On a Class of Fractional Obstacle Type Problems Related to the Distributional Riesz Derivative

In this work, we consider the fractional obstacle problem with a given obstacle $\psi$ in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^{d}$, such that $\mathbb{K}_\psi^s=\{v\in H^s_0(\Omega):v\geq\psi \text{ a.e. in }\Omega\}\neq\emptyset$, given by $$u\in\mathbb{K}_\psi^s:\quad\langle\mathcal{L}_Au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s_\psi,$$ for $F$ in $H^{-s}(\Omega)$, the dual space of the fractional Sobolev space $H^s_0(\Omega)$, $0<s<1$. The nonlocal operator $\mathcal{L}_A:H^s_0(\Omega)\to H^{-s}(\Omega)$ is defined with the distributional Riesz fractional derivatives and with a measurable, bounded, positive definite matrix $A(x):\mathbb{R}^d\to\mathbb{R}^{d\times d}$, by $\mathcal{L}_Au=-D^s\cdot(AD^su)$. We show that the corresponding bilinear form $\mathcal{E}_A(u,v)$ is a (not necessarily symmetric) Dirichlet form that corresponds to a nonlocal integral operator with a well defined integral kernel $k_A(x,y)$. We then consider obstacle-type problems involving $\mathcal{L}_A$ with one or two obstacles, as well as the $N$-membranes problem, thereby deriving several results, such as the weak maximum principle, comparison properties, approximation by bounded penalization and also the Lewy-Stampacchia inequalities, similarly to the classical obstacle problem which is obtained at the limit $s\to1$. This provides regularity of the solutions, including a global estimate in $L^\infty(\Omega)$, local H\"older regularity of the solutions when $A$ is symmetric, and local regularity in fractional Sobolev spaces when $A$ is the identity and $\mathcal{L}_A=(-\Delta)^s$ corresponds to fractional $s$-Laplacian obstacle-type problems. These novel results are complemented with the extension of the Lewy-Stampacchia inequalities to the order dual of $H^s_0(\Omega)$ and some remarks on the associated $s$-capacity and the $s$-fractional obstacle problem

On the Stability of the ss-Nonlocal pp-Obstacle Problem and their Coincidence Sets and Free Boundaries

We show that the solutions to the nonlocal obstacle problems for the nonlocal $-\Delta_p^s$ operator, when the fractional parameter $s\to\sigma$ for $0<\sigma\leq1$, converge to the solution of the corresponding obstacle problem for $-\Delta_p^\sigma$, being $\sigma=1$ the classical obstacle problem for the local $p$-Laplacian. We discuss the weak stability of the quasi-characteristic functions of coincidence sets of the solution with the obstacle, which is a strong convergence of their characteristic functions when $s\nearrow 1$ under a nondegeneracy condition. This stability can be shown also in terms of the convergence of the free boundaries, as well as of the coincidence sets, in Hausdorff distance when $s\nearrow 1$, under non-degeneracy local assumptions on the external force and a local topological property of the coincidence set of the limit classical obstacle problem for the local $p$-Laplacian, essentially when the limit coincidence set is the closure of its interior

On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $\Omega\subset\mathbb{R}^d$ with time-dependent Dirichlet boundary condition for the temperature $\vartheta = \vartheta(x, t)$, $\vartheta = g$ on $\Omega^c\times]0, T[$, and initial condition $\eta_0$ for the enthalpy $\eta = \eta(x, t)$, given in $\Omega\times]0, T[$ by $\frac{\partial \eta}{\partial t} +\mathcal{L}_A^s \vartheta = f\quad\text{ with }\eta\in \beta(\vartheta),$ where $\mathcal{L}_A^s$ is an anisotropic fractional operator defined in the distributional sense by $\langle\mathcal{L}_A^su, v\rangle = \int_{\mathbb{R}^d}AD^su\cdot D^sv\, dx,$ $\beta$ is a maximal monotone graph, $A(x)$ is a symmetric, strictly elliptic and uniformly bounded matrix, and $D^s$ is the distributional Riesz fractional gradient for 0 < s < 1 . We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $s\nearrow 1$ towards the classical local problem, the asymptotic behaviour as $t\to\infty$, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $\beta$.</p

