Social and Political Dimensions of Identity

We study the interior regularity of solutions to the Dirichlet problem Lu = g in Omega, u = 0 in R-nOmega, for anisotropic operators of fractional type Lu(x) = integral(+infinity)(0) dp integral(Sn-1) da(w) 2u(x) - u(x + rho w) - u(x - rho w)/rho(1+2s). Here, a is any measure on Sn-1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a is an element of C-infinity(Sn-1) and g is c(infinity)(Omega), solutions are known to be C-infinity inside Omega (but not up to the boundary). However, when a is a general measure, or even when a is L-infinity(s(n-1)), solutions are only known to be C-3s inside Omega. We prove here that, for general measures a, solutions are C1+3s-epsilon inside Omega for all epsilon > 0 whenever Omega is convex. When a is an element of L-infinity(Sn-1), we show that the same holds in all C-1,C-1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C3s+epsilon for any epsilon > 0 - even if g and Omega are C-infinity

Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension

Given a smoothly bounded domain $\Omega\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(\Omega)$ of solutions to the non-local equation $$(-\Delta)^\frac n2 u_k=\lambda_k u_ke^{\frac n2 u_k^2}\quad \text{in }\Omega,$$ where $\lambda_k\to\lambda_\infty \in [0,\infty)$, and $H^{\frac n2}_{00}(\Omega)$ denotes the Lions-Magenes spaces of functions $u\in L^2(\mathbb{R}^n)$ which are supported in $\Omega$ and with $(-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty(\Omega)$, a suitably rescaled subsequence $\eta_k$ converges to the function $\eta_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed non-local $Q$-curvature equation $$(-\Delta)^\frac n2 \eta =(n-1)!e^{n\eta}\quad \text{in }\mathbb{R}^n$$ recently studied by Da Lio-Martinazzi-Rivi\`ere when $n=1$, Jin-Maalaoui-Martinazzi-Xiong when $n=3$, and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if $\Lambda:=\limsup_{k\to \infty}\|(-\Delta)^\frac n4 u_k\|_{L^2}^2\ge \Lambda_1:= (n-1)!|S^n|$

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

H^s versus C^0-weighted minimizers

We study a class of semi-linear problems involving the fractional Laplacian under subcritical or critical growth assumptions. We prove that, for the corresponding functional, local minimizers with respect to a C^0-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the natural H^s-topology.Comment: 15 page

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756

A Gaseous Argon-Based Near Detector to Enhance the Physics Capabilities of DUNE

This document presents the concept and physics case for a magnetized gaseous argon-based detector system (ND-GAr) for the Deep Underground Neutrino Experiment (DUNE) Near Detector. This detector system is required in order for DUNE to reach its full physics potential in the measurement of CP violation and in delivering precision measurements of oscillation parameters. In addition to its critical role in the long-baseline oscillation program, ND-GAr will extend the overall physics program of DUNE. The LBNF high-intensity proton beam will provide a large flux of neutrinos that is sampled by ND-GAr, enabling DUNE to discover new particles and search for new interactions and symmetries beyond those predicted in the Standard Model

Long-baseline neutrino oscillation physics potential of the DUNE experiment

The sensitivity of the Deep Underground Neutrino Experiment (DUNE) to neutrino oscillation is determined, based on a full simulation, reconstruction, and event selection of the far detector and a full simulation and parameterized analysis of the near detector. Detailed uncertainties due to the flux prediction, neutrino interaction model, and detector effects are included. DUNE will resolve the neutrino mass ordering to a precision of 5σ, for all ΑCP values, after 2 years of running with the nominal detector design and beam configuration. It has the potential to observe charge-parity violation in the neutrino sector to a precision of 3σ (5σ) after an exposure of 5 (10) years, for 50% of all ΑCP values. It will also make precise measurements of other parameters governing long-baseline neutrino oscillation, and after an exposure of 15 years will achieve a similar sensitivity to sin22θ13 to current reactor experiments

Scintillation light detection in the 6-m drift-length ProtoDUNE Dual Phase liquid argon TPC

DUNE is a dual-site experiment for long-baseline neutrino oscillation studies, neutrino astrophysics and nucleon decay searches. ProtoDUNE Dual Phase (DP) is a 6  ×  6  ×  6 m 3 liquid argon time-projection-chamber (LArTPC) that recorded cosmic-muon data at the CERN Neutrino Platform in 2019-2020 as a prototype of the DUNE Far Detector. Charged particles propagating through the LArTPC produce ionization and scintillation light. The scintillation light signal in these detectors can provide the trigger for non-beam events. In addition, it adds precise timing capabilities and improves the calorimetry measurements. In ProtoDUNE-DP, scintillation and electroluminescence light produced by cosmic muons in the LArTPC is collected by photomultiplier tubes placed up to 7 m away from the ionizing track. In this paper, the ProtoDUNE-DP photon detection system performance is evaluated with a particular focus on the different wavelength shifters, such as PEN and TPB, and the use of Xe-doped LAr, considering its future use in giant LArTPCs. The scintillation light production and propagation processes are analyzed and a comparison of simulation to data is performed, improving understanding of the liquid argon properties

Low exposure long-baseline neutrino oscillation sensitivity of the DUNE experiment

The Deep Underground Neutrino Experiment (DUNE) will produce world-leading neutrino oscillation measurements over the lifetime of the experiment. In this work, we explore DUNE's sensitivity to observe charge-parity violation (CPV) in the neutrino sector, and to resolve the mass ordering, for exposures of up to 100 kiloton-megawatt-years (kt-MW-yr). The analysis includes detailed uncertainties on the flux prediction, the neutrino interaction model, and detector effects. We demonstrate that DUNE will be able to unambiguously resolve the neutrino mass ordering at a 3$\sigma$ (5$\sigma$) level, with a 66 (100) kt-MW-yr far detector exposure, and has the ability to make strong statements at significantly shorter exposures depending on the true value of other oscillation parameters. We also show that DUNE has the potential to make a robust measurement of CPV at a 3$\sigma$ level with a 100 kt-MW-yr exposure for the maximally CP-violating values \delta_{\rm CP}} = \pm\pi/2. Additionally, the dependence of DUNE's sensitivity on the exposure taken in neutrino-enhanced and antineutrino-enhanced running is discussed. An equal fraction of exposure taken in each beam mode is found to be close to optimal when considered over the entire space of interest