Second-order boundary estimates for solutions to singular elliptic equations in borderline cases

Let \Omega \subsetR^N be a bounded smooth domain. We investigate the effect of the mean curvature of the boundary \partial \Omega on the behaviour of the solution to the homogeneous Dirichlet boundary value problem for the equation \Delta u + f(u) = 0. Under appropriate growth conditions on f(t) as t approaches zero, we find asymptotic expansions up to the second order of the solution in terms of the distance from x to the boundary \partial \Omega

Steiner symmetry in the minimization of the first eigenvalue in problems involving the p-Laplacian

Let Ω ⊂ ℝN be an open bounded connected set. We consider the eigenvalue problem −Δpu = λρ|u|p−2u in Ω with homogeneous Dirichlet boundary condition, where Δp is the p-Laplacian operator and ρ is an arbitrary function that takes only two given values 0 < α < β and that is subject to the constraint ∫Ω ρdx = αγ +β(|Ω|−γ) for a fixed 0 < γ < |Ω|. The optimization of the map ρ ↦ λ1(ρ), where λ1 is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Ω and we show that the minimizers inherit the same symmetry

Symmetry and regularity of an optimization problem related to a nonlinear BVP

We consider the functional where uf is the unique nontrivial weak solution of the boundary-value problem where Ω ⊂ Rn is a bounded smooth domain. We prove a result of Steiner symmetry preservation and, if n = 2, we show the regularity of the level sets of minimizers

Stime sul bordo per soluzioni di equazioni ellittiche singolari

Nella prima parte di questa tesi viene studiata l’equazione Δu + f(u) = 0 in un dominio regolare (che soddisfa la proprietà della sfera interna e della sfera esterna) e limitato ­ di RN, dove f(t) è una funzione regolare positiva, decrescente, e che tende a infinito per t che tende a zero. Nella seconda parte della tesi si studia invece l’equazione Δu = f(u)in un dominio regolare e limitato ­Ω di RN, dove f(t) è una funzione regolare, crescente per t > 0 e che soddisfa la condizione di Keller-Osserman. Se si considera la condizione sul bordo u(x) ->(infinito) per x -> @Ω­, la soluzione di quest’altro problema di Dirichlet è chiamata soluzione blow-up (o esplosiva). In entrambi i casi si parla di equazione singolare

Problems for P-Monge-Ampere Equations

2010 Mathematics Subject Classification: 35A23, 35B51, 35J96, 35P30, 47J20, 52A40.We consider the homogeneous Dirichlet problem for a class of equations which generalize the p-Laplace equations as well as the Monge- Amp`ere equations in a strictly convex domain ⊂ Rn, n ≥ 2. In the sub-linear case, we study the comparison between quantities involving the solution to this boundary value problem and the corresponding quantities involving the (radial) solution of a problem in a ball having the same η1- mean radius as . Next, we consider the eigenvalue problem for such a p-Monge-Amp`ere equation and study a comparison between its principal eigenvalue (eigenfunction) and the principal eigenvalue (eigenfunction) of the corresponding problem in a ball having the same η1-mean radius as . Symmetrization techniques and comparison principles are the main tools used to get our results

Minimization of the first eigenvalue in problems involving the bi-laplacian

This paper concerns the minimization of the first eigenvalue in problems involvingthe bi-Laplacian under either homogeneous Navier boundary conditions or homogeneousDirichlet boundary conditions. Physically, in case of N = 2, our equation modelsthe vibration of a non homogeneous plate  which is either hinged or clamped alongthe boundary. Given several materials (with different densities) of total extension ||,we investigate the location of these materials inside  so to minimize the first modein the vibration of the corresponding plate.Keywords: bi-Laplacian, first eigenvalue, minimization.Este art´?culo trata de la minimizaci´on del primer autovalor en problemas relativosal bi-Laplaciano bajo condiciones de frontera homog´eneas de tipo Navier o Dirichlet.F´?sicamente, en el problema bi-dimensional, nuestra ecuacin modela la vibraci´on deuna placa inhomog´enea  fija con goznes a lo largo de su borde. Dados varios materiales(de diferentes densidades) y extensi´on total ||, investigamos cu´al debe serla localizaci´on de tales materiales en la placa para minimizar el primer modo de suvibraci´on.Palabras clave: bi-Laplaciano, primer autovalor, minimizaci´on

