A nonlinear parabolic problem with singular terms and nonregular data

We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form \begin{cases} \dys u_t - \Delta_p u = h(u)f+\mu & \text{in}\ \Omega \times (0,T),\\ u=0 &\text{on}\ \partial\Omega \times (0,T),\\ u=u_0 &\text{in}\ \Omega \times \{0\}, \end{cases} where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $u_0$ is a nonnegative integrable function, $\Delta_p$ is the $p$-Laplace operator, $\mu$ is a nonnegative bounded Radon measure on $\Omega \times (0,T)$ and $f$ is a nonnegative function of $L^1(\Omega \times (0,T))$. The term $h$ is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing $h$

On singular elliptic equations with measure sources

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is $$\begin{cases} -\Delta u = \frac{f(x)}{u^{\gamma}} +\mu & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\ \Omega, \end{cases}$$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$. Here $\gamma > 0$, $f$ is a nonnegative function on $\Omega$, and $\mu$ is a nonnegative bounded Radon measure on $\Omega$

A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an $L^2-$valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non--linear Feynman--Kac representation theorem under mild assumptions of differentiability

Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and L1L^1 terms

In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -\Delta_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\Omega$,} \newline u\geq 0 & \text{in $\Omega$,} \newline u=0 & \text{on $\partial \Omega$,} \ \end{cases} \end{equation*} in a domain $\Omega \subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N$, $g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a continuous function on $[0,\infty)$ (possibly blowing up at the origin). We show how the presence of regularizing terms $h$ and $g$ allows to prove existence of finite energy solutions for nonnegative data $f$ only belonging to $L^1(\Omega)$

A nonlinear parabolic problem with singular terms and nonregular data

We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form $$\begin{cases} \displaystyle u_t - \Delta_p u = h(u)f+\mu & \text{in}\ \Omega \times (0,T),\\ u=0 &\text{on}\ \partial\Omega \times (0,T),\\ u=u_0 &\text{in}\ \Omega \times \{0\}, \end{cases}$$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $u_0$ is a nonnegative integrable function, $\Delta_p$ is the $p$-laplace operator, $\mu$ is a nonnegative bounded Radon measure on $\Omega \times (0,T)$ and $f$ is a nonnegative function of $L^1(\Omega \times (0,T))$. The term $h$ is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing $h$

Bounded solutions for non-parametric mean curvature problems with nonlinear terms

In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain $\Omega$ of $\mathbb{R}^N$. The mean curvature, that depends on the location of the solution $u$ itself, is asked to be of the form $f(x)h(u)$, where $f$ is a nonnegative function in $L^{N,\infty}(\Omega)$ and $h:\mathbb{R}^+\mapsto \mathbb{R}^+$ is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. $h\equiv 1$. This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples

Check-rein technique for Achilles tendon elongation following conservative management for acute Achilles tendon ruptures: a two-year prospective clinical study

Background Following conservative management for acute Achilles tendon (AT) ruptures, the tendon may heal in continuity, and some patients may present with an elongated Achilles tendon-gastrosoleus complex. This study investigated the efficacy and feasibility of a novel minimally invasive technique, which we named "check-rein procedure", in patients with intact and elongated AT following conservative management for AT ruptures. Methods All patients who underwent the check-rein procedure for elongation of the gastrosoleus-AT complex by one experienced surgeon were prospectively enrolled. The AT resting angle (ATRA) and AT rupture score (ATRS) were assessed at baseline and repeated at 2-year follow-up, as were calf circumference and isometric plantarflexion strength of both ankles. Results Forty-three patients (43 procedures) were analysed. The mean time elapsed from injury to surgery was 28.7 +/- 7.9 weeks. The mean age at surgery was 38.5 +/- 5.7 years. At the last follow-up, ATRS, ATRA, isometric strength difference, and calf circumference of the affected side were increased (P &lt; 0.0001). The rate of the return to sport was 98% (42 of 43). No wound complications or rupture were experienced by any patient. Conclusion The check-rein technique for AT elongation after conservative management of AT tears is effective and feasible to restore tendon length and calf function. The surgical outcome was influenced by the preoperative performance status, and longer time elapsed from injury to surgery worsens the outcomes

