On the existence threshold for positive solutions of p-laplacian equations with a concave-convex nonlinearity

We study the following boundary value problem with a concave-convex nonlinearity: \begin{equation*} \left\{ \begin{array}{r c l l} -\Delta_p u & = & \Lambda\,u^{q-1}+ u^{r-1} & \textrm{in }\Omega, \\ u & = & 0 & \textrm{on }\partial\Omega. \end{array}\right. \end{equation*} Here $\Omega \subset \mathbb{R}^n$ is a bounded domain and $1<q<p<r<p^*$. It is well known that there exists a number $\Lambda_{q,r}>0$ such that the problem admits at least two positive solutions for $0<\Lambda<\Lambda_{q,r}$, at least one positive solution for $\Lambda=\Lambda_{q,r}$, and no positive solution for $\Lambda > \Lambda_{q,r}$. We show that $$\lim_{q \to p} \Lambda_{q,r} = \lambda_1(p),$$ where $\lambda_1(p)$ is the first eigenvalue of the p-laplacian. It is worth noticing that $\lambda_1(p)$ is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case $q=p$

8 minuti per salvare Chicago : Source Code di Duncan Jones

8 minuti per salvare Chicago - Source Code di Duncan Jones di Ilaria Parin

The eigenvalue problem for the ∞-Bilaplacian

We consider the problem of finding and describing minimisers of the Rayleigh quotient Λ∞:=infu∈W2,∞(Ω)∖{0}∥Δu∥L∞(Ω)∥u∥L∞(Ω), Λ∞:=infu∈W2,∞(Ω)∖{0}‖Δu‖L∞(Ω)‖u‖L∞(Ω), where Ω⊆RnΩ⊆Rn is a bounded C1,1C1,1 domain and W2,∞(Ω)W2,∞(Ω) is a class of weakly twice differentiable functions satisfying either u=0u=0 on ∂Ω∂Ω or u=|Du|=0u=|Du|=0 on ∂Ω∂Ω . Our first main result, obtained through approximation by LpLp -problems as p→∞p→∞ , is the existence of a minimiser u∞∈W2,∞(Ω)u∞∈W2,∞(Ω) satisfying {Δu∞∈Λ∞Sgn(f∞)Δf∞=μ∞ a.e. in Ω, in D′(Ω), {Δu∞∈Λ∞Sgn(f∞) a.e. in Ω,Δf∞=μ∞ in D′(Ω), for some f∞∈L1(Ω)∩BVloc(Ω)f∞∈L1(Ω)∩BVloc(Ω) and a measure μ∞∈M(Ω)μ∞∈M(Ω) , for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue Λ∞Λ∞ on the domain, establishing the validity of a Faber–Krahn type inequality: among all C1,1C1,1 domains with fixed measure, the ball is a strict minimiser of Ω↦Λ∞(Ω)Ω↦Λ∞(Ω) . This result is shown to hold true for either choice of boundary conditions and in every dimension

Molecular characterization of Bifidobacterium longum biovar longum NAL8 plasmids and construction of a novel replicon screening system

In this study, we performed molecular characterization and sequence analysis of three plasmids from the human intestinal isolate Bifidobacterium longum biovar longum NAL8 and developed a novel vector screening system. Plasmids pNAL8H (10 kb) and pNAL8M (4.9 kb) show close sequence similarity to and the same gene organization as the already characterized B. longum plasmids. The B. longum plasmid pNAC1 was identified as being most closely related to pNAL8L (3.5 kb). However, DNA sequence analysis suggested that direct repeat-rich sites could have promoted several recombination events to diversify the two plasmid molecules. We verified the likely rolling circle replication of plasmid pNAL8L and studied the phylogenetic relationship in all the Bifidobacterium plasmids fully sequenced to date based on in silico comparative sequence analysis of their replication proteins and iteron regions. Our transformation experiments confirmed that the ColE1 replication origin from high-copy-number pUC vectors could interfere with the replication apparatus of Bifidobacterium plasmids and give rise to false positive clones. As a result, we developed a system suitable for avoiding possible interference by other functional replication modules on the vector and for screening functional replicons from wild-type plasmids

Early respiratory viral infections in infants with cystic fibrosis

This article is made available for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.Background Viral infections contribute to morbidity in cystic fibrosis (CF), but the impact of respiratory viruses on the development of airway disease is poorly understood. Methods Infants with CF identified by newborn screening were enrolled prior to 4 months of age to participate in a prospective observational study at 4 centers. Clinical data were collected at clinic visits and weekly phone calls. Multiplex PCR assays were performed on nasopharyngeal swabs to detect respiratory viruses during routine visits and when symptomatic. Participants underwent bronchoscopy with bronchoalveolar lavage (BAL) and a subset underwent pulmonary function testing. We present findings through 8.5 months of life. Results Seventy infants were enrolled, mean age 3.1 ± 0.8 months. Rhinovirus was the most prevalent virus (66%), followed by parainfluenza (19%), and coronavirus (16%). Participants had a median of 1.5 viral positive swabs (range 0–10). Past viral infection was associated with elevated neutrophil concentrations and bacterial isolates in BAL fluid, including recovery of classic CF bacterial pathogens. When antibiotics were prescribed for respiratory-related indications, viruses were identified in 52% of those instances. Conclusions Early viral infections were associated with greater neutrophilic inflammation and bacterial pathogens. Early viral infections appear to contribute to initiation of lower airway inflammation in infants with CF. Antibiotics were commonly prescribed in the setting of a viral infection. Future investigations examining longitudinal relationships between viral infections, airway microbiome, and antibiotic use will allow us to elucidate the interplay between these factors in young children with CF

The prescribed mean curvature equation in weakly regular domains

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(\Lambda,r_{0})$-minimizers of the perimeter.Comment: 23 pages, 1 figure --- The results on the weak normal trace of vector fields have been now extended and moved in a self-contained paper available at: arXiv:1708.0139

Boundary and defect CFT: Open problems and applications

A review of Boundary and defect conformal field theory: open problems and applications, following a workshop held at Chicheley Hall, Buckinghamshire, UK, 7–8 Sept. 2017. We attempt to provide a broad, bird’s-eye view of the latest progress in boundary and defect conformal field theory in various sub-fields of theoretical physics, including the renormalization group, integrability, conformal bootstrap, topological field theory, supersymmetry, holographic duality, and more. We also discuss open questions and promising research directions in each of these sub-fields, and combinations thereof