An existence result for a nonlinear transmission problems

Let $\Omega^o$ and $\Omega^i$ be open bounded subsets of $\mathbb{R}^n$ of class $C^{1,\alpha}$ such that the closure of $\Omega^i$ is contained in $\Omega^o$. Let $f^o$ be a function in $C^{1,\alpha}(\partial\Omega^o)$ and let $F$ and $G$ be continuous functions from $\partial\Omega^i\times\mathbb{R}$ to $\mathbb{R}$. By exploiting an argument based on potential theory and on the Leray-Schauder principle we show that under suitable and completely explicit conditions on $F$ and $G$ there exists at least one pair of continuous functions $(u^o, u^i)$ such that $$\left\{ \begin{array}{ll} \Delta u^o=0&\text{in }\Omega^o\setminus\mathrm{cl}\Omega^i\,,\\ \Delta u^i=0&\text{in }\Omega^i\,,\\ u^o(x)=f^o(x)&\text{for all }x\in\partial\Omega^o\,,\\ u^o(x)=F(x,u^i(x))&\text{for all }x\in\partial\Omega^i\,,\\ \nu_{\Omega^i}\cdot\nabla u^o(x)-\nu_{\Omega^i}\cdot\nabla u^i(x)=G(x,u^i(x))&\text{for all }x\in\partial\Omega^i\,, \end{array} \right.$$ where the last equality is attained in certain weak sense. In a simple example we show that such a pair of functions $(u^o, u^i)$ is in general neither unique nor local unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we can prove the existence of at least one classical solution which is in addition locally unique

Shape analyticity and singular perturbations for layer potential operators

We study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image (Ω) of a reference set Ω and we present some real analyticity results for the dependence upon the map. Then we introduce a perforated domain Ω(ϵ) with a small hole of size ϵ and we compute power series expansions that describe the layer potentials on Ω(ϵ) when the parameter ϵ approximates the degenerate value ϵ = 0

Existence results for a nonlinear nonautonomous transmission problem via domain perturbation

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter

Mapping properties of weakly singular periodic volume potentials in Roumieu classes

The analysis of the dependence of integral operators on perturbations plays an important role in the study of inverse problems and of perturbed boundary value problems. In this paper, we focus on the mapping properties of the volume potentials with weakly singular periodic kernels. Our main result is to prove that the map which takes a density function and a periodic kernel to a (suitable restriction of the) volume potential is bilinear and continuous with values in a Roumieu class of analytic functions. This result extends to the periodic case of some previous results obtained by the authors for nonperiodic potentials, and it is motivated by the study of perturbation problems for the solutions of boundary value problems for elliptic differential equations in periodic domains

Environmental life cycle assessment of Italian mozzarella cheese: Hotspots and improvement opportunities

The present study investigated a cradle-to-grave life cycle assessment to estimate the environmental impacts associated with Italian mozzarella cheese consumption. The differences between mozzarella produced from raw milk and mozzarella produced from curd were studied, and differences in manufacturing processes have been emphasized in order to provide guidance for targeted improvements at this phase. Specifically, the third-largest Italian mozzarella producer was surveyed to collect site-specific manufacturing data. The Ecoinvent v3.2 database was used for secondary data, whereas SimaPro 8.1 was the modeling software. The inventory included inputs from farm activities to end of life disposal of wasted mozzarella and packaging. Additionally, plant-specific information was used to assign major inputs, such as electricity, natural gas, packaging, and chemicals to specific products; however, where disaggregated information was not provided, milk solids allocation was applied. Notably, loss of milk solids was accounted during the manufacture, moreover mozzarella waste and transport were considered during distribution, retail, and consumption phases. Feed production and animal emissions were the main drivers of raw milk production. Electricity and natural gas usage, packaging (cardboard and plastic), transport, wastewater treatment, and refrigerant loss affected the emissions from a farm gate-to-dairy plant gate perspective. Post-dairy plant gate effects were mainly determined by electricity usage for storage of mozzarella, transport of mozzarella, and waste treatment. The average emissions were 6.66 kg of CO2 equivalents and 45.1 MJ of cumulative energy demand/kg of consumed mozzarella produced directly from raw milk, whereas mozzarella from purchased curd had larger emissions than mozzarella from raw milk due to added transport of curd from specialty manufacturing plants, as well as electricity usage from additional processes at the mozzarella plant that are required to process the curd into mozzarella. Normalization points to ecotoxicity as the impact category most significantly influenced by mozzarella consumption. From a farm gate-to-grave perspective, ecotoxicity and freshwater and marine eutrophication are the first and second largest contributors of mozzarella consumption to average European effects, respectively. To increase environmental sustainability, an improvement of efficiency for energy and packaging usage and transport activities is recommended in the post-farm gate mozzarella supply chain