Sonoelastography in the diagnosis of tendinopathies: An added value

BACKGROUND: sonoelastography helps in the detection of abnormalities not yet evident on B-mode exam. METHODS: in this observational study, we report a collection of cases of symptomatic patients without alterations at ultrasound imaging but with evidence of pathological findings at sonoelastography. Patients, with clinical history suggestive for tendinopathies or surgically treated, and negative at the ultrasound exam, were submitted to sonoelastography. Out of 846, 632 patients with positive ultrasound exam were excluded. Sonoelastography was therefore performed in the remaining 214. RESULTS: the examination was positive in 168 cases: 78 patients were affected with shoulder diseases, while elbow pathology was observed in 31 subjects; patellar, Achilles and plantar fascia disorders were reported in 19, 27, and 13 patients, respectively. CONCLUSION: sonoelastography can reveal tendon abnormalities of clinical relevance in a high percentage of cases, where the ultrasound exam was negative, making the method a complementary tool to ultrasound evaluation

An Efficient Stochastic Linearisation Procedure for the Seismic Optimal Design of Passive Control Devices

The seismic response of structures is often enhanced by introducing passive control devices that can operate through the dissipation of the input energy or by modifying the dynamic characteristics of the main structure. The inherent non-linearities in the constitutive laws of some of them lead to computation difficulties and have limited the large-scale use and design of these devices. In this study, a procedure for the optimal design of multi passive control devices is proposed. The general case of linear Multi-Degree-Of-Freedom (MDOF) not-classically-damped structural systems controlled by Fluid Viscous Dampers (FVD) are investigated in a stochastic framework. The procedure consists of evaluation of the device optimal pattern by minimizing an objective function related to the dampers cost and subjected to a constraint on the structural behavior. For each configuration, the complete probabilistic characterization of the response is achieved by employing random vibration theory, Stochastic Linearisation (SL) techniques and a novel analytic model which provides closed-form PSD functions of earthquakes accelerations coherent to response spectra suggested by seismic codes. Exploiting this model, a procedure to speed up the Stochastic Linearisation technique by avoiding any numerical integration is proposed. Applications on MDOF building structures have been carried out to validate the proposed approach in terms of accuracy and reduction of the computational effort and to obtain optimal pattern of the passive control device coherently with the provisions of seismic building codes

Characterization of CRISPR-Cas Systems in Serratia marcescens Isolated from Rhynchophorus ferrugineus (Olivier, 1790) (Coleoptera: Curculionidae)

The CRISPR-Cas adaptive immune system has been attracting increasing scientific interest for biological functions and biotechnological applications. Data on the Serratia marcescens system are scarce. Here, we report a comprehensive characterisation of CRISPR-Cas systems identified in S. marcescens strains isolated as secondary symbionts of Rhynchophorus ferrugineus, also known as Red Palm Weevil (RPW), one of the most invasive pests of major cultivated palms. Whole genome sequencing was performed on four strains (S1, S5, S8, and S13), which were isolated from the reproductive apparatus of RPWs. Subtypes I-F and I-E were harboured by S5 and S8, respectively. No CRISPR-Cas system was detected in Si or S13. Two CRISPR arrays (4 and 51 spacers) were detected in S5 and three arrays (11, 31, and 30 spacers) were detected in S8. The CRISPR-Cas systems were located in the genomic region spanning from ybhR to phnP, as if this were the only region where CRISPR-Cas loci were acquired. This was confirmed by analyzing the S. marcescens complete genomes available in the NCBI database. This region defines a genomic hotspot for horizontally acquired genes and/or CRISPR-Cas systems. This study also supplies the first identification of subtype I-E in S. marcescens