Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

We consider the singular limit of a bistable reaction diffusion equation in the case when the velocity of the traveling wave solution depends on the space variable and converges to a discontinuous function. We show that the family of solutions converges to the stable equilibria off a front propagating with a discontinuous velocity. The convergence is global in time by applying the weak geometric flow uniquely defined through the theory of viscosity solutions and the level-set equation

On viscosity and equivalent notions of solutions for anisotropic geometric equations

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the euclidian space, and in Carnot groups is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the euclidian setting. These results simplify the handling of the singularities of the equation, for instance to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing for instance that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present

A comparison principle for the mean curvature flow equation with discontinuous coefficients

We study the level set equation in a bounded domain when the velocity of the interface is given by the mean curvature plus a discontinuous velocity. We prove a comparison principle for the initial-boundary value problem whose consequence is uniqueness of continuous solutions and well-posedness of the level set method

Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: the DecubICUs study

Funder: European Society of Intensive Care Medicine; doi: http://dx.doi.org/10.13039/501100013347Funder: Flemish Society for Critical Care NursesAbstract: Purpose: Intensive care unit (ICU) patients are particularly susceptible to developing pressure injuries. Epidemiologic data is however unavailable. We aimed to provide an international picture of the extent of pressure injuries and factors associated with ICU-acquired pressure injuries in adult ICU patients. Methods: International 1-day point-prevalence study; follow-up for outcome assessment until hospital discharge (maximum 12 weeks). Factors associated with ICU-acquired pressure injury and hospital mortality were assessed by generalised linear mixed-effects regression analysis. Results: Data from 13,254 patients in 1117 ICUs (90 countries) revealed 6747 pressure injuries; 3997 (59.2%) were ICU-acquired. Overall prevalence was 26.6% (95% confidence interval [CI] 25.9–27.3). ICU-acquired prevalence was 16.2% (95% CI 15.6–16.8). Sacrum (37%) and heels (19.5%) were most affected. Factors independently associated with ICU-acquired pressure injuries were older age, male sex, being underweight, emergency surgery, higher Simplified Acute Physiology Score II, Braden score 3 days, comorbidities (chronic obstructive pulmonary disease, immunodeficiency), organ support (renal replacement, mechanical ventilation on ICU admission), and being in a low or lower-middle income-economy. Gradually increasing associations with mortality were identified for increasing severity of pressure injury: stage I (odds ratio [OR] 1.5; 95% CI 1.2–1.8), stage II (OR 1.6; 95% CI 1.4–1.9), and stage III or worse (OR 2.8; 95% CI 2.3–3.3). Conclusion: Pressure injuries are common in adult ICU patients. ICU-acquired pressure injuries are associated with mainly intrinsic factors and mortality. Optimal care standards, increased awareness, appropriate resource allocation, and further research into optimal prevention are pivotal to tackle this important patient safety threat

Correction to: Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: the DecubICUs study

The original version of this article unfortunately contained a mistake

Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: the DecubICUs study

Funder: European Society of Intensive Care Medicine; doi: http://dx.doi.org/10.13039/501100013347Funder: Flemish Society for Critical Care NursesAbstract: Purpose: Intensive care unit (ICU) patients are particularly susceptible to developing pressure injuries. Epidemiologic data is however unavailable. We aimed to provide an international picture of the extent of pressure injuries and factors associated with ICU-acquired pressure injuries in adult ICU patients. Methods: International 1-day point-prevalence study; follow-up for outcome assessment until hospital discharge (maximum 12 weeks). Factors associated with ICU-acquired pressure injury and hospital mortality were assessed by generalised linear mixed-effects regression analysis. Results: Data from 13,254 patients in 1117 ICUs (90 countries) revealed 6747 pressure injuries; 3997 (59.2%) were ICU-acquired. Overall prevalence was 26.6% (95% confidence interval [CI] 25.9–27.3). ICU-acquired prevalence was 16.2% (95% CI 15.6–16.8). Sacrum (37%) and heels (19.5%) were most affected. Factors independently associated with ICU-acquired pressure injuries were older age, male sex, being underweight, emergency surgery, higher Simplified Acute Physiology Score II, Braden score 3 days, comorbidities (chronic obstructive pulmonary disease, immunodeficiency), organ support (renal replacement, mechanical ventilation on ICU admission), and being in a low or lower-middle income-economy. Gradually increasing associations with mortality were identified for increasing severity of pressure injury: stage I (odds ratio [OR] 1.5; 95% CI 1.2–1.8), stage II (OR 1.6; 95% CI 1.4–1.9), and stage III or worse (OR 2.8; 95% CI 2.3–3.3). Conclusion: Pressure injuries are common in adult ICU patients. ICU-acquired pressure injuries are associated with mainly intrinsic factors and mortality. Optimal care standards, increased awareness, appropriate resource allocation, and further research into optimal prevention are pivotal to tackle this important patient safety threat