Caracterização Anatômica de Folhas e Inflorescências de Espécies de Lavanda (lamiaceae) utilizadas como Medicinais no Brasil

http://dx.doi.org/10.5902/2179460X13654Plants of the genus Lavandula denominated communly as lavenders are aromatic originated in Europe, especially from the Mediterranean area and belonging to the Lamiceae group. There is little information about the anatomy of plants of this gender, so we prioritized to identify the anatomical characters of Lavandula angustifolia Mill and Lavandula dentata L. to assist their own identification. For the historical analysis it was performed the middle part of young leaves and inflorescences in full bloom, from transverse sections made ​​freehand with the help of razor and Styrofoam and the help of optical microscope and common digital camera (Olympus SP – 800UZ). It was also performed paradermical scraping with freehand from the abaxial surface of leaves and inflorescences and macerated dried material analysis of these organs. We sought to identify and classify the trichomes, the anatomical characteristics of leaves and inflorescences (flowers and bracts), and the powder of these agencies to compare the results obtained. In the anatomical characters examined, both from fresh material as the macerated material, there were similarities between the two species and morphology of epidermal cells, cuticle, stomata types, and heterogeneous dorsiventral mesophyll, and glandular trichomes in the leaves. L. dentata presents two larger secondary vessels conductors in the middle part of the leaves, while in angustifolia L. these vessels are all identical and with smaller gauges.http://dx.doi.org/10.5902/2179460X13654Plantas do gênero Lavandula denominadas comumente de alfazemas ou lavandas, são aromáticas originárias da Europa, especificamente da região do Mediterrâneo e pertencentes à Lamiaceae. Em função da escassez de informações acerca da anatomia de plantas deste gênero, objetivou-se conhecer os caracteres anatômicos de Lavandula angustifolia Mill. e Lavandula dentata L. para auxiliar na identificação das mesmas. Para a análise histológica foi utilizada a parte mediana de folhas adultas e de inflorescências (flores e brácteas) em plena floração, a partir de secções transversais feitas à mão livre com o auxílio de lâmina de barbear e isopor, visualizadas e analisadas a partir de microscópio óptico e câmera digital, marca Olympus SP – 800 UZ. Realizou-se raspagem paradérmica também à mão livre da face abaxial das folhas e inflorescências e análise de material seco macerado destes órgãos. Buscou-se identificar e classificar os tricomas, estudar as características anatômicas de folhas e inflorescências (flores e brácteas), além do pó destes órgãos para confrontar os resultados obtidos. Nos caracteres anatômicos analisados, tanto do material fresco quanto do material macerado, verificou-se semelhanças entre as duas espécies como morfologia das células epidérmicas, cutícula, tipos de estômatos, mesofilo dorsiventral e heterogêneo, tricomas tectores e glandulares observados nas folhas.  L. dentata apresentou dois vasos condutores secundários de maior porte na parte mediana das folhas, enquanto que em L. angustifolia todos estes vasos são menores e de calibres idênticos

Reply to: “Global Conservation of Phylogenetic Diversity Captures More Than Just Functional Diversity”

Academic biologists have long advocated for conserving phylogenetic diversity (PD), often (but not exclusively) on the basis that PD is a useful proxy for “feature diversity”, defined as the variety of forms and functions represented in set of organisms (see below for an extended discussion of this definition). In a recent paper, we assess the extent to which this proxy (which we coined the “phylogenetic gambit”) holds in three empirical datasets (terrestrial mammals, birds, and tropical marine fishes) when using functional traits and functional diversity (FD) to operationalize feature diversity. Owen et al. offer a criticism of our methods for quantifying feature diversity with FD and disagree with our conclusions. We are grateful that Owen et al. have engaged thoughtfully with our work, but we believe there are more points of agreement than Owen et al. imply

Prioritizing Phylogenetic Diversity Captures Functional Diversity Unreliably

In the face of the biodiversity crisis, it is argued that we should prioritize species in order to capture high functional diversity (FD). Because species traits often reflect shared evolutionary history, many researchers have assumed that maximizing phylogenetic diversity (PD) should indirectly capture FD, a hypothesis that we name the “phylogenetic gambit”. Here, we empirically test this gambit using data on ecologically relevant traits from \u3e15,000 vertebrate species. Specifically, we estimate a measure of surrogacy of PD for FD. We find that maximizing PD results in an average gain of 18% of FD relative to random choice. However, this average gain obscures the fact that in over one-third of the comparisons, maximum PD sets contain less FD than randomly chosen sets of species. These results suggest that, while maximizing PD protection can help to protect FD, it represents a risky conservation strategy

Conserving evolutionary history does not result in greater diversity over geological time scales

Alternative prioritization strategies have been proposed to safeguard biodiversity over macroevolutionary time scales. The first prioritizes the most distantly related species—maximizing phylogenetic diversity (PD)—in the hopes of capturing at least some lineages that will successfully diversify into the future. The second prioritizes lineages that are currently speciating, in the hopes that successful lineages will continue to generate species into the future. These contrasting schemes also map onto contrasting predictions about the role of slow diversifiers in the production of biodiversity over palaeontological time scales. We consider the performance of the two schemes across 10 dated species-level palaeo-phylogenetic trees ranging from Foraminifera to dinosaurs. We find that prioritizing PD for conservation generally led to fewer subsequent lineages, while prioritizing diversifiers led to modestly more subsequent diversity, compared with random sets of lineages. Importantly for conservation, the tree shape when decisions are made cannot predict which scheme will be most successful. These patterns are inconsistent with the notion that long-lived lineages are the source of new species. While there may be sound reasons for prioritizing PD for conservation, long-term species production might not be one of them