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Connor Mooney

Publications and source records attributed to Connor Mooney.

At least 37 records · Page 2Linked to original sources
arXiv2023

VERITAS discovery of very high energy gamma-ray emission from S3 1227+25 and multiwavelength observations

We report the detection of very high energy gamma-ray emission from the blazar S3 1227+25 (VER J1230+253) with the Very Energetic Radiation Imaging Telescope Array System (VERITAS). VERITAS observations of the source were triggered by the detection of a hard-spectrum GeV flare on May 15, 2015 with the Fermi-Large Area Telescope (LAT). A combined five-hour VERITAS exposure on May 16th and May 18th resulted in a strong 13$σ$ detection with a differential photon spectral index, $Γ$ = 3.8 $\pm$ 0.4, and a flux level at 9% of the Crab Nebula above 120 GeV. This also triggered target of opportunity observations with Swift, optical photometry, polarimetry and radio measurements, also presented in this work, in addition to the VERITAS and Fermi-LAT data. A temporal analysis of the gamma-ray flux during this period finds evidence of a shortest variability timescale of $τ_{obs}$ = 6.2 $\pm$ 0.9 hours, indicating emission from compact regions within the jet, and the combined gamma-ray spectrum shows no strong evidence of a spectral cut-off. An investigation into correlations between the multiwavelength observations found evidence of optical and gamma-ray correlations, suggesting a single-zone model of emission. Finally, the multiwavelength spectral energy distribution is well described by a simple one-zone leptonic synchrotron self-Compton radiation model.

astro-ph.HE↗
arXiv2022

Gradient estimates for the Lagrangian mean curvature equation with critical and supercritical phase

In this paper, we prove interior gradient estimates for the Lagrangian mean curvature equation, if the Lagrangian phase is critical and supercritical and $C^{2}$. Combined with the a priori interior Hessian estimates proved in [Bha21, Bha22], this solves the Dirichlet boundary value problem for the critical and supercritical Lagrangian mean curvature equation with $C^0$ boundary data. We also provide a uniform gradient estimate for lower regularity phases that satisfy certain additional hypotheses.

math.AP↗
arXiv2022

Singular structures in solutions to the Monge-Ampère equation with point masses

We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration.

math.AP↗
arXiv2022

Hilbert's $19^{\text{th}}$ problem revisited

In this survey article we revisit Hilbert's $19^{\text{th}}$ problem concerning the regularity of minimizers of variational integrals. We first discuss the classical theory (that is, the statement and resolution of Hilbert's problem in all dimensions). We then discuss recent results concerning the regularity of minimizers of degenerate convex functionals. Finally, we discuss some open problems. Exercises are included for the benefit of researchers who are entering the subject.

math.AP↗
arXiv2021

A proof by foliation that Lawson's cones are $A_Φ$-minimizing

We give a proof by foliation that the cones over $\mathbb{S}^k \times \mathbb{S}^l$ minimize parametric elliptic functionals for each $k,\,l \geq 1$. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.

math.AP↗
arXiv2020

Strict $2$-convexity of convex solutions to the quadratic Hessian equation

We prove that convex viscosity solutions to the quadratic Hessian inequality $σ_2(D^2u) \geq 1$ are strictly $2$-convex. As a consequence we obtain short proofs of smoothness and interior $C^2$ estimates for convex viscosity solutions to $σ_2(D^2u) = 1$, which were proven using different methods in recent works of Guan-Qiu \cite{GQ}, McGonagle-Song-Yuan \cite{MSY} and Shankar-Yuan \cite{SY2}.

math.AP↗
arXiv2020

Solutions to the Monge-Ampère equation with polyhedral and Y-shaped singularities

We construct convex functions on $\mathbb{R}^3$ and $\mathbb{R}^4$ that are smooth solutions to the Monge-Ampère equation $\det D^2u = 1$ away from compact one-dimensional singular sets, which can be Y-shaped or form the edges of a convex polytope. The examples solve the equation in the Alexandrov sense away from finitely many points. Our approach is based on solving an obstacle problem where the graph of the obstacle is a convex polytope.

math.AP↗
arXiv2019

Minimizers of convex functionals with small degeneracy set

We study the question whether Lipschitz minimizers of $\int F(\nabla u)\,dx$ in $\mathbb{R}^n$ are $C^1$ when $F$ is strictly convex. Building on work of De Silva-Savin, we confirm the $C^1$ regularity when $D^2F$ is positive and bounded away from finitely many points that lie in a $2$-plane. We then construct a counterexample in $\mathbb{R}^4$, where $F$ is strictly convex but $D^2F$ degenerates on the intersection of a Simons cone with $S^3$. Finally we highlight a connection between the case $n = 3$ and a result of Alexandrov in classical differential geometry, and we make a conjecture about this case.

