arXiv subjects
Publications and source records attributed to Mark Allen.
We consider a free boundary problem on three-dimensional cones depending on a parameter c and study when the free boundary is allowed to pass through the vertex of the cone. Combining analysis and computer-assisted proof, we show that when c is less than 0.43, the free boundary may pass through the vertex of the cone.
The International Virtual Observatory Alliance (IVOA) has developed and built, in the last two decades, an ecosystem of distributed resources, interoperable and based upon open shared technological standards. In doing so the IVOA has anticipated, putting into practice for the astrophysical domain, the ideas of FAIR-ness of data and service resources and the Open-ness of sharing scientific results, leveraging on the underlying open standards required to fill the above. In Europe, efforts in supporting and developing the ecosystem proposed by the IVOA specifications has been provided by a continuous set of EU funded projects up to current H2020 ESCAPE ESFRI cluster. In the meantime, in the last years, Europe has realised the importance of promoting the Open Science approach for the research communities and started the European Open Science Cloud (EOSC) project to create a distributed environment for research data, services and communities. In this framework the European VO community, had to face the move from the interoperability scenario in the astrophysics domain into a larger audience perspective that includes a cross-domain FAIR approach. Within the ESCAPE project the CEVO Work Package (Connecting ESFRI to EOSC through the VO) has one task to deal with this integration challenge: a challenge where an existing, mature, distributed e-infrastructure has to be matched to a forming, more general architecture. CEVO started its works in the first months of 2019 and has already worked on the integration of the VO Registry into the EOSC e-infrastructure. This contribution reports on the first year and a half of integration activities, that involve applications, services and resources being aware of the VO scenario and compatible with the EOSC architecture.
With some telescopes standing still, now more than ever simple access to archival data is vital for astronomers and they need to know how to go about it. Within European Virtual Observatory (VO) projects, such as AIDA (2008-2010), ICE (2010-2012), CoSADIE (2013-2015), ASTERICS (2015-2018) and ESCAPE (since 2019), we have been offering Virtual Observatory schools for many years. The aim of these schools are twofold: teaching (early career) researchers about the functionalities and possibilities within the Virtual Observatory and collecting feedback from the astronomical community. In addition to the VO schools on the European level, different national teams have also put effort into VO dissemination. The team at the Centre de Données astronomiques de Strasbourg (CDS) started to explore more and new ways to interact with the community: a series of blog posts on AstroBetter.com or a lunch time session at the virtual EAS meeting 2020. The Spanish VO has conducted virtual VO schools. GAVO has supported online archive workshops and maintains their Virtual Observatory Text Treasures. In this paper, we present the different formats in more detail, and report on the resulting interaction with the community as well as the estimated reach.
We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and $p$-Laplace type equations with a right hand side, as well as for the Laplace equation on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.
The European Open Science Cloud (EOSC) is in its early stages, but already some aspects of the EOSC vision are starting to become reality, for example the EOSC portal and the development of metadata catalogues. In the astrophysical domain already exists an open approach to science data: the Virtual Observatory view put in place by the International Virtual Observatory Alliance (IVOA) architecture of standards. The ESCAPE (European Science Cluster of Astronomy & Particle physics ESFRI research infrastructures) project has, among its tasks, to demonstrate that the VO architecture can be integrated within the EOSC building one and to provide guidelines to ESFRI partners (European Strategy Forum on Research Infrastructures) in doing this. This contribution reports on the progress of this integration after the first months of work inside ESCAPE.
We introduce a new boundary Harnack principle in Lipschitz domains for equations with right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the principle for divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.
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.
We give an example of a pair of nonnegative subharmonic functions with disjoint support for which the Alt-Caffarelli-Friedman monotonicity formula has strictly positive limit at the origin, and yet the interface between their supports lacks a (unique) tangent there. This clarifies a remark appearing in the literature (see \cite{cs05}) that the positivity of the limit of the ACF formula implies unique tangents; this is true under some additional assumptions, but false in general. In our example, blow-ups converge to the expected piecewise linear two-plane function along subsequences, but the limiting function depends on the subsequence due to the spiraling nature of the interface.
This paper proves Hölder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Hölder continuous.
