that the fields are smooth. We also consider material parameters which are not necessarily constant on either side of the temporal interface.

math.AP↗
arXiv2025

The Multi-marginal Monge Problem and an Application to Metasurfaces

This paper studies the multi-marginal Monge problem in the setting of compact metric spaces proving existence and uniqueness of solutions when the cost function is Lipschitz. We apply the results obtained to solve an optics problem involving metalenses, that is, we design a refracting-reflecting metalens that preserves given energy distributions.

math.AP↗
arXiv2024

Differentiability of monotone maps related to non-quadratic costs

The cost functions considered are $c(x,y)=h(x-y)$, with $h\in C^2(R^n)$, homogeneous of degree $p\geq 2$, with positive definite Hessian in the unit sphere. We consider monotone maps $T$ concerning that cost and establish local $L^\infty$-estimates of $T$ minus affine functions, which are applied to obtain differentiability properties of $T$ a.e. It is also shown that these maps are related to maps of bounded deformation, and further, differentiability and Hölder continuity properties are derived.

math.AP↗
arXiv2021

Metasurfaces and Optimal transport

This paper provides a theoretical and numerical approach to show existence, uniqueness, and the numerical determination of metalenses refracting radiation with energy patterns. The theoretical part uses ideas from optimal transport and for the numerical solution we study and implement a damped Newton algorithm to solve the semi discrete problem. A detailed analysis is carried out to solve the near field one source refraction problem and extensions to the far field are also mentioned.

math.AP↗
arXiv2020

Sobolev inequalities with jointly concave weights on convex cones

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form \[\left(\int_E |u(x)|^q\,ω(x) \,dx\right)^{1/q}\leq K_0\,\left(\int_E |\nabla u(x)|^p\,σ(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}\] where $p\geq 1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E\subseteq \mathbb R^n$ is an open convex cone, and $ω,σ:E\to (0,\infty)$ are two homogeneous weights verifying a general concavity-type structural condition. The constant $K_0= K_0(n, p, q, ω, σ) >0$ is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that $K_0$ is optimal in (WSI) if and only if $ω$ and $σ$ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to PDEs are also provided.

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↗
arXiv2019

On the Numerical Solution of the Near Field Refractor Problem

A numerical scheme is presented to solve the one source near field refractor problem to arbitrary precision and it is proved that the scheme terminates in a finite number of iterations. The convergence of the algorithm depends upon proving appropriate Lipschitz estimates for the refractor measure. The algorithm is presented in general terms and has independent interest.

math.NA↗
arXiv2017

Harnack's inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack's inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach from [DGL08] to obtain Harnack's inequality.

math.AP↗
arXiv2012

Interior second derivative estimates for solutions to the linearized Monge--Ampère equation

Let $Ω\subset \R^n$ be a bounded convex domain and $ϕ\in C(\barΩ)$ be a convex function such that $ϕ$ is sufficiently smooth on $\partialΩ$ and the Monge--Ampère measure $\det D^2ϕ$ is bounded away from zero and infinity in $Ω$. The corresponding linearized Monge--Ampère equation is \[ \trace(ΦD^2 u) =f, \] where $Φ:= \det D^2 ϕ~ (D^2ϕ)^{-1}$ is the matrix of cofactors of $D^2ϕ$. We prove a conjecture in \cite{GT} about the relationship between $L^p$ estimates for $D^2 u$ and the closeness between $\det D^2ϕ$ and one. As a consequence, we obtain interior $W^{2,p}$ estimates for solutions to such equation whenever the measure $\det D^2ϕ$ is given by a continuous density and the function $f$ belongs to $L^q(Ω)$ for some $q> \max{\{p,n\}}$.

math.AP↗