Global bifurcation for fractional pp-Laplacian and application

We prove the existence of an unbounded branch of solutions to the non-linear non-local equation $$(-\Delta)^s_p u=\lambda |u|^{p-2}u + f(x,u,\lambda) \quad\text{in}\quad \Omega,\quad u=0 \quad\text{in}\quad \mathbb{R}^n\setminus\Omega,$$ bifurcating from the first eigenvalue. Here $(-\Delta)^s_p$ denotes the fractional $p$-Laplacian and $\Omega\subset\mathbb{R}^n$ is a bounded regular domain. The proof of the bifurcation results relies in computing the Leray--Schauder degree by making an homotopy respect to $s$ (the order of the fractional $p$-Laplacian) and then to use results of local case (that is $s=1$) found in [17]. Finally, we give some application to an existence result.Comment: 38 page

FACTORS INFLUENCING BEST ANNUAL RACING TIME IN FINNISH HORSES

The fixed effects of year of race, season of race, sex, method of start, annual number of starts, length of race and racetrack were evaluated on best annual racing time in Finnish Horses. Data included 1,378 records for 554 horses by 206 sires. Five models were assumed within the age groups from 3 to 6 yr. The annual number of starts, method of start and season of race effects were interrelated. An increase in number of starts was associated with considerable improvement in a horse\u27s best annual racing time. Records should not, however, be adjusted for effect of annual number of starts because it would simultaneously account for part of genetic differences among horses. The largest estimates of heritability were obtained for best annual racing time when the model included the fixed year-season and sex effects. Corresponding to this model, the estimate of repeatability for best annual racing time over the four age groups was .60 ± .03. An example of best linear unbiased predictions of sires\u27 breeding values based on progeny records in one or several ages is presented

Fractional convexity

We introduce a notion of fractional convexity that extends naturally the usual notion of convexity in the Euclidean space to a fractional setting. With this notion of fractional convexity, we study the fractional convex envelope inside a domain of an exterior datum (the largest possible fractional convex function inside the domain that is below the datum outside) and show that the fractional convex envelope is characterized as a viscosity solution to a non-local equation that is given by the infimum among all possible directions of the $1-$dimensional fractional Laplacian. For this equation we prove existence, uniqueness and a comparison principle (in the framework of viscosity solutions). In addition, we find that solutions to the equation for the convex envelope are related to solutions to the fractional Monge-Ampere equation.Comment: 25 page

An automated cirrus classification

Cirrus clouds play an important role in determining the radiation budget of the earth, but many of their properties remain uncertain, particularly their response to aerosol variations and to warming. Part of the reason for this uncertainty is the dependence of cirrus cloud properties on the cloud formation mechanism, which itself is strongly dependent on the local meteorological conditions. In this work, a classification system (Identification and Classification of Cirrus or IC-CIR) is introduced to identify cirrus clouds by the cloud formation mechanism. Using re-analysis and satellite data, cirrus clouds are separated in four main types: orographic, frontal, convective and synoptic. Through a comparison to convection-permitting model simulations and back- trajectory based analysis, it is shown that these observation-based regimes can provide extra information on the cloud scale updraughts and the frequency of occurrence of liquid-origin ice, with the convective regime having higher updraughts and a greater occurrence of liquid-origin ice compared to the synoptic regimes. Despite having different cloud formation mecha- nisms, the radiative properties of the regimes are not distinct, indicating that retrieved cloud properties alone are insufficient to completely describe them. This classification is designed to be easily implemented in GCMs, helping improve future model-observation comparisons and leading to improved parametrisations of cirrus cloud processe

Ice crystal number concentration estimates from lidar–radar satellite remote sensing – Part 2: Controls on the ice crystal number concentration

The ice crystal number concentration (Ni) is a key property of ice clouds, both radiatively and microphysically. Due to sparse in situ measurements of ice cloud properties, the controls on the Ni have remained difficult to determine. As more advanced treatments of ice clouds are included in global models, it is becoming increasingly necessary to develop strong observational constraints on the processes involved. This work uses the DARDAR-Nice Ni retrieval described in Part 1 to investigate the controls on the Ni at a global scale. The retrieved clouds are separated by type. The effects of temperature, proxies for in-cloud updraft and aerosol concentrations are investigated. Variations in the cloud top Ni (Ni(top)) consistent with both homogeneous and heterogeneous nucleation are observed along with differing relationships between aerosol and Ni(top) depending on the prevailing meteorological situation and aerosol type. Away from the cloud top, the Ni displays a different sensitivity to these controlling factors, providing a possible explanation for the low Ni sensitivity to temperature and ice nucleating particles (INP) observed in previous in situ studies. This satellite dataset provides a new way of investigating the response of cloud properties to meteorological and aerosol controls. The results presented in this work increase our confidence in the retrieved Ni and will form the basis for further study into the processes influencing ice and mixed phase clouds

The Research Unit VolImpact: Revisiting the volcanic impact on atmosphere and climate – preparations for the next big volcanic eruption

This paper provides an overview of the scientific background and the research objectives of the Research Unit “VolImpact” (Revisiting the volcanic impact on atmosphere and climate – preparations for the next big volcanic eruption, FOR 2820). VolImpact was recently funded by the Deutsche Forschungsgemeinschaft (DFG) and started in spring 2019. The main goal of the research unit is to improve our understanding of how the climate system responds to volcanic eruptions. Such an ambitious program is well beyond the capabilities of a single research group, as it requires expertise from complementary disciplines including aerosol microphysical modelling, cloud physics, climate modelling, global observations of trace gas species, clouds and stratospheric aerosols. The research goals will be achieved by building on important recent advances in modelling and measurement capabilities. Examples of the advances in the observations include the now daily near-global observations of multi-spectral aerosol extinction from the limb-scatter instruments OSIRIS, SCIAMACHY and OMPS-LP. In addition, the recently launched SAGE III/ISS and upcoming satellite missions EarthCARE and ALTIUS will provide high resolution observations of aerosols and clouds. Recent improvements in modeling capabilities within the framework of the ICON model family now enable simulations at spatial resolutions fine enough to investigate details of the evolution and dynamics of the volcanic eruptive plume using the large-eddy resolving version, up to volcanic impacts on larger-scale circulation systems in the general circulation model version. When combined with state-of-the-art aerosol and cloud microphysical models, these approaches offer the opportunity to link eruptions directly to their climate forcing. These advances will be exploited in VolImpact to study the effects of volcanic eruptions consistently over the full range of spatial and temporal scales involved, addressing the initial development of explosive eruption plumes (project VolPlume), the variation of stratospheric aerosol particle size and radiative forcing caused by volcanic eruptions (VolARC), the response of clouds (VolCloud), the effects of volcanic eruptions on atmospheric dynamics (VolDyn), as well as their climate impact (VolClim)