arXiv subjects

Mariana Smit Vega Garcia

Publications and source records attributed to Mariana Smit Vega Garcia.

17 recordsLinked to original sources

arXiv2026

Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory

This paper continues the program that was initiated in \cite{Dav18} and continued in \cite{DSVG24}, where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. The articles \cite{Dav18} and \cite{DSVG24} address the constant-coefficient and variable-coefficient settings, respectively. Here, we focus on fractional operators. As shown in \cite{CS07}, \cite{NS16}, \cite{ST17}, fractional operators may be associated with certain degenerate operators via extension problems, so we study the corresponding class of degenerate operators. Our high-dimensional limiting technique is demonstrated through new proofs of three theorems for degenerate parabolic equations. Specifically, we establish the monotonicity of Almgren-type, Weiss-type, and Alt-Caffarelli-Friedman-type functionals in the degenerate parabolic setting. Each new parabolic proof in this article is based on a (new) related elliptic theorem and a careful limiting argument that is reminiscent of those from \cite{Dav18} and \cite{DSVG24}. Our proof of the degenerate parabolic Weiss-type monotonicity formula additionally uses an epiperimetric inequality for weakly $a$-harmonic functions, which we also prove. To the best of our knowledge, our Alt-Caffarelli-Friedman monotonicity result is new.

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arXiv2026

An Almgren monotonicity formula for discrete harmonic functions

The celebrated Almgren monotonicity formula for harmonic functions $u:\mathbb{R}^n \rightarrow \mathbb{R}$ says that its $L^2-$energy concentrated on a sphere of radius $r$, when measured in a suitable sense, is non-decreasing: if $u$ oscillates at a certain scale, it has even larger oscillations at a larger scale. We prove a discrete analogue of the Almgren monotonicity formula for harmonic functions on infinite combinatorial graphs $G=(V,E)$. Some applications are discussed.

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arXiv2026

Monotonicity formulas for harmonic functions on the infinite regular tree

We continue the program initiated in \cite{SVGS}. In this paper, we focus on the infinite $d-$regular tree, and prove the monotonicity of a weighted Dirichlet energy, a Weiss-type monotonicity formula, and a generalization of the Almgren monotonicity formula of \cite{SVGS} for $p\ge 1$. We also compute examples in the infinite $2-$ and $3-$regular trees.

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arXiv2025

Metrics on Signed Permutations with the Same Peak Set

Let $S^B_n$ be the Coxeter group of type B. We denote the set of indices where $σ\in S^B_n$ has a peak as $Peak(σ)$ and let $P^{B}(S;n)=\{σ\in S^{B}_n~|~ Peak(σ)=S\}$. In \cite{metrics}, Diaz-Lopez, Haymaker, Keough, Park and White considered metrics for unsigned permutations with the same peak set. In this paper, we generalize their result by studying Hamming, $l_{\infty}$, and the word metrics on $P^{B}(S;n)$ for all $S$. We also determine the minimum and maximum possible values that these metrics can achieve in these subsets of $S^B_n$.

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arXiv2024

Two-phase almost minimizers for a fractional free boundary problem

In this paper, we study almost minimizers to a fractional Alt-Caffarelli-Friedman type functional. Our main results concern the optimal $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries $F^+(u)=\partial\{u(\cdot,0)>0\}$ and $F^-(u)=\partial\{u(\cdot,0)<0\}$ cannot touch, that is, $F^+(u)\cap F^-(u)=\emptyset$. Lastly, we prove a flatness implies $C^{1,γ}$ result for the free boundary.

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arXiv2023

Mesas of Stirling permutations

Given a Stirling permutation w, we introduce the mesa set of w as the natural generalization of the pinnacle set of a permutation. Our main results characterize admissible mesa sets and give closed enumerative formulas in terms of rational Catalan numbers by providing an explicit bijection between mesa sets and rational Dyck paths.

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arXiv2023

Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory

This paper continues the study initiated in [B. Davey, Parabolic theory as a high-dimensional limit of elliptic theory, Arch Rational Mech Anal 228 (2018)], where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. In this article, we extend these ideas to the variable-coefficient setting. This generalized technique is demonstrated through new proofs of three important theorems for variable-coefficient heat operators, one of which establishes a result that is, to the best of our knowledge, also new. Specifically, we give new proofs of $L^2 \to L^2$ Carleman estimates and the monotonicity of Almgren-type frequency functions, and we prove a new monotonicity of Alt-Caffarelli-Friedman-type functions. The proofs in this article rely only on their related elliptic theorems and a limiting argument. That is, each parabolic theorem is proved by taking a high-dimensional limit of a related elliptic result.

