Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group

We prove new (sharp) Liouville-type properties via degenerate Hadamard three-sphere theorems for fully nonlinear equations structured over Heisenberg vector fields. As model examples, we cover the case of Pucci's extremal operators perturbed by suitable semilinear and gradient terms, extending to the Heisenberg setting known contributions valid in the Euclidean framework

Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton–Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton–Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem

A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci\u2019s extremal operators, some singular operators such as those modeled on the p- and 1e-Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature

On the existence and uniqueness of solutions to time-dependent fractional MFG

We establish existence and uniqueness of solutions to evolutive fractional mean field game systems with regularizing coupling for any order of the fractional Laplacian s 08 (0,1). The existence is addressed via the vanishing viscosity method. In particular, we prove that in the subcritical regime s > 1/2 the solution of the system is classical, while if s 64 1/2, we find a distributional energy solution. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We show uniqueness of solutions both under monotonicity conditions and for short time horizons

Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable

Mechanical ventilation during acute lung injury: current recommendations and new concepts

Despite a very large body of investigations, no effective pharmacological therapies have been found to cure acute lung injury. Hence, supportive care with mechanical ventilation remains the cornerstone of treatment. However, several experimental and clinical studies showed that mechanical ventilation, especially at high tidal volumes and pressures, can cause or aggravate ALI. Therefore, current clinical recommendations are developed with the aim of avoiding ventilator-induced lung injury (VILI) by limiting tidal volume and distending ventilatory pressure according to the results of the ARDS Network trial, which has been to date the only intervention that has showed success in decreasing mortality in patients with ALI/ARDS. In the past decade, a very large body of investigations has determined significant achievements on the pathophysiological knowledge of VILI. Therefore, new perspectives, which will be reviewed in this article, have been defined in terms of the efficiency and efficacy of recognizing, monitoring and treating VILI, which will eventually lead to further significant improvement of outcome in patients with ARDS

Evaluating the effectiveness of the Emergency Neurological Life Support educational framework in low-income countries.

BackgroundThe Emergency Neurological Life Support (ENLS) is an educational initiative designed to improve the acute management of neurological injuries. However, the applicability of the course in low-income countries in unknown. We evaluated the impact of the course on knowledge, decision-making skills and preparedness to manage neurological emergencies in a resource-limited country.MethodsA prospective cohort study design was implemented for the first ENLS course held in Asia. Knowledge and decision-making skills for neurological emergencies were assessed at baseline, post-course and at 6 months following course completion. To determine perceived knowledge and preparedness, data were collected using surveys administered immediately post-course and 6 months later.ResultsA total of 34 acute care physicians from across Nepal attended the course. Knowledge and decision-making skills significantly improved following the course (p=0.0008). Knowledge and decision-making skills remained significantly improved after 6 months, compared with before the course (p=0.02), with no significant loss of skills immediately following the course to the 6-month follow-up (p=0.16). At 6 months, the willingness to participate in continuing medical education activities remained evident, with 77% (10/13) of participants reporting a change in their clinical practice and decision-making, with the repeated use of ENLS protocols as the main driver of change.ConclusionsUsing the ENLS framework, neurocritical care education can be delivered in low-income countries to improve knowledge uptake, with evidence of knowledge retention up to 6 months