Boundary estimates for solutions to singular elliptic equations

We deal with the Dirichlet problem in a bounded smooth domain.[omissis

Measurement of Physical Activity and Sedentary Behavior by Accelerometry Among a Nationwide Sample from the KiGGS and MoMo Study: Study Protocol

Background: Currently, no nationwide objective physical activity data exists for children and adolescents living in Germany. The German Health Interview and Examination Survey for Children and Adolescents (KiGGS) and the Motorik-Modul study (MoMo) is a national cohort study that has incorporated accelerometers in its most recent data collection wave (wave 2, since 2014). This wave 2 marks the first nationwide collection of objective data on the physical activity of children and adolescents living in Germany. Objective: The purpose of this protocol is to describe the methods used in the KiGGS and MoMo study to capture the intensity, frequency, and duration of physical activity with accelerometers. Methods: Participants (N=11,003, aged 6 to 31 years) were instructed to wear an ActiGraph GT3X+ or wGT3X-BT accelerometer laterally on the right hip. Accelerometers were worn on consecutive days during waking hours, including at least 4 valid weekdays and 1 weekend day (wear time >8 hours) in the evaluation. A nonwear time protocol was also implemented. Results: Data collection was completed by October 2017. Data harmonization took place in 2018. The first accelerometer results from this wave were published in 2019, and detailed analyses are ready to be submitted in 2020. Conclusions: This study protocol provides an overview of technical details and basic choices when using accelerometers in large-scale epidemiological studies. At the same time, the restrictions imposed by the specified filters and the evaluation routines must be taken into account

The high frequency of manic symptoms in fibromyalgia does influence the choice of treatment?

BACKGROUND: Mood disorders were found associated with fibromyalgia (FM) and clinical studies have revealed the efficacy of antidepressant drugs in the treatment of FM. However no specific instruments to identify manic symptoms were used. OBJECTIVES: To assess the frequency of anxiety and mood disorders (particularly bipolar disorders and manic symptoms) in a consecutive sample of women affected by FM using standardized diagnostic tools and to compare the prevalence of these disorders with that observed in a sample of healthy controls from the general population. METHODS: Cases: consecutive series of women (N = 37, mean age 50.1 ± 21.0) attending a Rheumatology outpatient Unit at the University of Cagliari. Controls: 148 women, drawn from the data bank of an epidemiological study matched for sex and age with controls according to a randomisation "after blocks" method. The Italian version of the Composite International Diagnostic Interview Simplified were carried out by physicians. Psychiatric diagnosis was formulated according to DSM-IV criteria. The Italian version of the Mood Disorder Questionnaire (MDQ) was administered to identify manic symptoms and bipolar disorders. Diagnosis of FM were carried out by rheumatologist according to the criteria of American College of Rheumatology. RESULTS: Subjects with FM showed a higher comorbidity with Generalised Anxiety Disorder, Panic Disorder and Major Depressive Disorder than controls. The study showed a high frequency of manic symptoms (MDQ positive) in the sample of fibromyalgic patients (59%), approximately double that found in the control sample (P < 0.001). DISCUSSION: Clinical studies have shown the efficacy of antidepressants, especially tricyclic antidepressants, in the treatment of FM. The clinical difficulty in identifying hypomanic episodes is well known particularly where previous and not present episodes are concerned as in depressive patients. These data would suggest further studies on the subject are needed and more caution also in prescribing antidepressants in a population apparently at high risk for bipolar disorders