Since its inception in the early 2000, the Virtual Observatory (VO), developed as a collaboration of many national and international projects, has become a major factor in the discovery and dissemination of astronomical information worldwide. The IVOA has been coordinating all these efforts worldwide to ensure a common VO framework that enables transparent access to and interoperability of astronomy resources (data and software) around the world. The VO is not a magic solution to all astronomy data management challenges but it does bring useful solutions in many areas borne out by the fact that VO interfaces are broadly found in astronomy major data centres and projects worldwide. Astronomy data centres have been building VO services on top of their existing data services to increase interoperability with other VO-compliant data resources to take advantage of the continuous and increasing development of VO applications. VO applications have made multi-instrument and multi-wavelength science, a difficult and fruitful part of astronomy, somewhat easier. More recently, several major new astronomy projects have been directly adopting VO standards to build their data management infrastructure, giving birth to VO built-in archives. Embracing the VO framework from the beginning brings the double gain of not needing to reinvent the wheel and ensuring from the start interoperability with other astronomy VO resources. Some of the IVOA standards are also starting to be used by neighbour disciplines like planetary sciences. There is still quite a lot to be done on the VO, in particular tackling the upcoming big data challenge and how to find interoperable solutions to the new data analysis paradigm of bringing and running the software close to the data.
We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.
This document presents HiPS, a hierarchical scheme for the description, storage and access of sky survey data. The system is based on hierarchical tiling of sky regions at finer and finer spatial resolution which facilitates a progressive view of a survey, and supports multi-resolution zooming and panning. HiPS uses the HEALPix tessellation of the sky as the basis for the scheme and is implemented as a simple file structure with a direct indexing scheme that leads to practical implementations.
We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Holder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative approaches 1.
We study the regularity of minimizers to the functional \[ J(w)=\int_Ω a^{ij}w_iw_j + Qχ_{\{w>0\}}, \] over a bounded domain $Ω$ and among the class of nonnegative functions in $W^{1,2}(Ω)$ with prescribed boundary data. We assume that the coefficients $a^{ij}$ are only bounded and measurable and satisfy an ellipticity in condition. In two dimensions we prove that minimizers are Hölder continuous on subdomains. We also prove that in two dimensions a minimizer $u$ satisfies a linear growth condition from above and below near the free boundary $\partial \{u>0\}$.
We consider a free boundary problem on cones depending on a parameter c and study when the free boundary is allowed to pass through the vertex of the cone. We show that when the cone is three-dimensional and c is large enough, the free boundary avoids the vertex. We also show that when c is small enough but still positive, the free boundary is allowed to pass through the vertex. This establishes 3 as the critical dimension for which the free boundary may pass through the vertex of a right circular cone. In view of the well-known connection between area-minimizing surfaces and the free boundary problem under consideration, our result is analogous to a result of Morgan that classifies when an area-minimizing surface on a cone passes through the vertex.
We present results of an investigation into the formation of nitrogen-bearing molecules in the atmosphere of Titan. We extend a previous model (Li et al. 2015, 2016) to cover the region below the tropopause, so the new model treats the atmosphere from Titan's surface to an altitude of 1500 km. We consider the effects of condensation and sublimation using a continuous, numerically stable method. This is coupled with parameterized treatments of the sedimentation of the aerosols and their condensates, and the formation of haze particles. These processes affect the abundances of heavier species such as the nitrogen-bearing molecules, but have less effect on the abundances of lighter molecules. Removal of molecules to form aerosols also plays a role in determining the mixing ratios, in particular of HNC, HC3N and HCN. We find good agreement with the recently detected mixing ratios of C2H5CN, with condensation playing an important role in determining the abundance of this molecule below 500 km. Of particular interest is the chemistry of acrylonitrile (C2H3CN) which has been suggested by Stevenson et al. (2015) as a molecule that could form biological membranes in an oxygen-deficient environment. With the inclusion of haze formation we find good agreement of our model predictions of acrylonitrile with the available observations.
We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account "memory''. The precise model is \[ D_t^α u - \text{div}(u(-Δ)^{-σ} u) = f, \quad 0<σ<1/2. \] We pose the problem over $\{t\in {\mathbb R}^+, x\in {\mathbb R}^n\}$ with nonnegative initial data $u(0,x)\geq 0 $ as well as right hand side $f\geq 0$. We first prove existence for weak solutions when $f,u(0,x)$ have exponential decay at infinity. Our main result is Hölder continuity for such weak solutions.
We study solutions to a variational equation that models heat control on the boundary. This problem can be thought of as the two phase parabolic Signorini problem. We show that when the solution has a sign on the boundary, the study of the free boundary can be reduced to the study of the free boundary in the parabolic Signorini problem. We show that locally there is always a sign on the boundary; therefore, the optimal regularity of the solutions as well as the study of the free boundary is reduced to the one phase parabolic Signorini problem.