Polygenic burden in focal and generalized epilepsies

© The Author(s) (2019).Rare genetic variants can cause epilepsy, and genetic testing has been widely adopted for severe, paediatric-onset epilepsies. The phenotypic consequences of common genetic risk burden for epilepsies and their potential future clinical applications have not yet been determined. Using polygenic risk scores (PRS) from a European-ancestry genome-wide association study in generalized and focal epilepsy, we quantified common genetic burden in patients with generalized epilepsy (GE-PRS) or focal epilepsy (FE-PRS) from two independent non-Finnish European cohorts (Epi25 Consortium, n = 5705; Cleveland Clinic Epilepsy Center, n = 620; both compared to 20 435 controls). One Finnish-ancestry population isolate (Finnish-ancestry Epi25, n = 449; compared to 1559 controls), two European-ancestry biobanks (UK Biobank, n = 383 656; Vanderbilt biorepository, n = 49 494), and one Japaneseancestry biobank (BioBank Japan, n = 168 680) were used for additional replications. Across 8386 patients with epilepsy and 622 212 population controls, we found and replicated significantly higher GE-PRS in patients with generalized epilepsy of European-ancestry compared to patients with focal epilepsy (Epi25: P = 1.64×10-15; Cleveland: P = 2.85×10-4; Finnish-ancestry Epi25: P = 1.80×10-4) or population controls (Epi25: P = 2.35×10-70; Cleveland: P = 1.43×10-7; Finnish-ancestry Epi25: P = 3.11×10-4; UK Biobank and Vanderbilt biorepository meta-analysis: P = 7.99×10-4). FE-PRS were significantly higher in patients with focal epilepsy compared to controls in the non-Finnish, non-biobank cohorts (Epi25: P = 5.74×10-19; Cleveland: P = 1.69×10-6). European ancestry-derived PRS did not predict generalized epilepsy or focal epilepsy in Japanese-ancestry individuals. Finally, we observed a significant 4.6-fold and a 4.5-fold enrichment of patients with generalized epilepsy compared to controls in the top 0.5% highest GE-PRS of the two non-Finnish European cohorts (Epi25: P = 2.60×10-15; Cleveland: P = 1.39×10-2). We conclude that common variant risk associated with epilepsy is significantly enriched in multiple cohorts of patients with epilepsy compared to controls-in particular for generalized epilepsy. As sample sizes and PRS accuracy continue to increase with further common variant discovery, PRS could complement established clinical biomarkers and augment genetic testing for patient classification, comorbidity research, and potentially targeted treatment

A global metagenomic map of urban microbiomes and antimicrobial resistance

We present a global atlas of 4,728 metagenomic samples from mass-transit systems in 60 cities over 3 years, representing the first systematic, worldwide catalog of the urban microbial ecosystem. This atlas provides an annotated, geospatial profile of microbial strains, functional characteristics, antimicrobial resistance (AMR) markers, and genetic elements, including 10,928 viruses, 1,302 bacteria, 2 archaea, and 838,532 CRISPR arrays not found in reference databases. We identified 4,246 known species of urban microorganisms and a consistent set of 31 species found in 97% of samples that were distinct from human commensal organisms. Profiles of AMR genes varied widely in type and density across cities. Cities showed distinct microbial taxonomic signatures that were driven by climate and geographic differences. These results constitute a high-resolution global metagenomic atlas that enables discovery of organisms and genes, highlights potential public health and forensic applications, and provides a culture-independent view of AMR burden in cities.Funding: the Tri-I Program in Computational Biology and Medicine (CBM) funded by NIH grant 1T32GM083937; GitHub; Philip Blood and the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF grant number ACI-1548562 and NSF award number ACI-1445606; NASA (NNX14AH50G, NNX17AB26G), the NIH (R01AI151059, R25EB020393, R21AI129851, R35GM138152, U01DA053941); STARR Foundation (I13- 0052); LLS (MCL7001-18, LLS 9238-16, LLS-MCL7001-18); the NSF (1840275); the Bill and Melinda Gates Foundation (OPP1151054); the Alfred P. Sloan Foundation (G-2015-13964); Swiss National Science Foundation grant number 407540_167331; NIH award number UL1TR000457; the US Department of Energy Joint Genome Institute under contract number DE-AC02-05CH11231; the National Energy Research Scientific Computing Center, supported by the Office of Science of the US Department of Energy; Stockholm Health Authority grant SLL 20160933; the Institut Pasteur Korea; an NRF Korea grant (NRF-2014K1A4A7A01074645, 2017M3A9G6068246); the CONICYT Fondecyt Iniciación grants 11140666 and 11160905; Keio University Funds for Individual Research; funds from the Yamagata prefectural government and the city of Tsuruoka; JSPS KAKENHI grant number 20K10436; the bilateral AT-UA collaboration fund (WTZ:UA 02/2019; Ministry of Education and Science of Ukraine, UA:M/84-2019, M/126-2020); Kyiv Academic Univeristy; Ministry of Education and Science of Ukraine project numbers 0118U100290 and 0120U101734; Centro de Excelencia Severo Ochoa 2013–2017; the CERCA Programme / Generalitat de Catalunya; the CRG-Novartis-Africa mobility program 2016; research funds from National Cheng Kung University and the Ministry of Science and Technology; Taiwan (MOST grant number 106-2321-B-006-016); we thank all the volunteers who made sampling NYC possible, Minciencias (project no. 639677758300), CNPq (EDN - 309973/2015-5), the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science – MOE, ECNU, the Research Grants Council of Hong Kong through project 11215017, National Key RD Project of China (2018YFE0201603), and Shanghai Municipal Science and Technology Major Project (2017SHZDZX01) (L.S.

Analysis of Outcomes in Ischemic vs Nonischemic Cardiomyopathy in Patients With Atrial Fibrillation A Report From the GARFIELD-AF Registry

IMPORTANCE Congestive heart failure (CHF) is commonly associated with nonvalvular atrial fibrillation (AF), and their combination may affect treatment strategies and outcomes