math.AP↗
arXiv2019

Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces

Motivated by the inequality $\|f+g\|_{2}^{2} \leq \|f\|_{2}^{2}+2\|fg\|_{1}+\|g\|^{2}_{2}$, Carbery (2006) raised the question what is the "right" analogue of this estimate in $L^{p}$ for $p \neq 2$. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an $L^{p}$ version of this inequality by providing upper bounds for $\|f+g\|_{p}^{p}$ in terms of the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and $\|fg\|_{p/2}^{p/2}$ when $p \in(0,1] \cup [2,\infty)$, and lower bounds when $p \in (-\infty,0) \cup (1,2)$, thereby proving (and improving) the suggested possible inequalities of Carbery. We continue investigation in this direction by refining the estimates of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for $\|f + g\|_p^p$ also when $p \in (-\infty,0) \cup (1,2)$ and lower bounds when $p \in (0,1] \cup [2,\infty)$. For $p \in [1,2]$ we extend our upper bounds to any finite number of functions. In addition, we show that all our upper and lower bounds of $\|f+g\|_{p}^{p}$ for $p \in \mathbb{R}$, $p\neq 0$, are the best possible in terms of the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and $\|fg\|_{p/2}^{p/2}$, and we characterize the equality cases.

math.AP↗
arXiv2019

A proof of the Krylov-Safonov theorem without localization

The Krylov-Safonov theorem says that solutions to non-divergence uniformly elliptic equations with rough coefficients are Hölder continuous. The proof combines a basic measure estimate with delicate localization and covering arguments. Here we give a "global" proof based on convex analysis that avoids the localization and covering arguments. As an application of the technique we prove a $W^{2,\,ε}$ estimate where $ε$ decays with the ellipticity ratio of the coefficients at a rate that improves previous results, and is optimal in two dimensions.

math.AP↗
arXiv2018

The Monge-Ampère Equation

In this survey article we discuss the interior and boundary regularity of Alexandrov solutions to $\det D^2u = 1$. We include some topics which it seems were not recently revisited in similar articles, including Calabi's interior $C^3$ estimate, and the approaches of Cheng-Yau and Lions to obtain classical solutions to the Dirichlet problem. The survey grew from two mini-courses given by the author in May 2018. One was for "Advanced Lectures in Nonlinear Analysis" at l'Università degli Studi di Torino, and the other for the Oxford PDE CDT.

math.AP↗
arXiv2018

Regularity results for the equation $u_{11}u_{22} = 1$

We study the equation $u_{11}u_{22} = 1$ in $\mathbb{R}^2$. Our results include an interior $C^2$ estimate, classical solvability of the Dirichlet problem, and the existence of non-quadratic entire solutions. We also construct global singular solutions to the analogous equation in higher dimensions. At the end we state some open questions.

math.AP↗
arXiv2017

An obstacle problem for conical deformations of thin elastic sheets

A developable cone ("d-cone") is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance $ε$. Starting from a nonlinear model depending on the thickness $h > 0$ of the sheet, we prove a $Γ$-convergence result as $h \rightarrow 0$ to a fourth-order obstacle problem for curves in $\mathbb{S}^2$. We then describe the exact shape of minimizers of the limit problem when $ε$ is small. In particular, we rigorously justify previous results in the physics literature.

math.AP↗
arXiv2017

Dimension of the minimum set for the real and complex Monge-Ampère equations in critical Sobolev spaces

We prove that the zero set of a nonnegative plurisubharmonic function that solves $\det (\partial \overline{\partial} u) \geq 1$ in $\mathbb{C}^n$ and is in $W^{2, \frac{n(n-k)}{k}}$ contains no analytic sub-variety of dimension $k$ or larger. Along the way we prove an analogous result for the real Monge-Ampère equation, which is also new. These results are sharp in view of well-known examples of Pogorelov and Błocki. As an application, in the real case we extend interior regularity results to the case that $u$ lies in a critical Sobolev space (or more generally, certain Sobolev-Orlicz spaces).

math.AP↗
arXiv2016

The sharp quantitative Euclidean concentration inequality

The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square of the volume of the symmetric difference between the set and a ball. This sharp result is strictly related to the physically significant problem of understanding near maximizers in the Riesz rearrangement inequality with a strictly decreasing radially decreasing kernel. Moreover, it implies as a particular case the sharp quantitative Euclidean isoperimetric inequality from \cite{fuscomaggipratelli}.

math.AP↗