Some new results on reaction-diffusion equations and geometric flows

In this thesis we discuss the asymptotic behavior of the solutions of scaled reaction-diffusion equations in the unbounded domain Rn × (0 + ∞), in the cases when such a behavior is described in terms of moving interfaces. As first class of asymptotic problems we consider the singular limit of bistable reaction-diffusion equations in the case when the velocity of the traveling wave equation depends on the space variable, i.e. cε = cε(x), and it satisfies, in some suitable sense, cε/ετ → α, as ε → 0+, where α is a discontinuous function and τ is an integer that can be equal to 0 or 1. The second part of the thesis concerns semilinear reaction-diffusion equations with diffusion term of type tr(Aε(x)D2uε), where tr denotes the trace operator, Aε = σεσtε for some matrix map σε : Rn → Rn×(m+n) and Aε converges to a degenerate matrix. In order to establish such results rigorously, we modify and adapt to our problems the ”geometric approach” introduced by G. Barles and P. E. Souganidis for solving problems in Rn, and then partially revisited by G. Barles and F. Da Lio for reaction-diffusion equations in bounded domains. When it is possible we always consider the question of the well posedness of the Cauchy problems governing the motion of the fronts that describe the asymptotics we considerIn questa tesi discutiamo il comportamento asintotico delle soluzioni di equazioni di reazione-diffusione nel dominio illimitato Rn × (0,+∞) nei casi in cui tale comportamento sia descritto da un’interfaccia in movimento. Come primo tipo di problemi asintotici consideriamo il limite singolare di equazioni di reazione-diffusione bistabili nel caso in cui la velocità dell’onda viaggiante dipenda dalla variabile di stato, cioè cε = cε(x), e sia soddisfatto, al tendere di ε a zero e in qualche modo opportuno, cε/ετ → α, laddove α è una funzione discontinua e τ è un intero che può essere uguale a 0 o a 1. La seconda parte della tesi riguarda equazioni di reazione-diffusione semilineari e aventi termini di diffusione del tipo tr(Aε(x)D2uε), laddove tr denota l’operatore traccia, Aε = σεσtε per qualche funzione σε : Rn → Rn×(m+n) e Aε converge ad una matrice degenere. Al fine di provare tali risultati in modo rigoroso, abbiamo modificato e adattato "l’approccio geometrico" introdotto da G. Barles e P. E. Souganidis per risolvere problemi in Rn e in seguito parzialmente rivisto dallo stesso G. Barles assieme a F. Da Lio per equazioni di reazione-diffusione in domini limitati. Laddove possibile abbiamo sempre considerato la questione della buona posizione dei problemi di Cauchy che governano il moto dei fronti che descrivono le asintotiche da noi considerat

Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

We consider the singular limit of a bistable reaction diffusion equation in the case when the velocity of the traveling wave solution depends on the space variable and converges to a discontinuous function. We show that the family of solutions converges to the stable equilibria off a front propagating with a discontinuous velocity. The convergence is global in time by applying the weak geometric flow uniquely defined through the theory of viscosity solutions and the level-set equation

Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients

none2We study the Cauchy problem for a homogeneous and not necessarily coercive Hamilton-Jacobi-Isaacs equation with an x-dependent, piecewise continuous coefficient. We prove that under suitable assumptions there exists a unique and continuous viscosity solution. The result applies in particular to the Carnot-Caratheodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the Hormander condition. Our results are also of interest in connection with geometric flows with discontinuous velocity in anisotropic media with a non-euclidian ambient space.mixedDE ZAN C; SORAVIA P.DE ZAN, Cecilia; Soravia, Pierpaol