Regularity of solutions to a fractional elliptic problem with mixed Dirichlet-Neumann boundary data

In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian

Deterministic KPZ-type equations with nonlocal “gradient terms”

The main goal of this paper is to prove existence and non-existence results for deterministic Kardar–Parisi–Zhang type equations involving non-local “gradient terms”. More precisely, let Ω ⊂ RN, N≥ 2 , be a bounded domain with boundary ∂Ω of class C2. For s∈ (0 , 1) , we consider problems of the form {(-Δ)su=μ(x)|D(u)|q+λf(x),inΩ,u=0,inRN\Ω,(KPZ)where q> 1 and λ> 0 are real parameters, f belongs to a suitable Lebesgue space, μ∈ L∞(Ω) and D represents a nonlocal “gradient term”. Depending on the size of λ> 0 , we derive existence and non-existence results. In particular, we solve several open problems posed in [Abdellaoui in Nonlinearity 31(4): 1260-1298 (2018), Section 6] and [Abdellaoui in Proc Roy Soc Edinburgh Sect A 150(5): 2682-2718 (2020), Section 7]. © 2022, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature

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

Effects of sampling intensity and biomass levels on the precision of acoustic surveys in the Mediterranean Sea

Acoustic surveys represent the standard fishery-independent method worldwide for evaluating the biomass and spatial distribution of small pelagic fish populations. Considering the peculiarities of the spatial behaviour of pelagic fishes, the efficiency of the survey design in determining their biomass and spatial distribution is related to its ability to capture the portion of the patches accounting for larger part of the total biomass. Yet, the spatial structure of the patches could be strongly influenced by ecosystem characteristics as well as by changes in total biomass related to a density-dependent mechanism. This is of particular interest for anchovy and sardine which are known for their wide fluctuations and high sensitivity to the environment. In this study, we analysed the efficiency of acoustic surveys, targeting European anchovy (Engraulis encrasicolus) and European sardine (Sardina pilchardus), in 10 different areas of the Mediterranean Sea across three years of different biomass levels. Using the geostatistical Coefficient of Variation (CVgeo) of the average occurrence probability of high/medium density values, we showed different patterns in terms of survey design efficiency among areas and species. Anchovy usually showed lower CVgeo than sardine, but in the Alboran Sea. In 4 out of 20 cases, CVgeo values showed a consistent decrease with increasing biomass while in the remaining cases the CVgeo did not follow any clear pattern suggesting the presence of important environmental effects. Higher survey design efficiency was found in high productive sectors influenced by river run-off, letting us to hypothesize that higher productivity along with the presence of well-localized enrichment mechanisms could favour a spatially consistent distribution and coherent organization of fish population leading to higher precision estimates with a given transect design. While most surveys displayed CVgeo close to 10% or less even at low biomass, evidencing generally good performances of the survey design, a few areas exhibited higher CVgeo, yielding discussions about a potential need to decrease the inter-transect distance, always keeping in mind that survey should be as synoptic as possible.Versión del edito

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

Enantiospecific sp(2)-sp(3) coupling of secondary and tertiary boronic esters

The cross-coupling of boronic acids and related derivatives with sp² electrophiles (the Suzuki–Miyaura reaction) is one of the most powerful C–C bond formation reactions in synthesis, with applications that span pharmaceuticals, agrochemicals and high-tech materials. Despite the breadth of its utility, the scope of this Nobel prize-winning reaction is rather limited when applied to aliphatic boronic esters. Primary organoboron reagents work well, but secondary and tertiary boronic esters do not (apart from a few specific and isolated examples). Through an alternative strategy, which does not involve using transition metals, we have discovered that enantioenriched secondary and tertiary boronic esters can be coupled to electron-rich aromatics with essentially complete enantiospecificity. As the enantioenriched boronic esters are easily accessible, this reaction should find considerable application, particularly in the pharmaceutical industry where there is growing awareness of the importance of, and greater clinical success in, creating biomolecules with three-dimensional architectures