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arXiv2023

Pinnacle sets of signed permutations

Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type $A$ symmetric group have been widely studied. In this article we define the pinnacle set of signed permutations of types $B$ and $D$. We give a closed formula for the number of type $B$/$D$ admissible pinnacle sets and answer several other related enumerative questions.

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arXiv2022

Branch Points for (Almost-)Minimizers of Two-Phase Free Boundary Problems

We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt- Caffarelli-Friedman type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis- Spolaor-Velichkov on the structure of branch points in the free boundary of stationary solutions.

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arXiv2020

Regularity for almost-minimizers of variable coefficient Bernoulli-type functionals

In [David-Toro 15] and [David-Engelstein-Toro 19], (some of) the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli in [Alt-Caffarelli 81] and Alt, Caffarelli and Friedman in [Alt-Caffarelli-Friedman 84]. In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to [DT15, DET19. AC 81] and [ACF84] which deal only with the "Laplacian" setting). We prove Lipschitz regularity up to, and across, the free boundary, generalizing the results of [David-Toro 15] to the variable coefficient setting.

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arXiv2020

Almost minimizers for the thin obstacle problem with variable coefficients

We study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.

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arXiv2019

On a Bernoulli-type overdetermined free boundary problem

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation for a fixed $1 0$ is a given constant, then there exists a unique convex domain $Ω$ with $K\subset Ω$ and a function $u$ which is $\mathcal{A}$-harmonic in $Ω\setminus K$, has continuous boundary values $1$ on $\partial K$ and $0$ on $\partialΩ$, such that $|\nabla u|=c$ on $\partial Ω$. Moreover, $\partialΩ$ is $C^{1,γ}$ for some $γ>0$, and it is smooth provided $\mathcal{A}$ is smooth in $\mathbb{R}^n \setminus \{0\}$. We also show that the super level sets $\{u>t\}$ are convex for $t\in (0,1)$.

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arXiv2018

The fractional unstable obstacle problem

We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation \[ (-Δ)^s u = χ_{\{u>c\}} \] for $0 c\}$. We study the behavior of the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that when $s\leq 1/2$ certain symmetric solutions are stable; however, when $s>1/2$ these solutions are not stable and therefore not minimizers of the corresponding functional.

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arXiv2016

Higher regularity of the free boundary in the parabolic Signorini problem

We show that the quotient of two caloric functions which vanish on a portion of an $H^{k+ α}$ regular slit is $H^{k+ α}$ at the slit, for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+α}$ if the slit is assumed to be space-time $C^{1, α}$ regular. This can be thought of as a parabolic analogue of a recent important result in [DSS14a], whose ideas inspired us. As an application, we show that the free boundary near a regular point of the parabolic thin obstacle problem studied in [DGPT] with zero obstacle is $C^{\infty}$ regular in space and time.

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arXiv2016

Singularities in axisymmetric free boundaries for ElectroHydroDynamic equations

We consider singularities in the ElectroHydroDynamic equations. In a regime where we are allowed to neglect surface tension, and assuming that the free surface is given by an injective curve and that either the fluid velocity or the electric field satisfies a certain non-degeneracy condition, we prove that either the fluid region or the gas region is asymptotically a cusp. Our proofs depend on a combination of monotonicity formulas and a non-vanishing result by Caffarelli and Friedman. As a by-product of our analysis we also obtain a special solution with convex conical air-phase which we believe to be new.

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arXiv2015

Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift

We establish the $C^{1+γ}$-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research, where two of us established the optimal interior regularity of solutions.

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arXiv2015

An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients

In this paper we establish the $C^{1,β}$ regularity of the regular part of the free boundary in the Signorini problem for elliptic operators with variable Lipschitz coefficients. This work is a continuation of the recent paper [GSVG14], where two of us established the interior optimal regularity of the solution. Two of the central results of the present work are a new monotonicity formula and a new epiperimetric inequality.

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